c-analytic geometry i: gaga theorems - … c-analytic geometry i: gaga theorems 3 pregeometries and...

DERIVED C-ANALYTIC GEOMETRY I: GAGA THEOREMS

MAURO PORTA

Abstract. We further develop the foundations of derived C-analytic geometryintroduced in [11] by J. Lurie. We introduce the notion of coherent sheaf on aderived C-analytic space. Moreover, building on the previous joint work with T.Y. Yu [20], we introduce the notion of proper morphism of derived C-analyticspaces. We show that these definitions are solid by proving a derived version ofGrauert’s proper direct image theorem and of the GAGA theorems. The proofsrely on two main ingredients: on one side, we prove a comparison result betweenthe ∞-category of higher Deligne-Mumford analytic stacks introduced in [20]and the ∞-category of derived C-analytic spaces. On the other side, we carryout a careful analysis of the analytification functor introduced in [11] and provethat the canonical map Xan → X is flat in the derived sense. This is part of aseries of articles [18, 19] that will soon appear.

Contents

Introduction 21. The setup 13

1.1. The C-analytic pregeometry 131.2. Weak Morita equivalences of pregeometries 15

2. Local analytic rings 212.1. Functional spectrum 212.2. Quotients of local analytic rings 242.3. Analytic algebras 272.4. Pregeometries of germs 292.5. Strict models 30

3. The functor of points 333.1. The geometric context 343.2. Geometric stacks 373.3. Truncations of derived C-analytic spaces 39

2010 Mathematics Subject Classification. Primary 14A20; Secondary 32C35 14F05.Key words and phrases. derived analytic stack, derived complex geometry, infinity category,

analytic stack, Grauert’s theorem, analytification, GAGA, derived GAGA, derived algebraicgeometry.

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3.4. Essential surjectivity 434. Coherent sheaves on a derived C-analytic space 445. The Grauert theorem for derived stacks 456. The analytification functor 47

6.1. Relative spectrum functor 486.2. Comparison with the classical analytification 496.3. Flatness I 526.4. Analytification of germs 546.5. Flatness II 566.6. Computing the analytification 59

7. GAGA for derived Deligne-Mumford stacks 617.1. GAGA 1 627.2. GAGA 2 64

8. Extension to Artin stacks 66Appendix A. Flat morphisms 67References 68

Introduction

Since its appearance, derived algebraic geometry has been proved an useful toolto deal with situations where the techniques of classical algebraic geometry couldn’tyield a full understanding of the involved mathematical phenomena. Strikingexamples are given by the work of B. Toen on derived Azumaya algebras [23] andby the approach to the geometric Langlands program of D. Gaitsgory [7]. Otherapplications can be found in Gromov-Witten theory, see for example the papers[21, 16]. On the other side, derived algebraic geometry has been an active researchfield on its own, as the recent works on shifted symplectic and Poisson structures[17, 3] demonstrate.

However, before reaching the foundations nowadays accepted [26, 12, 13, 14],there has been several different approaches and attempts, see e.g. [4]. There hasbeen recent activity toward the possibility of having a derived version of C-analyticgeometry, but the state of the art still resembles very much the one of derivedalgebraic geometry a decade ago. In [2], the authors propose an approach toanalytic geometry from the point of view of relative algebraic geometry [25]. Asfar as we understand, this would amount to take the affine objects to be simplicialind-Banach rings. This is indeed a reasonable proposal, which is, however, quitedifferent from the perspective we are adopting in the present work.

DERIVED C-ANALYTIC GEOMETRY I: GAGA THEOREMS 3

Pregeometries and derived geometry. The approach to derived C-analyticgeometry used in this article is due to J. Lurie and relies on his general theory ofpregeometries developed in [12]. He proposed to use it to lay foundations of derivedC-analytic geometry in [11, §11, §12]. As this way of constructing a category ofderived objects differs significantly from the perhaps more familiar one of [26], itseems worth to describe the general ideas.

The procedure of [26] consists in taking as input an ambient “linear”∞-categoryC endowed with a symmetric monoidal structure and satisfying certain basicassumptions. Out of this ambient ∞-category C one obtains a category of affineobjects Aff, which precisely coincides with the∞-category of commutative monoidsin C (with respect to the tensor structure). Then, one endows this category of affineobjects with some geometrical extra structure, such as a Grothendieck topologyτ and a collection of “smooth” morphisms P. These data are required to satisfya certain amount of compatibility conditions (we refer to [26, § 1.3.2] for theprecise formulation of these ideas). At this point, one can form the ∞-category ofhypercomplete sheaves Sh(Aff, τ)∧, and inside this category it is possible to isolateobjects that behave much like schemes, algebraic spaces, Deligne-Mumford stacksetc. known as geometric stacks. All of this is applied to derived algebraic geometryby taking C to be the ∞-category sModA of simplicial A-modules, where A is any(discrete) commutative ring. Commutative monoids in sModA can be identifiedwith simplicial commutative A-algebras, whose underlying ∞-category preciselycoincides with the (opposite of the) category of affine derived schemes.

However, in developing derived analytic geometry it is not so clear what theambient ∞-category C should be. Possible attempts are the category of simplicialind-Banach spaces, or of simplicial bornological spaces (we refer to [1] where it isshown that it is indeed possible to endow such categories with the structure ofHAG contexts). Still, the framework of [26] can be successfully applied as soonas the category of affine objects is known. Together with T. Y. Yu, in [20] weshowed that the Grauert proper direct image theorem and the GAGA theoremshold for (underived) higher Artin analytic stacks (see also [24] for a different kindof applications in a very similar framework). But, again, it is not at all clear whatshould the affine objects of derived C-analytic geometry look like, though someversion of analytic rings [6], could yield a workable setting.

The approach taken by J. Lurie in [12] is quite different and it can be used todeal with derived C-analytic geometry, and we hope that this paper will contributeto show the solidity of the foundations laid in [11]. The essential idea of [12] is thatwe do not need an ambient linear category to develop derived algebraic geometry,nor we need to start with a well-known (∞-)category of affine objects: we only need

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a category C of classical geometrical objects that we declare to be smooth, togetherwith a collection of smooth morphisms between such objects. With only this, it ispossible to construct the “free category of derived C-objects” simply by requiringthat pullbacks along smooth morphisms have to be classical (i.e. computed in C).The correct way of formalizing this striking idea is given with the language ofpregeometries. We refer the reader to Section 1.1 (and obviously to [12]) for a moredetailed technical account on this notion. For the time being, let us say that apregeometry T is an∞-category with finite products, equipped with a Grothendiecktopology τ and a collection of admissible morphisms (these data are required tosatisfy some mutual compatibility conditions, see Definition 1.1). The reader couldthink of this as a (multisorted) Lawvere theory equipped with some geometricalextra structure, and indeed the case of multisorted Lawvere theories is covered bythe theory of [12] when the Grothendieck topology is discrete and the admissiblemorphisms are precisely the equivalences. As in the case of a (multisorted) Lawveretheory, there is a notion of a “model” for T. This concept is referred to in [12] asT-structures. More specifically, if X is an ∞-topos (the reader might want to thinkto the ∞-category of spaces S or to sheaves of spaces on some topological space), aT-structure on X is a product preserving functor F : T → X satisfying the followingadditional conditions:

(1) the functor F preserves the pullbacks in C of the form

Y ′ Y

X ′ X

f

where f is an admissible morphism.(2) Whenever a family of morphisms Ui → X generates a covering sieve, the

induced morphism ∐F (Ui)→ F (X)

is an (effective) epimorphism in the ∞-topos X.

The full subcategory of Fun(T,X) spanned by T-structures will be denoted StrT(X).However, as in the case of locally ringed spaces, we do not really wish to workwith all the morphisms of T-structures, but only with those that have a sufficientlyregular behavior on the stalks of the topos X. This can be achieved with the notionof local transformation: a morphism of T-structures α : O→ O′ is said to be local


if for every admissible morphism f : U → V in T the induced square

O(U) O(V )

O′(U) O′(V )

O(f)

αU αV

O′(f)

is a pullback in X. The subcategory of StrT(X) spanned by local morphisms willbe denoted by Strloc

T (X).Before going any further, it is important to discuss a couple of examples, on

which the intuition is based. The first important situation is the one of a discretepregeometry (that is, of a multisorted Lawvere theory).

Example 1. Let k be a (discrete) commutative ring. We let Tdisc(k) be the oppositecategory of finitely presented free k-algebras. Admissible morphisms are preciselyisomorphisms and the Grothendieck topology is the discrete one. In this case,StrTdisc(k)(S) = Strloc

Tdisc(k)(S) can be identified with the underlying ∞-category ofsimplicial k-algebras. If char(k) = 0 we can further interpret this as the categoryof E∞ connective Hk-algebras, but it is important to remark that in positive (ormixed) characteristic we recover the simplicial formalism rather than the spectralone. Similarly, if X is an ∞-topos, StrTdisc(k)(X) = Strloc

Tdisc(k)(X) can be identifiedwith the ∞-category of sheaves of simplicial commutative algebras over X.

Thus, interpreting in spaces gave us back the ∞-category considered in [26]to start the construction of derived algebraic geometry. However, this proceduredoes not provide us with the geometrical extra structure needed in [26]. Let ustry to modify the pregeometry Tdisc(k) in order to encode the Zariski or the etaletopology:

Example 2. Let k be a (discrete) commutative ring. We let TZar(k) (resp. Tet(k))be the category of standard Zariski open (resp. etale) maps to An

k . We say that amorphism is admissible if it is an open Zariski immersion (resp. an etale map), andwe consider the Zariski (resp. etale) topology. In this case, the forgetful functors

StrTZar(k)(S)→ StrTdisc(S), StrTet(k)(S)→ StrTdisc(S)are fully faithful and the essential image is the collection of simplicial k-algebras Asuch that π0(A) is local (resp. strictly henselian). Moreover,

StrlocTet(k)(S)→ Strloc

TZar(k)(S)

is fully faithful, and a morphism f : A → B in StrTZar(k)(S) lies in StrlocTZar(k)(S) if

and only if the induced π0(f) : π0(A)→ π0(B) is a (local) morphism of local rings.The same analysis can be carried over a generic ∞-topos.

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Perhaps surprisingly, this example shows that when we interpret TZar (or Tet) inspaces S we do not recover the full∞-category of simplicial commutative rings. Weonly get back the ones with good locality behavior with respect to the Grothendiecktopology we took into account. This suggests that the way of constructing thecategory of free T-objects out of a pregeometry T has to be slightly more complicated.As we said at the beginning, the category we are looking for should be determinedby the requirement that pullbacks along admissible morphisms can be computedin T, and it is freely generated by T otherwise. This leads to the key of geometry,which is an ∞-category G having all finite limits and equipped with the verysame geometrical data of a pregeometry (that is, a Grothendieck topology and acollection of admissible morphism). The notion of a structure for a geometry G ismodified accordingly: StrG(X) is now the full subcategory of Fun(G,X) spanned byfinite limit preserving functors which takes τ -covering to effective epimorphisms.

It is easy to imagine what a universal geometry generated by a pregeometry T

should be; namely, a continuous functor of Grothendieck sites ϕ : T → G whichmoreover preserves products and admissible pullbacks, and satisfies the followinguniversal property: for every ∞-topos X, composition with ϕ should induce anequivalence

StrG(X)→ StrT(X)

In this situation, we say that ϕ exhibits G as a geometric envelope of T. It canbe shown that it always exists (see [12, Lemma 3.4.3]). It is considerably harderthan Example 1 to show that when T = TZar, the geometric envelope is preciselythe ∞-category of finitely presented simplicial commutative rings, equipped withthe (derived) Zariski topology. Nevertheless, it is true (see [12, Proposition 4.2.3]).Even more remarkably, it can be shown [12, Proposition 4.3.15] that the geometricenvelope of Tet is the ∞-category of finitely presented simplicial commutative ringsequipped with the (derived) etale topology.

So far we described only the procedure that allows to recover the affine objects,which, as we discussed at the beginning, is the only thing we need as the subsequentglobalization step can be handled with the techniques of [26]. However, it isimportant to remark here that the framework of [12] comes with a gluing procedurethat allows the author to introduce a structured space point of view on derivedgeometry. Roughly speaking, this consists in defining an∞-category of T-structuredtopoi and then in isolating a full subcategory of T-schemes inside. These categoriesare respectively denoted Top(T) and Sch(T), but we refer to the introduction of[18] for an expository account of these ideas. The reader who wish to see thedetails can consult [12, Definitions 1.4.8, 2.3.9]. In this work we will take the latter


point of view, but some comparison result with the approach stemming from [26]is provided (see Section 3).

The analytic pregeometry and the analytification functor. We are left toexplain which pregeometry should lead to a meaningful notion of derived C-analyticspace. Following J. Lurie [11, 11.2, 11.3], we define Tan to be the category of opensubsets of Cn, and declare a morphism to be admissible if it is an open immersion.Let us explain what Tan-structures roughly look like. We hope this descriptionwill help the intuition of the reader in moving through this paper. For simplicity,we will think of Tan-structures in Set rather than in the ∞-category of spacesS. First of all, it is important to observe that Tan-structures are very similar to(local) C∞-rings as they are introduced for example in [22]. A similar approachto C-analytic geometry has been taken by [6], with the difference that they weretaking a larger class of admissible pullbacks. Roughly speaking, the idea behind thisLawvere-style approach to C-analytic geometry is that the affine objects ought tobe related to (commutative) Banach algebras, but that the very topological natureof such objects prevents their category from having good categorical properties.However, we don’t necessarily need the full structure of Banach algebra to give aworking definition of C-analytic spaces. What is really important is that Banachalgebras admit a so-called holomorphic functional calculus, which is a formal wayof encoding the action of the algebra of holomorphic functions on some open subsetU ⊂ Cn on a given (commutative) Banach algebra A. Unraveling the definitions, itemerges that a Tan-structure in Set is precisely the given of a commutative ring Atogether with the choice of a subset A(U) of An for every open subset U ⊂ Cn anda choice of a map A(U) → A for every holomorphic function U → C, satisfyingbeing compatible with the compositions.

As we anticipated, following J. Lurie we will define the category of derivedC-analytic spaces dAnC as a full subcategory of Top(Tan) (see Definition 1.3 forthe precise formulation). We warn the reader that the category dAnC is differentfrom the category Sch(Tan) we introduced before. A careful comparison of the twohas been carried out by J. Lurie in [11, Corollary 12.22, Proposition 12.23].

The last important idea of [12] that plays a major role in this paper is the notionof relative spectrum, which is used in [11, Remark 12.26] in order to define theanalytification functor. The starting point is the universal property of the classicalanalytification functor described by Grothendieck in [9, Expose XII]. Recall that ifX is a scheme locally of finite presentation over C, then an analytification of Xis the given of a C-analytic space Xan together with a morphism of locally ringedspaces Xan → X inducing isomorphisms

HomAnC(Y,Xan) ' HomLRingSpaces(Y,X)

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for every C-analytic space Y . This idea generalizes directly to the derived setting.J. Lurie constructed in [12, Theorem 2.1.1] a far reaching generalization of the ana-lytification functor, which is known as relative spectrum associated to a morphismof pregeometries ϕ : T → T′. To put it simply, he proved that the natural forgetfulfunctor

Top(T′)→ Top(T)(which sends a T′-structured∞-topos (X,O) in the T-structured∞-topos (X,Oϕ))admits a right adjoint, denoted SpecT′T . Furthermore he showed that this functorrespects the subcategories of schemes. Then, [11, Corollary 12.22] allows to seethat SpecTan

Tettakes the category of derived Deligne-Mumford stacks to dAnC. This

will therefore be the analytification functor we will be using through this paper.As its definition is abstract (and the proof of the existence rather indirect), somework is required to show that SpecTan

Tetenjoys the good properties everyone would

expect. This is one of the main parts of this article.

The main results. We now explain the content of the current paper. In Section 1we begin by reviewing in a less expository way the notion of pregeometry andof derived C-analytic space. We take the opportunity to summarizing the mainresults of [11]. Next, in Section 1.2, where we introduce the notion of weak Moritaequivalence of pregeometries. This is a rather technical notion that neverthelessplays an important role in the study of the properties of germs of derived C-analyticspaces. The main result of this section asserts that the∞-categories Strloc

T (X)/O arealways presentable for a pregeometry T and a T-structure O on X (while Strloc

T (X)is not, in general, presentable).

In Section 2 we show an important structure theorem for Tan-structures withvalues in spaces: namely, we show that except for the null Tan-structure, they arealways canonically augmented over C. This is a reminiscence of the well-knownfact in functional analysis asserting that a commutative C-Banach algebra which isa field is in fact isomorphic to C. We use this fact to give a different description ofthe ∞-category of Tan-structures in S. It is easy to provide a model categoricalpresentation for this alternative description, and we exploit this fact to carry out afew basic computations that will prove useful in dealing with the analytificationfunctor.

With Section 3 we enter in the main body of the article. The goal is to provide anfunctor of points description of derived C-analytic spaces and use this to (partially)compare them with the higher analytic stacks introduced in [20]. To be moreprecise, we introduce a category of derived Stein spaces Stnder

C and we endow itwith a Grothendieck topology τ and a collection of morphisms Pet. We then prove:


Theorem 3 (Proposition 3.6 and Theorem 3.7). There exists a fully faithful functor

φ : dAnC → Sh(StnderC , τ)

Let now dAnlocC be the full subcategory of dAnC spanned by those derived C-

analytic spaces (X,OX) for which X is an n-localic ∞-topos for some integern. If X ∈ dAnloc

C , then φ(X) is a geometric stack with respect to the context(Stnder

C , τ,Pet) (see [20, Definitions 2.11 and 2.15]). Vice-versa, geometric stacksfor the context (Stnder

C , τ,Pet) constitute the essential image of the restrictionφ : dAnloc

C → Sh(StnderC , τ).

Remark 4. An∞-topos is n-localic if it is a category of sheaves on some n-category.We redirect the reader to [10, § 6.4.5] for the definition and the main properties ofn-localic ∞-topoi.

Remark 5. An analogous statement holds in the case of algebraic stacks, in whichcase the result can be thought as a precise comparison between the notion ofderived Deligne-Mumford stack in the sense of [26] and the one introduced in [12].This precise formulation seems to be a folklore result and it can somehow be foundscattered through the DAG series of J. Lurie. Nevertheless, a concentrated proofcan be found in [18, Theorem 1.7].

We end Section 3 by discussing in some details the notion of truncations ofderived C-analytic spaces and the basic properties enjoyed by this operation. Weobtain the following comparison result:

Corollary 6 (Corollary 3.10). Suppose X ∈ dAnC is a truncated derived C-analyticspace belonging to dAnloc

C . Then φ(X) is a higher analytic Deligne-Mumford stackin the sense of [20].

In Section 4 we simply introduce the notion of coherent sheaf on a derivedC-analytic space, and we continue the comparison with [20] by proving:

Theorem 7 (Proposition 4.3). Suppose X ∈ dAnC is a truncated C-analytic spacesatisfying the same finiteness conditions of Theorem 3. Then the ∞-category ofcoherent sheaves on X is equivalent to the ∞-category of coherent sheaves on φ(X)introduced in [20].

In Section 5, we show that we inherit from [20] a notion of proper morphismfor derived Deligne-Mumford stacks. Using the comparison results Theorem 3 andTheorem 7, we obtain the first main result of this article: a derived version ofGrauert’s proper direct image theorem.

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Theorem 8 (Proposition 5.5). Let f : X → Y be a proper morphism of derivedC-analytic spaces, both satisfying the same finiteness conditions of Theorem 3.Then the derived direct image functor Rf∗ : OX-Mod → OY -Mod takes Coh+(X)to Coh+(Y ).

Section 6 is the veritable heart of the article. Here, we carry over a detailedanalysis of the analytification functor introduced in [11, Remark 12.26]. We cansummarize the main results of this section as follows:

Theorem 9 (Proposition 6.8 and Corollary 6.20). Let X be a derived Deligne-Mumford stack which is locally of finite presentation over C. Then:

(1) (flatness) There is a natural map Xan → X in the category of E∞-ringedtopoi which is flat in the derived sense.

(2) (comparison) If X is truncated, then φ(Xan) coincides with the analytifi-cation of the functor of points associated to X, this analytification beingunderstood in the sense of [20]. In particular, when X is a scheme of finitepresentation over C, Xan can be identified with the usual analytification inthe sense of Serre, cf. [9, Expose XII].

The flatness part is perhaps the most technical part of this article. The proof relieson the computations made in Section 2. As immediate consequence of this result,we obtain a rather explicit description of the analytification of a derived Deligne-Mumford stack in terms of the analytification of its truncation (see Corollary 6.21for a precise statement).

Flatness unlocks moreover the two GAGA theorems, which we can now prove byreducing to the analogous results proven in [20]:

Theorem 10 (Derived GAGA-1, Theorem 7.5). Let f : X → Y be a propermorphism of derived Deligne-Mumford stacks locally of finite presentation over C.Then for every F ∈ Coh+(X) the canonical analytification map

(Rf∗(F))an → Rf an∗ (Fan)

is an equivalence.

Theorem 11 (Derived GAGA-2, Theorem 7.7). Let X be a proper Deligne-Mumford stack locally of finite presentation over C. Then the analytificationfunctor induces an equivalence of ∞-categories

Coh(X)→ Coh(Xan)

Remark 12. In [20] the second GAGA theorem is stated only for the categoriesCohb. However, the hard part of the proof is to deal with the case of sheaves in


the heart Coh♥(X). When [20] appeared, T. Y. Yu and I weren’t aware that thesame proof could also yield this stronger result.

We conclude the article with the short Section 8, where we explain how the mainresults of this article can be extended to derived Artin stacks.Future work. This paper is part of my ongoing Ph.D. thesis at the universityof Paris Diderot. It is part of a larger program of exploration of derived analyticgeometry. In some sense, it is also a natural continuation of [20].

It will soon be followed by two related papers, [18, 19]. In [18] we carry out asimilar analysis to the one of Section 3 in the setting of derived algebraic geometry.In other words, we set a precise comparison between the notion of derived Deligne-Mumford stack of [26] and the one of [12]. This is certainly a folklore result whichis nevertheless difficult to find in the literature. As the proof is subtler than theone we carry over in Section 3, we feel like it deserves to be discussed at lengthand separately.

On the other side, in [19] we will deal more extensively with the notion of(coherent) sheaf of modules over an analytic space. There are in fact at leasttwo reasonable definitions for this category, and both are important to have areasonably complete theory. A large part of [19] will be devoted to the discussion ofa comparison between these two notions. We plan to apply this theory to provide aworkable theory of square-zero extensions and Postnikov truncations in the settingof derived C-analytic geometry.

About longer term projects, we plan to concentrate on a couple of very naturalquestions that this article leaves somehow open. Namely, the proof of the firstGAGA theorem Theorem 7.5 can be easily extended (both in the derived andin the underived setting) to the unbounded category of coherent sheaves underthe additional requirement for the maps f and f an to be of finite cohomologicaldimension. On the algebraic side one can reason by noetherian induction using thegeneric flatness lemma [20, Lemma 8.5] to show that properness implies boundedcohomological dimension. On the analytic side the picture seems slightly morecomplicated. As it seems an interesting question on its own, we will investigate itfurther.

Another question left aside both from this paper and from [20] is a version ofboth GAGA theorems relative to a C-analytic base (note that instead the underivedrigid analytic version has been dealt with in [20]). This problem is interesting as itis often needed in the practical problems of computing the analytifications of givengeometric stacks. We will therefore continue to pursue the matter.

On a even longer term program, and for sake of an ongoing joint project withT. Y. Yu we are currently investigating the properties of the analytic cotangent

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complex and its relation with a version of Artin representability theorem for derivedC-analytic spaces.

Finally, we also plan to apply the derived GAGA theorems to the study of someexamples of derived non abelian mixed hodge structures, following a recent proposalof Simpson, Toen, Vaquie and Vezzosi and their kind suggestion.

Last but not least, Jacob Lurie has very recently informed us that some of theresults in this paper will also appear in a forthcoming draft of his new book. Thetwo authors worked on this topics independently, so the two final versions willprobably differ. As an example, our GAGA Theorems 7.5 and 7.7 are plannedto appear in such a first draft under the additional assumption that the involvedstacks are actually spectral algebraic spaces. As J. Lurie informed us, he plannedto include our version for derived Deligne-Mumford stacks in a later version of theaforementioned book.

Conventions. Throughout this paper we will work freely with the notion of (∞, 1)-category. We will refer to such objects simply as ∞-categories. Sometimes, it willbe necessary to consider (n, 1)-categories. We refer to [10, §2.3.4] for the basictheory of such objects. In the article, we will denote them simply by n-categories(no confusion will arise because there will be no need of considering objects such as(∞, n)-categories throughout this paper).

For reasons of practical convenience, we chose to work within the frameworkof [10] and of [15]. When citing from these sources, we will suppress the wordsDefinition, Lemma, Proposition etc. The notation S will be reserved to the ∞-category of spaces. In [10, 6.3.1.5] two categories of ∞-topoi are introduced, LTopand RTop. If not specified otherwise, we will denote by Top the ∞-category RTop.

We will denote by CAlgC the∞-category of connective HC-algebras. Equivalently,this can be identified with the underlying∞-category of simplicial C-algebras. Sincewe work within the derived framework most of the time, we prefer to reserve thenotation B ⊗A C for the derived tensor product. Whenever the underived one isneeded, we will denote it by TorA0 (B,C).

Acknowledgments. I tried to make my intellectual debt to J. Lurie and hismonumental work as evident as possible since the very introduction, and I wouldlike to emphasize it once more here. I am deeply thankful to my advisor GabrieleVezzosi for suggesting me this very interesting research topic, for his kindness andgenerosity in suggesting many possible further developments. I would like to expressmy gratitude toward Carlos Simpson, who, directly and indirectly, encouraged mein moving on during this project. Finally, I am very grateful to V. Melani, M.Robalo, G. Vezzosi and T. Y. Yu for countless many stimulating discussions and forvery helpful comments that helped me while writing this paper. I would also like


to thank V. F. Zenobi for introducing me to the idea of the holomorphic functionalcalculus.

1. The setup

1.1. The C-analytic pregeometry. This section is mainly for the convenienceof the reader. We review the basic definitions and results of [11].

Definition 1.1 ([12, Def. 1.2.1]). Let C be an ∞-category. An admissibilitystructure on C consists of the following data:

(1) a subcategory Cad ⊂ C containing every object of C. Morphisms in Cad willbe referred to as admissible morphisms.

(2) a Grothendieck topology on C which is generated by admissible morphismsin the following sense: every covering sieve C

(0)/X ⊂ C/X contains a covering

sieve which is generated by admissible morphisms Uα → X.These data are required to satisfy the following compatibilities:

(1) whenever f : U → X is an admissible morphism and g : X ′ → X is anymorphism, the pullback

U ′ X ′

U X

f ′

g

f

exists and f ′ is again an admissible morphism.(2) Given a commutative triangle in C

X Y

Z

h

f

g

such that g and h are admissible, the same goes for f .(3) Admissible morphisms are stable under retracts.

Definition 1.2 ([12, Def. 3.1.1]). A pregeometry T consists of an∞-category withfinite products and an admissibility structure on it.

We let Tan be the pregeometry defined as follows: the underlying∞-category is the(nerve of) the category of open subsets of Cn. We say that a morphism is admissibleif it is an open immersion and we endow Tan with the analytic (Grothendieck)topology. Following [11] we introduce the notion of derived C-analytic space:

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Definition 1.3 ([11, Def. 12.3]). A derived C-analytic space is a Tan-structured∞-topos (X,OX) such that

(1) there exists a covering Ui of X such that X/Ui is the∞-topos of a topologicalspace Xi.

(2) The pair (Xi, π0(OalgX |Ui)) is a classical C-analytic space.

(3) For every i, the sheaves πi(OalgX |Ui) are coherent as π0(Oalg

X |Ui)-modules.We will denote by dAnC the ∞-category of derived C-analytic spaces.

We summarize the basic results about dAnC in the following proposition:

Proposition 1.4. (1) The category dAnC has pullbacks, and the forgetful func-tor dAnC → Top commutes with them.

(2) There is a canonical functor Φ: AnC → dAnC going from classical C-analytic spaces to derived C-analytic spaces which is fully faithful. Moreover,it commutes with pullbacks along local biholomorphisms.

(3) The natural forgetful functor dAnC → Top(Tet) commutes with pullbacksalong closed immersions.

Proof. These results are obtained in [11, §12]. The functor Φ is defined as follows:if X is a C-analytic space, Φ(X) = (Sh(X),OX), where Sh(X) is the ∞-topos of(a priori non-hypercomplete) sheaves on X, and OX is the functor

OX : Tan → Sh(X)

defined byOX(U) := Hom(U,−) : Opens(X)→ S

where Hom(U, V ) for V an open in X denotes the set of holomorphic maps fromU to V . In particular, we see that OX is 0-truncated.

Remark 1.5. The functor Φ: AnC → dAnC factorizes by construction through thefull subcategory of derived C-analytic spaces (X,OX) such that OX is 0-truncatedand X is 0-localic. It can be shown that every such derived C-analytic space lies inthe essential image of Φ.

To ease future reference, we collect here also the definitions of etale morphismand closed immersion of derived C-analytic spaces.

Definition 1.6. Let f : (X,OX) → (Y,OY) be a morphism of derived C-analyticspaces. We will say that:

(1) ([12, Definition 2.3.1]) f is etale if the underlying geometric morphism of∞-topoi f−1 : Y X : f∗ is etale in the sense of [10, § 6.3.5] and the inducedmorphism f−1OY → OX is an equivalence.


(2) ([11, Definition 1.1]) f is a closed immersion if the underlying geometricmorphism of ∞-topoi f−1 : Y X : f∗ is a closed immersion in the senseof [10, 7.3.2.7] and the induced morphism f−1OY → OX is an effectiveepimorphism.

Let Tdisc = Tdisc(C) be the pregeometry discussed in the Introduction, Example 1.There is a natural morphism of pregeometries Tdisc → Tan which induces a forgetfulfunctor

Top(Tan)→ Top(Tdisc)Accordingly to the notations introduced in [11], if (X,OX) ∈ Top(Tan), we willdenote by (X,Oalg

X ) its image under this functor. Observe that OalgX can be identified

with a sheaf of connective HC-algebras on X.

1.2. Weak Morita equivalences of pregeometries. Let T be a pregeometryand let X be an ∞-topos. In general, the ∞-category of T-structures on X isnot a presentable ∞-category. The problem is that StrT(X) and Strloc

T (X) are notcocomplete (see [12, Prop. 1.5.1] for a more detailed discussion), and the ultimatereason for this is to be found in the compatibilities that the objects of StrT(X) arerequired to have with the Grothendieck topology of T. This could be an issue indeveloping derived C-analytic geometry. However, it is still true in general that,whenever O ∈ StrT(X) is a T-structure, the overcategory Strloc

T (X)/O is a presentable∞-category. The goal of this subsection is to provide a proof of this statement.In the economy of the present work, the relevance of this section is mainly for itsapplication to the structure of local analytic rings, that will be discussed in thenext section.

We begin with a definition:

Definition 1.7. Let ϕ : T′ → T be a morphism of pregeometries. We will saythat ϕ is a weak Morita equivalence if for every ∞-topos X and every T-structureO ∈ StrT(X) the restriction functor

StrlocT (X)/O → Strloc

T′ (X)/Oϕis an equivalence of ∞-categories.

Remark 1.8. In [12, §3.2] J. Lurie introduced the notion of Morita equivalenceof pregeometries in order to understand which modifications of the admissibilitystructure on a pregeometry T gives rise to the same ∞-category of T-structuredtopoi. The notion we introduced above is quite different in the spirit. It is in factmeant to be an intermediate step between Morita equivalences and the other notion,not yet appeared, of stable Morita equivalences. The latter is of great importancefor the theory of modules for analytic structures, and we will come back on the

16 MAURO PORTA

subject in [19]. Some of the results collected here which do not find an immediateapplication are stated for sake of future reference.

Lemma 1.9. Let T be a pregeometry and let X be an ∞-topos. For every O ∈StrT(X) the canonical functor

j : StrlocT (X)/O → StrT(X)/O

is fully faithful.

Proof. The functor StrlocT (X)/O → StrT(X) is faithful, and therefore j is faithful as

well. Let A,B ∈ StrlocT (X)/O and denote by p : A → O, q : B → O the structural

morphisms. If we show that every morphism α : A→ B is a local morphism, thefullness of j will follow at once. Let f : U → V be an admissible morphism in T

and consider the commutative diagram

A(U) B(U) O(U)

A(V ) B(V ) O(V )

αU qU

αV qV

The horizontal composites are equivalent to pU and pV respectively. Since p andq are local morphisms, we see that the right square as well as the outer one arepullback. It follows that the left square is a pullback too, completing the proof.

Lemma 1.10. Let C be an ∞-category and let X be an object in C. There is afaithful functor

C/X → Fun(∆1,C)

whose essential image is given by those arrows with target equivalent to X. Moreover,a morphism in Fun(∆1,C) belongs to C/X if and only its restriction to the target isequivalent to the identity of X.

Proof. The simplicial model for C/X of [10, 1.2.9.2] provides us with the desiredfunctor. Moreover, C/X can be exhibited as the (homotopy) pullback diagram

C/X Fun(∆1,C)

∆0 C

ev1

X

from which the lemma follows.


Proposition 1.11. Let T be a pregeometry and let X be an ∞-topos. Let O ∈StrT(X) be a T-structure. There exists a faithful functor

StrlocT (X)/O → Fun(T ×∆1,X) (1.1)

whose essential image consists of those functors F : T × ∆1 → X satisfying thefollowing conditions:

(1) the restriction F1 := F |T×1 is equivalent to O;(2) the restriction F0 := F |T×0 commutes with products;(3) for every admissible morphism f : U → V in T the induced square

F0(U) F0(V )

F1(U) F1(V )

is a pullback square.

Moreover, let F,G ∈ StrlocT (X)/O and let α : F → G be a morphism between them in

Fun(T ×∆1,X). Then α belongs to StrlocT (X)/O if and only if the restriction α|T×0

is equivalent to the identity of O.

Proof. We can factor the functor (1.1) as

StrlocT (X)/O u−→ StrT(X)/O v−→ Fun(T ×∆1,X).

Lemma 1.9 shows that u is fully faithful. The first statement and the last onefollow therefore from Lemma 1.10. It moreover clear that all the objects in theessential image of (1.1) satisfy conditions (1) to (3). Let F : T × ∆1 → X be afunctor satisfying these conditions. Using (1) and (3) we see that this functordetermines a local morphism F0 → O in Fun(T,X). To conclude, we only need toshow that F0 ∈ StrT(X). Since F0 commutes with products in virtue of condition(2), we only need to check that F0 commutes with admissible pullbacks and takescoverings in T to effective epimorphisms. Let

U ′ U

V ′ V

f ′ f

18 MAURO PORTA

be a pullback square in T, where f : U → V is an admissible morphism. Considerthe commutative diagram

F0(U ′) F0(U) O(U)

F0(V ′) F0(V ) O(V )

and observe that the right square is a pullback because f is admissible. Moreover,the outer square can be also factored as

F0(U ′) O(U ′) O(U)

F0(V ′) O(V ′) O(V )

The left square is a pullback because O ∈ StrT(X), and the left square is a pullbackbecause f ′ : U ′ → V ′ is an admissible pullback. It follows that F0 preservesadmissible pullbacks.

Now let Ui → U be an admissible covering in T. Since X is an ∞-topos, wesee that there is a pullback square

∐F0(Ui) F0(U)

∐O(Ui) O(U)

Since the bottom line is an effective epimorphism, the result now follows from thestability of effective epimorphisms under pullbacks.

Proposition 1.11 allows to produce several examples of weak Morita equivalences.

Definition 1.12. Let T be a pregeometry. The associated discrete pregeometry Tdis the discrete pregeometry having the same underlying∞-category of T. The asso-ciated semi-discrete pregeometry Tsd is the pregeometry having the same underlying∞-category and the same admissible morphisms of T, but discrete Grothendiecktopology.

Observe that we have morphisms of pregeometries Td → Tsd → T.

Corollary 1.13. Let T be a pregeometry. The morphism Tsd → T is a weak Moritaequivalence.


Proof. Let X be an∞-topos and let O ∈ StrT(X). Denote by O′ the restriction of Oalong Tsd → T. Proposition 1.11 allows to identify both Strloc

T (X)/O and StrlocTsd

(X)/O′with the same subcategory of Fun(T ×∆1,X). Therefore, they are equivalent.

Lemma 1.14. Let T be a pregeometry in which the Grothendieck topology is discrete.Then for every ∞-topos X, the ∞-category StrT(X) is presentable.

Proof. This is a direct consequence of [10, Lemmas 5.5.4.18, 5.5.4.19].

Proposition 1.15. Let T be a pregeometry and let X be an ∞-topos. For everyO ∈ StrT(X) the ∞-category Strloc

T (X)/O is presentable.

Proof. In virtue of Corollary 1.13 we can assume that the Grothendieck topologyon T is discrete. In this case, Lemma 1.14 shows that StrT(X) is a presentable∞-category. [12, Theorem 1.3.1] shows that there exists a factorization system(SX

L , SXR) on StrT(X) such that SX

R is the collection of morphisms in StrlocT (X).

Let D be the full subcategory of Fun(∆1, StrT(X)) spanned by the elements ofSXR. [10, Lemma 5.2.8.19] shows that D is a localization of Fun(∆1, StrT(X)). Since

filtered colimits commutes with pullbacks, we see that D is closed under filteredcolimits in Fun(∆1, StrT(X)). In other words, D is an accessible localization ofthe latter category. Since StrT(X) is a presentable ∞-category, [10, Proposition5.5.1.2] shows that D is presentable as well. Observe now that Strloc

T (X)/O fits intoa pullback square

StrlocT (X)/O D

O StrT(X)

ev1

[10, Theorem 5.5.3.18] shows that StrlocT (X)/O is presentable, thus completing the

proof.

For future convenience, we record the following easy consequence:

Corollary 1.16. Let T be a pregeometry and let X be an ∞-topos. For everyO ∈ StrT(X) the category of O-modules Sp(Strloc

T (X)/O) is a presentable ∞-category.

Proof. This follows from Proposition 1.15 and [15, Proposition 1.4.4.4.(1)].

Proposition 1.17. Let T be a pregeometry and let X be an ∞-topos. Let O ∈StrT(X) be a T-structure. The functor Strloc

T (X)/O → Fun(T × ∆1,X) createsconnected limits and sifted colimits.

Proof. Let K be a simplicial set and let F : K → Fun(T × ∆,X) be a functorfactorizing through Strloc

T (X)/O. Let F : K. → Fun(T ×∆1,X) be a limit diagram

20 MAURO PORTA

and set G := F (v0), where v0 denotes the initial vertex of K/. G0 commutes withproduct and G1 is the limit of the constant diagram associated to O. Since siftedcolimits are connected, we see that G1 is equivalent to O. Moreover, if f : U → V

is an admissible morphism in T, the diagram

G0(U) G0(V )

G1(U) G1(V )

is the limit of pullback diagrams and it is therefore a pullback diagram by itself.Let now K be a sifted simplicial set and let F : K → Fun(T×∆,X) be a functor

factorizing through StrlocT (X)/O. Let F : K/ → Fun(T×∆1,X) be a colimit diagram

and set G := F (v∞), where v∞ denotes the final object of K/. Since K is sifted,we see that G0 commutes with products. As before, we see that G1 is equivalent toO. Let f : U → V be an admissible morphism in T. Since X is an ∞-topos, we seethat

G1(U)×G1(V ) G0(V ) ' O(U)×O(V ) G0(V )' O(U)×O(V ) colim

K(F0(V ))

' colimK

(O(U)×O(V ) F0(V )) ' colimK

F0(U) ' G0(U)

We therefore conclude that G belongs to the StrlocT (X)/O.

Corollary 1.18. Let ϕ : T′ → T be a morphism of pregeometries. Let O ∈ StrT(X)be a T-structure. The restriction functor

Φ: StrlocT (X)/O → Strloc

T (X)/Oϕcommutes with limits and sifted colimits.

Proof. Observe that the relevant functor takes the final object to the final objectby the very construction. It will be therefore sufficient to show that it commuteswith connected limits and sifted colimits. Consider the commutative diagram

StrlocT (X)/O Strloc

T′ (X)/Oϕ

Fun(T ×∆1,X) Fun(T′ ×∆1,X)

Proposition 1.17 shows that the vertical morphisms creates both connected limitsand sifted colimits, while Fun(T ×∆1,X)→ Fun(T′ ×∆1,X) preserves all (small)limits and (small) colimits. The proof is therefore complete.


Corollary 1.19. Let ϕ : T′ → T be a morphism of pregeometries. Let O ∈ StrT(X)be a T-structure. The restriction functor

StrlocT (X)/O → Strloc

T (X)/Oϕis a right adjoint.

2. Local analytic rings

The goal of this section is to analyze in detail the category StrlocTan(S). We

introduce a special notation:

Definition 2.1. The∞-category of local analytic rings is the∞-category StrlocTan(S).

We will denote it by AnRingC.

2.1. Functional spectrum. Let us fix some notations. We will denote by H0 theTan-structure defined by

H0(U) = HomTan(∗, U) ' U ∈ S

where U is thought as a simple (discrete) set. Observe that H0 is indeed a Tan-structure and that it is moreover discrete. If O ∈ AnRingC, then we see that, bythe Yoneda lemma, one has

MapAnRingC(H0, F ) ' F (∗) ' ∗

In particular, H0 is an initial object in AnRingC. On the other side the categoryAnRingC has a final object, which we will denote by 0. We will refer to it as thenull analytic ring. It is the constant functor associated to ∗ ∈ S.

The main goal of this section is to prove the following result:

Theorem 2.2. Let StrTan(S)′ be the full subcategory of StrTan(S) spanned by everyobject but the null analytic ring. Then H0 is a final object in StrTan(S)′. Moreover,for every O ∈ StrTan(S)′ the canonical map O→ H0 is a local transformation.

Corollary 2.3. One has StrTan(S)′ ' StrTan(S)/H0 ' AnRingC/H0. In particular,all these categories are presentable.

Proof. The two equivalences follow directly from the theorem. The last assertion isa consequence of Proposition 1.15.

Remark 2.4. We also see that the above corollary, together with Lemma 1.9,implies that Strloc

Tan(S) ' StrTan(S).

Definition 2.5. We will refer to the ∞-category StrlocTan(S)/H0 as the ∞-category

of local analytic rings. We will denote it AnRinglocC .

22 MAURO PORTA

Observe that since H0 is 0-truncated, any map O → H0 factors uniquely asO→ τ≤0O→ H0. Observe also that since Tan is compatible with n-truncations (see[12, Definition 3.3.2] for the definition and [11, Proposition 11.4] for a proof), τ≤0O

is a Tan-structure in S, and the morphism O→ τ≤0O is a local morphism. Therefore,to prove Theorem 2.2 it is sufficient to deal with discrete local analytic rings. Wewill denote by AnRing0

C the full subcategory of AnRingC spanned by discreteobjects. The above considerations show that the first statement of Theorem 2.2 isin fact equivalent to prove the following key result:

Proposition 2.6. Let O ∈ AnRing0C and suppose O 6= 0. Then Oalg is a local ring

with residue field C.

Remark 2.7. Observe that this result would be false if we replaced Tan with TZar.

We will need several lemmas.

Lemma 2.8. Let O ∈ AnRinglocC and suppose O 6= 0. Then O(∅) = ∅.

Proof. Since we have an admissible morphism ∅ → ∗, we see that O(∅)→ O(∗)is a monomorphism. Therefore O(∅) has cardinality at most 1. Suppose bycontradiction that O(∅) has one element. Then, consider the commutative triangle

∅

C C,ta

where ta denotes the translation by the element a ∈ C. Applying O we wouldget another commutative triangle, and the morphism O(∅)→ O(C) would selectan element f ∈ O(C) such that f + a = f . This is impossible as soon as a 6= 0.Therefore O(∅) = ∅.

Let O ∈ AnRing0C be a non-null discrete analytic ring and let f ∈ Oalg. Define

the spectrum of f as follows:

σO(f) = σ(f) := a ∈ C | f − a 6∈ (Oalg)×

Lemma 2.9. Let f ∈ Oalg. Then a 6∈ σ(f) if and only if f ∈ O(C∗a), whereC∗a := C \ a.

Proof. When a = 0, we will simply write C∗ instead of C0. If f ∈ O(C∗a) thenf − a ∈ O(C∗) and therefore f − a ∈ (Oalg)×. Conversely, suppose that f − a is


invertible. Consider the pullback square

C∗ × C∗ C× C

C∗ C

The horizontal morphisms are open immersions, and therefore the induced square

O(C∗)× O(C∗) O(C)× O(C)

O(C∗) O(C)

is a pullback square as well. Let g be an inverse for f−a. Then the pair ((f−a, g), 1)determines an element of the above pullback. This means that both f − a and g

factors through O(C∗), completing the proof.

Lemma 2.10. If f ∈ Oalg then σ(f) 6= ∅.

Proof. Suppose by contradiction that σ(f) = ∅ for some f ∈ Oalg. Then for everya ∈ C, f − a is invertible, that is, f ∈ O(C∗a) for every a ∈ C. Fix a ∈ C and write

C∗a =⋃n>0

(C \D

(a,

1n

))

It follows that there exists ε > 0 such that f ∈ O(C \D(a, ε)). Since O(∅) = ∅, weconclude that f 6∈ O(D(a, ε)). But then we can cover C with open subsets Ui suchthat f 6∈ O(Ui) for every i, which is a contradiction.

Lemma 2.11. Suppose that O ∈ AnRing0C and let f ∈ Oalg. Then σ(f) has at

most one point.

Proof. Suppose that a, b ∈ σ(f), with a 6= b. Then f 6∈ O(C∗a) and f 6∈ O(Cb), whichimplies that f ∈ O(D(a, ε)) ∩ O(D(b, η)) for every ε, η > 0. This is a contradictionsince O(∅) = ∅.

Corollary 2.12. Let O ∈ AnRing0C be a non-null discrete analytic ring. For every

f ∈ Oalg, σ(f) consists precisely of one point.

At this point we obtain a well defined function of sets

σ : Oalg → C

Lemma 2.13. One has σ(f + g) = σ(f) + σ(g) and σ(fg) = σ(f)σ(g).

24 MAURO PORTA

Proof. Set σ(f) = a and σ(g) = b. It will be sufficient to check that a+b ∈ σ(f+g).Suppose that there exists h ∈ Oalg such that (f + g − a− b)h = 1. Then the idealgenerated by f − a and g − b is the unit ideal. However, O is local and both f − aand g − b belongs to the maximal ideal. This is therefore impossible. Similarly,suppose there exists h such that (fg − ab)h = 1. Then (f − a)gh+ ah(g − b) = 1,and the proof proceeds as for the addition.

We can now prove Proposition 2.6:

Proof of Proposition 2.6. Let O ∈ AnRingC be a non-null analytic structure. Wededuce from Lemma 2.13 that σ : Oalg → C is a morphism of rings. Lemma 2.9shows that the kernel of σ is precisely the set of non invertible elements of Oalg. Itfollows that ker(σ) has to be the unique maximal ideal of Oalg. Since σ is evidentlysurjective, the proof is complete.

The next result achieves the proof of Theorem 2.2.

Proposition 2.14. Let O1 ∈ AnRing0C be a non-null discrete local analytic ring.

Then Oalg1 is canonically augmented toward C and the morphism σ1 : O1 → C

comes from a local morphism s1 : O1 → H0. Moreover, if O2 is any other non-nulldiscrete local analytic ring and f : O1 → O2 is any morphism between them, onehas s2 f = s1.

Proof. Let us prove that O1 → H0 is a natural transformation. This is equivalentto prove that if f ∈ O(U) then σ(f) ∈ U . Let a 6∈ U . Then there exists ε > 0 suchthat f 6∈ O(D(a, ε)), which means that f ∈ O1(C∗a). Therefore f − a is invertible,i.e. a 6∈ σ(f). Therefore σ(f) ∈ U .

Let us now prove that O1 → H0 is a local morphism. We claim that if f ∈ O1(C)and σ(f) ∈ U , then f ∈ O1(U). Let a = σ(f), so that f 6∈ O1(C∗a). SinceC = C∗a ∪ U , we conclude that f ∈ O1(U). This completes the proof.

Finally, we prove the last statement. Let x ∈ O1(C), a = σ1(x). Then x − ais non-invertible. This means that x − a belongs to O1(D(0, ε)) for every ε > 0.Then f(x − a) = f(x) − a belongs to O2(D(0, ε)) for every ε > 0. It followsthat σ2(f(x) − a) ∈ D(0, ε) for every ε > 0. Therefore σ2(f(x) − a) = 0, i.e.σ2(f(x)) = a = σ1(x).

Corollary 2.15. For any non-null local analytic ring O ∈ AnRingC (not necessarilydiscrete), the canonical morphism H0 → O is a local morphism.

2.2. Quotients of local analytic rings. This subsection and the next one arenot, strictly speaking, needed for this work. Nevertheless, we chose to include themfor two reasons: on one side, they help in building a 1-categorical intuition on what


kind of objects (local) analytic rings are, and on the other side the results collectedhere will be used in [19].

Let A ∈ AnRing0C and let I ⊂ Aalg be a (proper) ideal. Observe that I is

contained in the maximal ideal of A and therefore σ(I) = 0.

Construction 2.16. We will construct an analytic ring A/I : T 0an → Set (we are

using the results of Section 2.4). Define

(A/I)(p,Cn) := a ∈ (Aalg/I)n | σ(a) = p

Observe that this is well defined. Indeed, if a = b, then a− b ∈ In and thereforeσ(a− b) = 0. Let now ϕ : (p,Cn)→ (q,Cm) be a germ of holomorphic map. Set

(A/I)(ϕ)(a) := A(ϕ)(a)

This is well defined. Indeed, if a = b, then we can write a = b+ x, where x ∈ In.Therefore

A(ϕ)(a) = A(ϕ)(b) +∑

A

(∂f

∂zi

)(b)xi +

∑A(gij)(b+ x)xixj

and thereforeA(ϕ)(a) = A(ϕ)(b)

It follows from the construction that A/I commutes with products. In particular,we see that A/I defines a local analytic ring. Moreover, the natural projectionmap p : A → A/I is a morphism of local analytic rings. Observe finally that(A/I)alg = Aalg/I.

Lemma 2.17. Let f : A→ B be a morphism of discrete local analytic rings. Thefollowing are equivalent:

(1) the induced map A(0,C)→ B(0,C) is surjective;(2) the induced map A(0,Cn)→ B(0,Cn) is surjective for every n ≥ 0;(3) the induced map A(p,Cn)→ B(p,Cn) is surjective for every n and every p.

Proof. The implications 3.⇒ 2. and 2.⇒ 1. are trivial. Suppose that A(0,C)→B(0,C) is surjective. Since both A and B commute with products, 2. follows atonce. Since (p,Cn) and (0,Cn) are isomorphic, 3. follows as well.

Definition 2.18. We will say that a morphism of discrete local analytic ringsf : A→ B is surjective if it satisfies the equivalent conditions of the above lemma.

Lemma 2.19. Let f : A→ B be a surjective map of discrete local analytic ringsand set I := ker(Aalg → Balg). Then for every other discrete local analytic ring C,we have

Hom(B,C) = g : A→ C | galg(I) = 0

26 MAURO PORTA

Proof. Let g : A → C be such that galg(I) = 0. Define g : B → C as follows. Ifb ∈ B(p,Cn), we can choose an element a ∈ A(p,Cn) such that f(a) = b. Define

g(b) := g(a)

If a′ ∈ A(p,Cn) is another element satisfying f(a′) = b, we see that a−a′ ∈ A(0,Cn)and a− a′ ∈ I. Therefore f(a− a′) = 0, i.e. f(a) = f(a′). This shows that g is welldefined. Let us show that g is a natural transformation. If ϕ : (p,Cn)→ (q,Cm) isa germ of a holomorphic map, we can consider the diagram

A(p,Cn) B(p,Cn) C(p,Cn)

A(q,Cm) B(q,Cm) C(q,Cm)

Since the maps A(p,Cn)→ B(p,Cn) and A(q,Cm)→ B(q,Cm) are surjective, inorder to check that the square on the right commutes, it is sufficient to check thatthe outer rectangle does. This follows from the hypothesis that g is a naturaltransformation.

Let A ∈ AnRing0C and let I ⊂ Aalg be an ideal. Then the canonical map

A→ (A/I) is surjective and therefore it is characterized by the universal propertyof the previous lemma. Vice-versa, if A → B is a surjective morphism of localanalytic rings and I := ker(Aalg → Balg), then B ' A/I.

For the next result, we will need a bit of notation. As we discussed in theintroduction, analytic rings are a way of axiomatizing the holomorphic functionalcalculus enjoyed by commutative Banach C-algebras. For this reason, we choose todenote the pushouts B∐AC in AnRing0

C by B⊗AC.

Corollary 2.20. Suppose that f : A→ B is a surjective morphism of local analyticrings. For every other morphism g : A→ C, we have

(B⊗AC)alg ' Balg ⊗Aalg Calg

In other words, the functor (−)alg commutes with pushouts in which at least one ofthe two maps is surjective.

Proof. Write B = A/I. Then

Hom(B⊗AC,R) = g : C → R | galg(ICalg) = 0

In other words, B⊗AC = C/ICalg. It follows that (B⊗AC)alg = Calg/ICalg =Balg ⊗Aalg Calg.


Remark 2.21. This result, whose counterpart for Banach algebras is very well-known, has been generalized to a great extent by J. Lurie in [11]. We can withoutany doubt say that this generalization is the most important idea contained there.The precise way of formulating this uses once again the language of pregeometries,and the relevant notion is that of unramified transformation of pregeometries (seeloc. cit., Definition 10.1) The condition of being unramified is a rather simple test fora morphism of pregeometries (see loc. cit., Remark 10.2 for a simplified formulation),and yet it has the powerful consequence expressed in loc. cit., Proposition 10.3.The proof of this striking result passes through the highly technical Proposition2.2, which is perhaps an interesting result on its own.

2.3. Analytic algebras. We will denote by LRingSpaces the (1-)category of lo-cally ringed topological spaces.

Let An = Cz1, . . . , zn be the algebra of germs of holomorphic functions around0 ∈ Cn. This is an analytic algebra in the sense of Malgrange. We define anenhancement of An to the setting of analytic rings as follows. Define Hn : Tan → Setby setting:

Hn(U) := germs of holomorphic functions (0,Cn)→ U= MapLRingSpaces((S, An), U)

It is clear that Hn defines a functor Tan → Set and that moreover Hn(C) = An.

Lemma 2.22. The functor Hn is a Tan-structure on Set.

Proof. It follows directly from the definition that Hn commutes with all the limitsthat exist in Tan. In particular, it commutes with products and with admissiblepullbacks. Let now Ui ⊂ U be an open cover of U . The morphisms Ui ⊂ U

are open immersions in the category of locally ringed spaces, and therefore theyenjoy the following universal property: a morphism Z → U from a locally ringedspace Z factors through Ui if and only if it factors topologically. Therefore, wesee that for every morphism (S, An) → U , there exists an index i such that thismorphism factorizes through Ui. It follows that the map ∐

Hn(Ui) → Hn(U) issurjective.

Our next goal is to characterize Hn with a universal property.

Proposition 2.23. Let B ∈ AnRing0C. Then

HomAnRing0C(Hn, B) = (b1, . . . , bn) ∈ B(C) | σ(b1) = · · · = σ(bn) = 0

Proof. Suppose given a natural transformation ϕ : Hn → B. Let us denote byz1, . . . , zn the germs at 0 of the coordinate functions on Cn. That is, we have

28 MAURO PORTA

z1, . . . , zn ∈ Hn(C) = Hn. Their image via ϕ define elements b1, . . . , bn ∈ B.Moreover, ?? shows that σ(bi) = σ(ϕ(zi)) = ϕ(zi) = 0.

Conversely, suppose given elements b1, . . . , bn ∈ B(C) such that σ(b1) = · · · =σ(bn) = 0, we can define a morphism ϕ : Hn → B as follows. Let U ∈ Tan. Anelement f ∈ Hn(U) can be represented by a holomorphic function f : V → U forsome open neighborhood V of 0 ∈ Cn. Since the functional spectrum of b1, . . . , bnis zero, we see that

(b1, . . . , bn) ∈ B(V )We therefore define

ϕU(f) := B(f)(b1, . . . , bn) ∈ B(U)

If g : V ′ → U is another representation of f , then we can suppose that V ′ ⊂ V andthat g = f |V ′ . In this case, we have a commutative triangle

B(V ′) B(V )

B(U)B(g)

B(f)

and (b1, . . . , bn) ∈ B(V ′). This shows that the definition of ϕU (f) does not dependon the choice of the representation of f . It is straightforward to check that ϕUdefines a natural transformation. Finally, the two constructions we defined areclearly one the inverse of each other.

Corollary 2.24. The coproduct of Hn and Hm in AnRing0C is Hn+m.

Let now I be an ideal of Hn. This corresponds to a germ of C-analytic space(0, X), together with a map of germs j : (0, X) → (0,Cn). Define a functorHn/I : Tan → Set as follows:

(Hn/I)(U) := germs of holomorphic maps (0, X)→ U

Clearly, Hn/I ∈ AnRing0C. Moreover, (Hn/I)(C) = Hn/I. Composition with j

produces a natural transformation π : Hn → Hn/I.

Proposition 2.25. Let B ∈ AnRing0C. We have

Hom(Hn/I,B) = ϕ : Hn → B | ϕalg(I) = 0

Proof. Given ψ : Hn/I → B we obtain ϕ := ψ π : Hn → B, and it is clear thatϕalg(I) = 0. Conversely, if ϕ : Hn → B is such that ϕalg factors through Hn/I,then define ψ : Hn/I → B as follows. Fix U ∈ Tan and let f ∈ (Hn/I)(U) be agerm of a holomorphic function (0, X)→ U . Using Oka principle, we can represent


f as a holomorphic map V → U , where V is some open neighborhood of 0 in Cn.In other words, f is the restriction of some f ∈ Hn(U). Set

ψU(f) := ϕU(f)It is clear that this definition doesn’t change if we shrink V . On the other hand,suppose that g : V → U is another extension of the germ f : (0, X) → U . Thenf − g ∈ Hn(U) ⊂ Hn(C)m belong to the ideal Im. Therefore ϕU(f) = ϕU(g) byhypothesis.

Remark 2.26. This proposition allows to identify Hn/I with the categoricalquotient of Hn by the ideal I, as defined in the previous subsection.Corollary 2.27. The category of analytic algebras in the sense of Malgrangeembeds fully faithfully in AnRing0

C.Remark 2.28. This is a low-tech version of the fully faithful embedding of [11,Theorem 12.8].2.4. Pregeometries of germs. Corollary 2.3 is to some extent a very surprisingresult, at least for two reasons: it doesn’t hold when the topos is different fromS, and it doesn’t hold for a general pregeometry, even for X = S. We will showin this subsection that there is a deeper reason for this result. Namely, we willdescribe Strloc

Tan(S) as algebras for a suitable (multi-sorted) Lawvere theory. Wecould therefore deduce Corollary 2.3 directly from Lemma 1.14.

Let us define the category T 0an as follows:

(1) the objects of T 0an are pairs (n, p) where n is a natural number and p ∈ Cn.

(2) A morphism from (n, p) to (m, q) is a germ of holomorphic function from(p,Cn) to (q,Cm).

Observe that the category T 0an has products. Indeed, (n, p) × (m, q) = (n +

m, (p, q)). Let us say that a morphism f : (n, p)→ (m, q) in T 0an if it can be repre-

sented by an open immersion. In particular we have n = m. Let f : (n, p0)→ (n, p1)be an admissible morphism and let g : (m, q) → (n, p1) be any other morphism.Then

(m, q) (m, q)

(n, p0) (n, p1)

id

g + p0 − p1 g

f

is a pullback diagram in T 0an. In particular, we see that T 0

an can be equipped with astructure of semi-discrete pregeometry. However, since all admissible morphismsare isomorphisms, we see that T 0

an is in fact a discrete pregeometry. In other words,T 0

an is a (multi-sorted) Lawvere theory.

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Theorem 2.29. There exists an equivalence of ∞-categories StrTan(S) ' StrT 0an(S).

Sketch of a proof. Define a functorStrT 0

an(S)→ StrTan(S)by sending O0 ∈ StrT 0

an(S) to the functor O : Tan → S defined by

O(U) :=∐p∈U

O0(n, p)

It is clear that O is a functor and that it depends functorially on O0. Finally, itcommutes with admissible pullbacks and it takes coverings to effective epimor-phisms.

On the other direction, if O ∈ StrTan(S) is a Tan-structure on S, we defineO0 : T 0

an → S as follows:O(n, p) := O(Cn)×Cn p

where the map defining this homotopy fiber product is O(Cn)→ π0(O)(Cn) σ−→ Cn.It is straightforward to extend this definition on morphisms. Finally, it is clearthat these two functors are mutually inverse.

2.5. Strict models. It will be of some importance to develop in this section anexplicit presentation for the ∞-category Strloc

Tan(S). In virtue of Theorem 2.29, wecan replace Tan with the discrete pregeometry T 0

an. In particular, we haveStrloc

Tan(S) = Fun×(T 0an, S)

where the right hand side is the category of functors that preserve finite products.Consider now the 1-category sAnRing := Funct×(T 0

an, sSet) of (strict) functorspreserving finite products. Invoking [10, 5.5.9.1, 5.5.9.2], we see that defining amorphism f : F → G in sAnRing to be:

(1) a fibration if it is an objectwise fibration;(2) a weak equivalence if it is an objectwise weak equivalence.

we obtain a model structure on sAnRing whose underlying ∞-category is preciselyFun×(T 0

an, S).

Remark 2.30. Observe that Theorem 2.29 implies in particular that StrTan(Set) 'StrT 0

an(Set). We therefore deduce an equivalence of 1-categoriesStrTan(sSet) ' Funct×(T 0

an, sSet)Under this equivalence, the object Hn can be identified with the functor T 0

an →sSet corepresented by the germ (0,Cn). The universal property explained inProposition 2.23 becomes therefore a direct consequence of the Yoneda lemma.

We will need a better understanding of this model structure.


Lemma 2.31. Let f0 : K0 → L0 and f1 : K1 → L1 be Kan fibrations (resp. weakequivalences) of simplicial sets. Then f0×f1 : K0×K1 → L0×L1 is a Kan fibration(resp. a weak equivalence).

Proof. The statement on weak equivalences follows at once from the fact that thegeometric realization functor commutes with products and so do the homotopygroups. The statement on Kan fibrations follows immediately from the characteri-zation with the right lifting property against anodyne maps.

Proposition 2.32. Let f : R → S be a morphism in sAnRing. The followingconditions are equivalent:

(1) the induced morphism R(0,C)→ S(0,C) is a fibration (resp. a weak equiv-alence);

(2) the induced morphism R(0,Cn) → S(0,Cn) is a fibration (resp. a weakequivalence) for every n;

(3) the induced morphism R(p,Cn) → S(p,Cn) is a fibration (resp. a weakequivalence) for every n and every p ∈ Cn.

Proof. The implications 3. ⇒ 2. and 2. ⇒ 1. are obvious. Suppose now thatR(0,C)→ S(0,C) is a fibration (resp. a weak equivalence). Since both R and S

strictly preserve products, 2. follows from the previous lemma. Finally, 3. followsbecause we have isomorphisms (0,Cn) ' (p,Cn).

Corollary 2.33. Every object in sAnRing is fibrant.

Proof. Observe that the germ (0,C) has an abelian group structure (given bypointwise sum of germs of holomorphic functions). Since every F ∈ sAnRingcommutes with products, we see that F (0,C) is a simplicial group and therefore it isa Kan complex. At this point, the conclusion follows directly from Proposition 2.32.

Consider the forgetful functor

U : sAnRing→ sSet

given by U(R) := R(0,C).

Remark 2.34. Observe that this functor is quite different from the underlyingalgebra functor Φ. Indeed, Φ(R) = ∐

p∈CR(p,C).

U commutes with limits and sifted colimits. In particular, since both sAnRingand sSet are presentable, it follows that it has a left adjoint, which we denote by

H− : sSet→ sAnRing

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It follows from Proposition 2.32 that U is a right Quillen functor, so that H− a Uis a Quillen adjunction.

Proposition 2.35. The collection of morphisms I := H∂∆n → H∆nn∈Nis a set of generating cofibrations for sAnRing. The family of morphisms J :=HΛn

i → H∆nn∈N,0≤i≤n is a set of generating trivial cofibrations for sAnRing.

Proof. We already remarked that H− is a left Quillen functor. It follows thatall the morphisms in J are trivial cofibrations and all the morphisms in I arecofibrations. Let now p : R→ S a morphism in sAnRing. Observe that the liftingproblems

HΛni R

H∆n S

p

H∂∆n R

H∆n S

p

are respectively equivalent to the lifting problems

Λni U(R)

∆n U(S)

U(f)

∂∆n U(R)

∆n U(S)

U(f)

In particular, we see that p has the right lifting property with respect to maps in J(resp. I) if and only if U(p) is a fibration (resp. a trivial fibration). This completesthe proof.

Lemma 2.36. Let In be the set with n elements. Then π0(HIn) = Hn.

Proof. The two universal properties match.

Proposition 2.37. For every n, we have

πi(Φ(H∆n)) =

Cz if n = 00 otherwise.

where Cz denotes the algebra of germs of holomorphic functions around 0 ∈ C.

Proof. The morphism ∆0 → ∂∆n selecting the 0-th vertex is an acyclic cofibration.It follows that Φ(H∂∆n) is weakly equivalent to Φ(H∆0). It is thereforesufficient to prove the proposition for n = 0. The case i = 0 follows directly fromthe previous lemma. If i ≥ 1, we first observe that it is sufficient to show thatπi(H∆0(0,C)) = 0. For a simplicial set K with finitely many simplexes in everydegree, the n-simplexes of H∆0(0,C) are in bijection with Cz1, . . . , zm0, where


the variables zj correspond to the n-simplexes of K. In particular, we see thatH∆0 is the constant simplicial set associated to Cz0. The proof is thereforecomplete.

Proposition 2.38. We have π0(Φ(H∂∆n)) = Cz.

Proof. It will be enough to prove that π0(H∂∆n(0,C)) can be identified withthe set of germs of holomorphic maps (0,C)→ (0,C). We can explicitly representH∂∆n(0,C) as a simplicial algebra whose 1-skeleton is

Cz0,0, z0,1, . . . , zi,j, . . . , zn−1,n0 → Cz0, . . . , zn0

where the variable zij corresponds to the edge of ∂∆n connecting the vertex i tothe vertex j (hence i ≤ j) and

d0(zi,j) = zi, d1(zi,j) = −zj, s0(zi) = zi,i

Applying Dold-Kan and computing the 0-th cohomology of the resulting complex,we deduce that

π0(H∂∆n(0,C)) = Cz0, . . . , zn0/(zi + zj)0≤i<j≤n ' Cz0

3. The functor of points

With Definition 1.3 we are giving a “structured space” perspective on derivedC-analytic spaces. However, in the practice, it is often important to rely on thedual perspective of the functor of points. The goal of this section is precisely tointroduce this alternative point of view and to prove a comparison result withDefinition 1.3. Concretely, this means that we will have to discuss the followingpoints:

(1) we have to exhibit a geometric context (C, τ,P) (in the precise sense of[20, Definition 2.11]) in such a way that dAnC can be exhibited as a fullsubcategory of Sh(C, τ).

(2) We will have to identify the essential image of (a large subcategory of)dAnC with the geometric stacks in the sense of [20, Definition 2.15].

In dealing with the first point of the list, one could try to invoke [12, Theorem2.4.1]. Unfortunately, this result is not sufficient for our purposes: indeed, there isdiscrepancy between the Tan-schemes and the derived C-analytic spaces (see [11,Corollary 12.22 and Proposition 12.23] for a detailed discussion of this difference).Nevertheless, the same idea of the proof in loc. cit. applies to our context, as weare going to discuss.

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3.1. The geometric context.

Definition 3.1. We let StnderC be the full subcategory of dAnC spanned by those

derived C-analytic spaces (X,OX) such that (X, π0OalgX ) is of the form Φ(S), with S

a Stein space. We will refer to StnderC as the category of derived (C-analytic) Stein

spaces.

The notion of etale morphism of Tan-structured topoi (see Definition 1.6) induces aGrothendieck topology τ on the∞-category Stnder

C . We define a functor φ : dAnC →PSh(Stnder

C ) as follows:

dAnCj−→ Fun(dAnop

C , S)→ Fun((StnderC )op, S)

Our first task is to show that φ is fully faithful. To achieve this, we will show thatit can be factored as

φ : dAnC → Sh(StnderC , τ)

and that moreover φ is a fully faithful functor. We will need a couple of preliminaryfacts.

Lemma 3.2. Let (X,OX) be a derived C-analytic space. Then X is hypercomplete.

Proof. Let Y be an ∞-topos. The hypercompletion of Y is defined to be the fullsubcategory Y∧ of Y spanned by hypercomplete objects. Moreover, Y is said to behypercomplete if the inclusion Y∧ ⊂ Y is a categorical equivalence. Therefore, itfollows from [10, 6.5.2.21 6.5.2.22] that being hypercomplete is local on Y, in thesense that if there are objects Ui ∈ Y such that the map ∐Ui → 1Y is an effectiveepimorphism and each ∞-topos Y/Ui is hypercomplete, then the same goes for Y.

In the case of our interest, the definition of derived C-analytic space allows usto choose objects ∐Ui ∈ X such that each (X/Ui , π0O

algX |Ui) is a local model for

C-analytic spaces. In particular, we can identify X/Ui ' Sh(Xi), where Xi is alocally compact Hausdorff space of finite covering dimension. It follows that thehomotopy dimension of Xi is finite and therefore that X/Ui is hypercomplete.

Let X = (X,OX) be a derived C-analytic space and suppose that X is 0-localic.We will denote by Xtop the underlying topological space of X (see [12, 2.5.21]) Asevery point in a C-analytic space has a Stein open neighbourhood, we can findan hypercover U• of Xtop made by open Stein subsets of Xtop. Each open subsetof Xtop defines a (discrete) object of the ∞-topos X = Sh(Xtop). Therefore, thesimplicial object U• determines by composition with the Yoneda embedding asimplicial object in Xtop, which we will still denote U•. It follows directly from thedefinitions and from the criterion [10, 7.2.1.14] that U• is an hypercover of X. Let


us denote by Xn the derived C-analytic space defined as

Xn := (X/Un ,OX|Un)

The universal property of etale morphisms of structured Tan-topoi (see [12, Remark2.3.4]) shows that we can arrange the Xn into a simplicial object X• in the ∞-category Top(Tan). We claim that the geometric realization of the diagram X•

in Top(Tan) (and hence in dAnC) coincides precisely with the original derived C-analytic space X. To prove this it is sufficient to show the following two assertions:

(1) one has |X/U•| ' X;(2) Let jn∗ : X/U• → X be the given geometric morphism. Then in Strloc

Tan(X) onehas

OX ' lim←−∆jn∗OX|Un

The first assertion is a consequence of the general descent theory for ∞-topoi(see [10, Theorem 6.1.3.9]) and the fact that in X one has

|U•| ' U

as it follows from Lemma 3.2 and from [10, 6.5.3.12]. As for the second statement,since Strloc

Tan(X) is closed under limits in Fun(Tan,X), it will be enough to show thatfor every V ∈ Tan, in X one has

OX(V ) ' lim←−∆jn∗OX(V )|Un

Since X = Sh(Xtop) is closed under limits in PSh(Xtop), we see that it is enoughto prove that whenever W is an open of Xtop one has

OX(V )(W ) ' lim←−∆

OX(V )(Un ×W )

Since U• ×W is an hypercover of W , this statement is equivalent to say that eachOX(V ) ∈ X is hypercomplete, which is obvious since X is itself hypercomplete invirtue of Lemma 3.2.

Summarizing, we proved the following:

Lemma 3.3. Let X = (X,OX) be a derived C-analytic space such that X is 0-localic.Then there exists an hypercover U• in X such that each derived C-analytic spaceXn := (X/Un ,OX|Un) is a derived Stein space. Moreover, for each such hypercover,the geometric realization of X• in dAnC is X.

Corollary 3.4. The Grothendieck topology τ on StnderC is subcanonical. Moreover, if

X = (X,OX) is a derived C-analytic space, then φ(X) belongs to the hypercompletionof Sh(Stnder

C , τ).

36 MAURO PORTA

One can prove something more general. Indeed, the notion of etale morphismof Tan-structures defines also a Grothendieck topology τ ′ on dAnC and, with thesame arguments used above, one can prove that the Yoneda embedding

j : dAnC → PSh(dAnC)

factors through Sh(dAnC, τ′). This has the following useful consequence:

Corollary 3.5. Let X = (X,OX) be a derived C-analytic space and let p : U →1X be an effective epimorphism. Let U• be the Cech nerve of p and set Xn :=(X/Un ,OX|Un). Then in Sh(Stnder

C , τ) one has

φ(X) ' colim∆

φ(X•)

Proof. Let us temporarily denote by ψ : dAnC → Sh(dAnC, τ′) the functor obtained

by factorizing the Yoneda embedding j : dAnC → PSh(dAnC). The above discussionmakes clear that the relation

ψ(X) ' colim∆

ψ(X•)

holds in Sh(dAnC, τ′). The morphism of sites (Stnder

C , τ) → (dAnC, τ′) is both

continuous and cocontinuous. It follows from [20, Lemma 2.30] that the restrictionalong this functor is a left adjoint. In particular, it commutes with colimits, sothat the proof is complete.

Proposition 3.6. The functor φ is fully faithful.

Proof. Let X, Y ∈ dAnC and consider the natural map

ψX,Y : MapdAnC(X, Y )→ MapSh(Stnder

C ,τ)(φ(X), φ(Y ))

Keeping Y fixed, consider the full subcategory C of dAnC spanned by those X forwhich ψX,Y is an equivalence.

Choose objects Ui of X = (X,OX) in such a way that p : ∐Ui → 1X is an effectiveepimorphism and that each X/Ui is 0-localic. Let U := ∐

Ui and let U• be the Cechnerve of p. Finally, set Xn := (X/Un ,OX|Un). It follows from Corollary 3.5 that

φ(X) ' colim∆

φ(Xn)

We can therefore reduce ourselves to the case where X is 0-localic. InvokingLemma 3.3 and repeating the above argument, we can further reduce to the casewhere X ∈ Stnder

C , and the lemma is now a restatement of the Yoneda lemma.


3.2. Geometric stacks. We know turn to the second main goal of this section,that is, the characterization of the essential image of φ : dAnC → Sh(Stnder

C , τ).First of all, let us observe that letting Pet be the collection of etale morphisms inStnder

C , the triple (StnderC , τ,Pet) becomes a geometric context in the sense of [20,

Definition 2.11]. We will refer to geometric stacks relative to this context as derivedDeligne-Mumford analytic stacks. We will denote by DM the full subcategory ofSh(Stnder

C , τ)∧ spanned by derived Deligne-Mumford analytic stacks. We will furtherdenote by DMn the full subcategory of DM spanned by n-geometric stacks. Onthe other side, we will denote by dAn≤nC the full subcategory of dAnC spanned bythose derived C-analytic spaces (X,OX) for which X is an n-localic topos. Finally,we will denote by dAnloc

C the reunion of all the subcategories dAn≤nC as n rangesthrough the integers. With these notations, we can formulate the main result ofthis section:

Theorem 3.7. For every n ≥ 0, the functor φ : dAnC → Sh(StnderC , τ) restricts to

an equivalence dAn≤nC ' DMn.

Remark 3.8. The previous theorem (when combined with Corollary 3.10) providesa large generalization of the comparison result [11, Theorem 12.8]. See howeverRemark 12.25 loc. cit. for something going in this direction.

The following lemma will be used repeatedly in the proof of Theorem 3.7.

Lemma 3.9. Let f : U → V be a morphism in Tan. Let us write SpecTan(U) =(XU ,OU) and SpecTan(V ) = (XV ,OV ). The following conditions are equivalent:

(1) f is etale (resp. an etale monomorphism);(2) the induced morphism (f∗, ϕ) : SpecTan(U)→ SpecTan(V ) is an etale mor-

phism of derived C-analytic spaces (resp. is etale and XU ' (XV )/W whereW is a discrete object of XV ).

Proof. Suppose first that f is etale. Then the assertion follows immediately from [12,Example 2.3.8]. If moreover f is a monomorphism, then XU can be identified withthe topos associated to an open subsetW of V top and therefore XU ' (XV )/W andWis clearly a discrete object of XV . Vice-versa, suppose that (f∗, ϕ) : SpecTan(U)→SpecTan(V ) is an etale morphism of derived C-analytic spaces. Then f can berecovered entirely from the geometric morphism f∗ : XU → XV and the verydefinition of etale morphisms of ∞-topoi makes clear that f has to be a localhomeomorphism, which is injective if XU is the etale subtopos of XV associated toa discrete object. Since it was a holomorphic map to begin with, it also followsthat f is a local biholomorphism, thus completing the proof.

The proof of Theorem 3.7 naturally splits into two parts:

38 MAURO PORTA

(1) to show that whenever X ∈ dAn≤n+1C the sheaf φ(X) is an n-geometric

stack;(2) to show that every n-geometric stack arises in this way.

The rest of this subsection will be devoted to the proof of the first step. We will dealwith the second one in Section 3.4, after having discussed the notion of truncationat length in Section 3.3.

Corollary 3.4 shows that φ factors in fact through Sh(StnderC , τ)∧, and therefore

we only have to prove that φ(X) admits an n-atlas and that the diagonal of φ(X)is (n− 1)-representable. In order to simplify the proof, it will be convenient for therest of this section to introduce the category dAn0

C of 0-localic derived C-analyticspaces. We will endow dAn0

C with the etale topology σ and we will let Qet be thecollection of etale morphisms. The inclusion

(StnderC , τ,Pet)→ (dAn0

C, σ,Qet)

is a morphism of geometric contexts, and every object in dAn0C defines a 1-geometric

stack for the Stein context. Lemma 3.3 shows that every object in dAn0C admits an

atlas coming from StnderC , and the morphism of sites (Stnder

C , τ)→ (dAn0C, σ) is easily

seen to be cocontinuous. Therefore we can apply [20, Lemma 2.36] and temporarilywork with the geometric stacks for the second context. This has an importantadvantage that we are going to describe. The geometric context (dAn0

C, σ,Qet)is closed under σ-descent in the sense that whenever we are given a morphismF → G in Sh(dAn0

C, τ)∧ with G being representable, if there exists a τ -coveringGi → G (with Gi being representable) such that each base change Gi ×G F isrepresentable, then the same goes for F . This is easy to see: indeed, we haveGi = φ(Ui) for certain Ui ∈ dAn0

C and G = φ(Y ). Since φ is fully faithful in virtueof Proposition 3.6, the morphisms Gi → G are represented by morphisms Ui → Y

which are etale by Lemma 3.9. Let U := ∐Ui, p : U → Y the total morphism and

U• the Cech nerve of p. Since φ commutes with limits and with disjoint unions, wesee that φ(U•) is the Cech nerve of ∐Gi → G. By hypothesis, each level of thesimplicial object φ(U•)×G F is representable. Since φ is fully faithful, we can forma simplicial object V • in dAn0

C in such a way that φ(V •) ' φ(U•)×GF . Lemma 3.9shows that all the face maps in V • etale. As consequence, the geometric realizationof V • exists in dAn0

C. Let us denote by X this colimit. Corollary 3.5 shows that

φ(X) ' |φ(V •)| ' |φ(U•)×G F | ' F.

Thus the proof of the claim is completed.The importance of this fact is the following: since the geometric context

(dAn0C, σ,Qet) is closed under σ-descent, the tool of groupoid presentations becomes

available for geometric stacks on this context. In particular, the requirement on


the representability of the diagonal in the definition of geometric stack is nowsuperfluous (cf. [26, Remark 1.3.3.2]). Therefore, in proving that φ(X) is geometricwith respect to this context, we will only need to show that it admits an n-atlas.Taking into account the shift of the geometric level coming from [20, Lemma 2.36],we have now to show that whenever n ≥ −1 and X ∈ dAn≤n+1

C , then φ(X) isn-geometric.

Proof of Theorem 3.7, Step 1. Let X be an (n+1)-localic derived C-analytic space.We will prove the statement by induction on n. If n = −1, then φ(X) coincides withthe representable sheaf associated to X itself. Therefore, φ(X) is (−1)-geometricand hence the base of the induction holds.

Let now n ≥ 0 and suppose that the statement has already been proved forn-localic derived C-analytic spaces. Fix X = (X,OX) ∈ dAn≤n+1

C . Choose aneffective epimorphism ∐

Ui → 1X in such a way that ∞-topos X/Ui is 0-localic. Weclaim that each Ui is n-truncated. Assuming for the moment this fact, we see thatthe morphisms φ(Ui)→ φ(X) are etale and the total morphism ∐

φ(Ui)→ φ(X)is an effective epimorphism in Sh(Stnder

C , τ)∧. Since each Ui is n-truncated, weconclude that the morphisms φ(Ui) → φ(X) are representable by sheaves of theform φ(Y ) with Y ∈ dAn≤nC . The inductive hypothesis implies that each φ(Y ) is(n− 1)-geometric and therefore we obtain that φ(X) is n-geometric.

We are therefore left to prove the claim. To do so, we replace X with t0(X) :=(X, π0OX). We can review the latter as a G0

an-structured topos, where G0an is a

0-truncated geometric envelope for the pregeometry Tan. [12, Lemma 2.6.19] showsthat (X, π0OX) is an (n+ 1)-truncated object of Top(G0

an). Let F be the functor onStnC represented by t0(X) and let Fi the one represented by t0(Ui). It is enoughto show that the fibers of Fi(S)→ F (S) are n-truncated for every Stein space S.Since t0(X) is (n+ 1)-truncated, we see that F (S) is (n+ 1)-truncated. On theother side, Fi(S) is 0-truncated. The assertion now follows from the long exactsequence of homotopy groups.

Before discussing the essential surjectivity, we will need a digression on thetruncation functor for derived C-analytic spaces.

3.3. Truncations of derived C-analytic spaces. Let (Stn, τ0,P0) be the geo-metric context introduced in [20, §3.2]. Observe that there is a continuous morphismof geometric contexts u : (Stn, τ0,P0)→ (Stnder

C , τ,Pet). This functor is fully faith-ful in virtue of [11, Theorem 12.8]. It follows that there exists a fully faithfulfunctor

us : Sh(Stn, τqet)→ Sh(StnderC , τqet)

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which moreover preserves geometric stacks (we refer to [20, §2.4] for a discussion ofthe notation employed). Conversely, using Theorem 3.7 we have:

Corollary 3.10. Let X = (X,OX) be an n-localic 0-truncated derived C-analyticspace. Then φ(X) = (X,OX) ∈ Sh(Stnder

C , τ0) belongs to the essential image of usand is a higher analytic Deligne-Mumford stack in the sense of [20].

Proof. We already know that φ(X) is geometric. To prove that φ(X) belongs tothe essential image of us, we can proceed by induction on n. When n = 0, thestatement is clear, and the induction step follows from the construction of the atlasof φ(X) given in the proof of Theorem 3.7 and the fact that us commutes withgeometric realizations of etale groupoids (being a left adjoint).

One of the most basic and yet useful constructions in derived geometry is thetruncation of a derived object. This often allows to reduce the proofs to the classicalsetting, where they can be handled with different techniques. As we will see, thisis exactly the case for many of the main results of this article. For this reason, weintroduce now the truncation functor t0. Roughly speaking this has simply to bethe functor sending a derived C-analytic space (X,OX) into (X, τ≤0OX) (that thelatter is still a derived C-analytic space is a consequence of [12, Proposition 3.3.3]and of [11, Proposition 11.4]). However, in order to construct this as an ∞-functorwe will need to describe it in a rather different fashion.

As it is consequence of a much more general fact concerning geometries, let usswitch for a short while to this setting. Let G be a geometry (e.g. any geometricenvelope for Tan) and let X be an ∞-topos. Inside StrG(X) we can look for thefull subcategory spanned by n-truncated objects. Let us denote it by StrG(X)≤n.A very natural question is whether the latter category can be obtained as thecategory of structures for a suitable modification of G. This is indeed the case;the relevant object is referred to as the n-stub of G and its existence is guaranteedby [12, Proposition 1.5.11]. Let us denote it by G≤n. By construction, it comesequipped with a morphism of geometries G → G≤n. When n = 0 we can askwhether the relative spectrum functor associated to such a morphism coincideswith the functor t0 we are trying to define. This is probably not true withoutadditional hypotheses on G. The point is that, in general, it is not true that if Ois a G-structure on an ∞-topos X then τ≤n O is again a G-structure. There is asufficient condition for this to be true, though: it happens when the geometry iscompatible with n-truncations (see [12, Definition 3.3.2]). Under this condition weare able to prove:


Proposition 3.11. Let G be a geometry compatible with n-truncations and letG→ G≤n be an n-stub for G. Then

SpecG≤nG : Top(G)→ Top(G≤n)

coincides on objects with the assignment

(X,OX) 7→ (X, τ≤nOX)

Proof. Let (X,OX) be a G-structured topos. It follows from [12, Proposition 3.3.3]that τ≤nOX is a G-structure on X. Since τ≤nOX is n-truncated, [12, Proposition1.5.14] shows that it defines a G≤n-structure on X. The morphism OX → τ≤nOX isa local morphism because G is compatible with n-truncations. Therefore, it definesa well defined morphism

pn : (X, τ≤nOX)→ (X,OX)

in Top(G). We claim that for every (Y,OY) ∈ Top(G≤n), the canonical morphism

MapTop(G≤n)((Y,OY), (X, τ≤nOX))→ MapTop(G)((Y,OY), (X,OX))

is a homotopy equivalence. Indeed, we have a commutative diagram of fibersequences

MapStrlocG≤n

(Y)(f−1τ≤nOX,OY) MapStrlocG (Y)(f−1OX,OY)

MapTop(G≤n)((Y,OY), (X, τ≤nOX)) MapTop(G)((Y,OY), (X,OX))

MapTop(Y,X) MapTop(Y,X)id

where both the fibers are computed over the geometric morphism f−1 : X Y : f∗.Since f−1 is left exact, we see that

f−1τ≤nOX ' τ≤nf−1OX

Finally, since the functor StrlocG≤n

(Y) → StrlocG (Y) is fully faithful (in virtue of [12,

Proposition 1.5.14]), we see that the top horizontal morphism is a homotopyequivalence. The proof is therefore complete.

Definition 3.12. Let G be a geometry compatible with 0-truncations. We willrefer to the functor SpecG≤0

G as the truncation functor and we will denote it tG0 , orsimply by t0 when the geometry G is clear from the context.

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The analytic pregeometry Tan is compatible with n-truncations for every n ≥ 0in virtue of [11, Proposition 11.4]. Therefore this allows to introduce the truncationfunctor for derived C-analytic spaces. Let us denote by dAn0

C the full subcategoryof dAnC spanned by the 0-truncated derived C-analytic spaces. Similarly, letus denote by Top0(Tan) the full subcategory of Top(Tan) spanned by 0-truncatedTan-structured topoi. Then we have (cf. [26, Proposition 2.2.4.4]):

Proposition 3.13. Let i : dAn0C → dAnC be the natural inclusion functor. Then:

(1) the functor t0 : Top(Tan) → Top0(Tan) restricts to a functor t0 : dAnC →dAn0

C;(2) the functor i is left adjoint to the functor t0;(3) the functor i is fully faithful.

Proof. It follows from inspection that t0 respects the category of derived C-analyticspaces. Therefore the points (1) and (2) follow immediately. As for (3), the resultfollows from the description of the unit of this adjunction given in Proposition 3.11and the fact that the truncation functor τ≤0 of any ∞-topos is idempotent.

We close this section with a couple of important remark. Let X = (X,OX) be ann-localic 0-truncated derived C-analytic space. If Y → X is an etale morphism ofderived C-analytic spaces, we see that Y has to be 0-truncated. This observationtogether with the fully faithfulness of the functor φ proved in Proposition 3.6 yieldsthe following result (cf. [26, Proposition 2.2.4.4.(4)]):

Corollary 3.14. Let X = (X,OX) be an n-localic 0-truncated derived C-analyticspace. Then the small etale site (Stnder

C/X)et is canonically equivalent to the smalletale site (StnC/φ(X))et.

We also have the following useful equivalence, very familiar to the panorama ofderived algebraic geometry (cf. [26, Corollary 2.2.2.9]):

Proposition 3.15. Let X = (X,OX) be a derived C-analytic space and let t0(X) =(X, π0OX) be its truncation. The base change along the morphism t0(X) → X

induces an equivalence

(StnderC/X)et ' (Stnder

C/t0(X))et

Proof. The universal property of etale morphisms of Tan-structured topoi describedin [12, Example 2.3.4] shows that both sides can be identified with the full sub-category of X spanned by those objects U ∈ X such that X/U ' Sh(S) for sometopological space S which is the underlying space of a C-analytic Stein space.


3.4. Essential surjectivity. We are now ready to complete the proof of The-orem 3.7. We will switch again to the context (dAn0

C, τ,Pet). We will need todiscuss a variation of Proposition 3.15. Let X be a geometric stack for the context(dAn0

C, τ,Pet). We will denote by Xet its small etale site, that is the full subcate-gory of (dAn0

C)/X spanned by etale morphisms (we refer to [20, Remark 2.17] for adefinition). Furthermore, we have a fully faithful inclusion of sites

j : AnC → dAn0C

which is cocontinuous in virtue of the same observation used to prove Corollary 3.14.In particular, it induces a restriction functor on the level of geometric stacks. Wewill denote such functor again by t0.Proposition 3.16. Let X be a geometric stack for the context (dAn0

C, τ,Pet). Thenthe functor t0 : Xet → (t0(X))et is an equivalence of sites.Proof. We prove this by induction on the geometric level of X. If X is (−1)-representable, this follows from Proposition 3.15. Suppose now that X is n-geometric and that the statement holds true for (n− 1)-geometric stacks. Choosean etale n-groupoid presentation U• for X. Recall that this means that U• is agroupoid object in the∞-category dSt, that each Um is (n− 1)-geometric and thatthe map U0 → X is (n− 1)-etale. Since t0 commutes with products in virtue ofProposition 3.13 and takes effective epimorphisms to effective epimorphisms by [10,7.2.1.14], we see that V • := t0(U•) is a groupoid presentation for t0(X).

Now, let Y → t0(X) be an etale map. We see that Y ×t0(X) V• → V • is an etale

map (i.e. it is a map of groupoids which is etale in each degree). By the inductivehypothesis, we obtain a map of simplicial objects Z• → U•, which is such that

t0(Z•) = Y ×t0(X) V•

Since Y ×t0(X) V• was a groupoid, the same goes for Z• (here we use again the

equivalence guaranteed by the inductive hypothesis). The geometric realization ofZ• provides us with an etale map Z → X. Since t0 preserves effective epimorphisms,we conclude that t0(Z) = Y . This construction is functorial in Y , and it providesthe inverse to the functor t0.

If X is a n-geometric stack with respect to the context (dAn0C, τ,Pet), its trun-

cation is a (n + 1)-truncated, as it follows from the same argument given in [26,Lemma 2.1.1.2]. It follows that the mapping spaces in Xet are (n+ 1)-truncatedand therefore Xet itself is equivalent to an (n + 1)-category. It follows that thecategory of (non hypercomplete) ∞-sheaves X := Sh(Xet, τ) is an (n+ 1)-localictopos. Consider the composition

Tan × (Xet)op → dAn0C × (dAn0

C)op y−→ S

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where the last arrow is the functor classifying the Yoneda embedding (see [15, §5.2.1]). This induces a well defined functor

OX : Tan → PSh(Xet)

and Corollary 3.4 shows that is hypercomplete.

Lemma 3.17. Keeping the above notations, OX commutes with products, admissiblepullbacks and takes τ -covers to effective epimorphisms. In other words, OX definesa Tan-structure on X.

Proof. The functor Tan → dAn0C commutes with products and admissible pullbacks

by Proposition 1.4. Moreover, it takes τ -covers to effective epimorphisms in virtueof Corollary 3.5. At this point, the conclusion is straightforward.

If Ui → X is an etale atlas of X, each Ui defines an object Vi in X. Unravelingthe definitions, we see that the Tan-structured topos (X/Vi ,OX |Vi) is canonicallyisomorphic to Ui ∈ dAn0

C itself. Therefore X ′ := (X,OX) is a derived C-analyticspace.

We are left to prove that φ(X ′) ' X. We can proceed by induction on thegeometric level n of X. If n = −1, the statement is clear. If n ≥ 0, the first part ofthe proof of Theorem 3.7 shows that the Cech nerve of ∐Ui → X is a groupoidpresentation for φ(X ′). Since φ commutes with Cech nerves of etale maps and theirrealizations (in virtue of Corollary 3.5), we conclude that φ(X ′) is equivalent to Xitself. The proof of Theorem 3.7 is now achieved.

4. Coherent sheaves on a derived C-analytic space

In this section we introduce the notion of coherent sheaf on a derived C-analyticspace. Let us start by recalling that given an ∞-topos X it is possible to definethe category of D(Ab)-valued sheaves as in [12, §1.1]. This ∞-category inheritsa symmetric monoidal structure from the one of D(Ab). If OX is E∞-ring on X,we can therefore form the ∞-category OX-Mod of OX-modules. We refer to [14,Proposition 2.1.3] for the main properties of this ∞-category.

Definition 4.1. Let X be a derived C-analytic space. The ∞-category Coh(X) isthe full subcategory of Oalg

X -Mod spanned by those OalgX -modules F such that the

cohomology sheaves Hi(F) are locally on X coherent sheaves of π0(OalgX )-modules.

We let Coh+(X) be the full subcategory of Coh(X) spanned by those coherentO

algX -modules F such that Hi(F) = 0 for all i 0 (in cohomological notation).

We let Coh+(X) be the full subcategory of Coh(X) spanned by those coherentO

algX -modules F such that Hi(F) = 0 for all |i| 0.


Clearly, both Coh+(X) and Cohb(X) are stable subcategories of OX-Mod. More-over, the cohomology sheaves Hi allow to define a t-structure on Coh(X), and onehas

Coh♥(X) ' Coh♥(t0(X))as it follows combining Proposition 3.15 and [14, Remark 2.1.5].

Remark 4.2. It could seem that Definition 4.1 is to some extent arbitrary becauseit doesn’t take at all into account the additional analytic structure of OX. This isnot quite true, but a justification of this fact is beyond the scope of the presentarticle. We will come back to this subject in [19].

In order to freely use the results proved in [20] for underived higher analyticstacks, we will need to compare the two categories of sheaves on them. We alreadymoved a first step in this direction with Corollary 3.14. We will now complete thetask as follows. Let X = (X,OX) be a derived C-analytic space. It follows fromthe very definition of etale morphism of derived C-analytic space that there existsa fully faithful functor

(StnderC/X)et → X

defined by sending an etale map Y → X to the object U ∈ X such that Y ' X/U .Since X admits an atlas made of derived Stein spaces, we conclude that X isequivalent to the ∞-category of sheaves on ((Stnder

C/X)et, τ). Therefore we obtain:

Proposition 4.3. Let X = (X,OX) be a localic 0-truncated derived C-analyticspace. Then the ∞-category of OX-modules (resp. of coherent OX-modules) on X iscanonically equivalent to the one on φ(X) in the sense of [20, §5.1].

5. The Grauert theorem for derived stacks

In this short section we explain how the Grauert theorem proved in [20] forunderived higher Artin analytic stacks induces an analogous theorem for derivedC-analytic spaces.

Definition 5.1. Let f : X → Y be a morphism of derived C-analytic spaces andsuppose that X and Y are n-localic for some n. We will say that f is proper ift0(f) : t0(X)→ t0(Y ) is proper as morphism of higher Deligne-Mumford analyticstacks (in the sense of [20, Definition 4.8]).

Remark 5.2. As the formulation of a reasonable definition of proper map betweenhigher analytic stacks has perhaps been the newest concept introduced in [20], it isworth of recalling it. In this remark, we will limit ourselves to the C-analytic case.The definition is given by induction on the geometric level of the map f : X → Y

and it relies on the feebler notion of weakly proper map. In the C-analytic setting,

46 MAURO PORTA

one can say that a morphism of analytic stacks f : X → Y with Y representable isweakly proper if for every Stein open subset W b Y and every atlas Uii∈I of Xthere exists a finite subset I ′ ⊂ I such that W ×Y Uii∈I′ is an atlas for W ×Y X.This definition is vaguely reminiscent of the topological notion of compact space.It is not quite the definition adopted in [20], but it is equivalent: see Definition 4.5loc. cit. for the original one and Lemma 6.2 for the equivalence with the one wereported here.

Once this notion is established, we say that a morphism f : X → Y of higherC-analytic stacks is proper if it is weakly proper and separated (i.e. the diagonal,whose geometric level is strictly less than the one of f , is proper). For example,the stack BG with G a C-analytic Lie group is proper if only if the group G wascompact to begin with.

Remark 5.3. It follows from [20, Lemma 4.12] and Proposition 3.13 that propermorphisms are stable under base change. It can be further proved that they arestable under composition.

Lemma 5.4. Let X be a derived C-analytic space and let i : t0(X) → X be theinclusion of its truncation. Then Ri∗ takes Coh+(t0(X)) to Coh+(X) and it is ofcohomological dimension 0.

Proof. Since X is a derived Deligne-Mumford stack we can represent it as (X,OX),where X is the small etale topos of X. With this notation, we can identify t0(X)with (X, π0(OX)). The functor Ri∗ is therefore identified with the forgetful functoralong OX → π0(OX). It follows immediately that Ri∗ is of cohomological dimension0 (in fact, it is t-exact). If F ∈ Coh+(t0(X)), then each Hi(F) is a coherentπ0(OX)-modules. Hence, it follows from the definitions that F ∈ Coh+(X).

With these definitions it is immediate to prove the following:

Proposition 5.5. Let f : X→ Y be a proper morphism of derived Deligne-Mumfordstacks. Suppose moreover that t0(Y) is locally noetherian. Then the derived push-forward

Rf∗ : OX-Mod→ OY-Mod

takes the full subcategory Coh+(X) to Coh+(Y ).

Proof. Let C be the full subcategory of OX-Mod spanned by those OX-modules F

such that Rf∗(F) ∈ Coh+(Y ). We make the following remarks:(1) C is closed under loops and suspensions in OX-Mod: this is obvious, since

Rf∗ is an exact functor of stable ∞-categories;


(2) C is closed under extensions in OX-Mod: again, this follows from the factthat Rf∗ takes fiber sequences to fiber sequences (being an exact functor ofstable ∞-categories);

(3) C contains Coh♥(X). Indeed, we have the following commutative square:

t0(X) t0(Y)

X Y

f0

i j

f

which induces a commutative square

π0(OX)-Mod π0(OY)-Mod

OX-Mod OY-Mod

Rf0∗

Ri∗ Rj∗Rf∗

If F ∈ Coh♥(X), we can write F = Ri∗(F′) with F′ ∈ Coh♥(t0(X)), asit follows from [14, Remark 2.1.5]. Therefore [20, Theorem 5.11] showsthat Rf0∗(F′) ∈ Coh+(t0(Y)), and Lemma 5.4 shows that Rj∗(Rf0∗(F′)) ∈Coh+(Y).

To conclude the proof, we only need to observe that if τ≤nF ∈ Coh+(X) ∩ C forevery n, then F ∈ C. For a fiber sequence

τ≤nF → F → τ>nF

As Rf∗ is an exact functor of stable ∞-categories, we have a fiber sequence

Rf∗(τ≤nF)→ Rf∗F → Rf∗(τ>nF)

As Rf∗ is left t-exact, we see that Hi(Rf∗(τ>nF)) vanishes whenever i ≤ n. There-fore the long exact sequence of cohomology groups shows that

Hi(Rf∗(τ≤nF))→ Hi(Rf∗F)

is an equivalence for every i ≤ n. Since τ≤nF ∈ Coh+(F), letting n vary, we obtainthat the cohomology sheaves of Rf∗F are coherent. In other words, F ∈ C, and theproof is now complete.

6. The analytification functor

As we explained in the introduction, this section contain the most importantresult of this article, from a technical point of view. Following [11, Remark 12.26],

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we consider the morphism of pregeometries Tet → Tan induced by the classicalanalytification functor of [9, Expose XII]. This morphism induces a forgetful functor

(−)alg : Top(Tan)→ Top(Tet)

defined informally by the rule (X,OX) 7→ (X,OalgX ). It follows from the general

theory of [12, §2.1] that this functor admits a right adjoint, denoted by SpecTanTet

.Moreover, [12, Proposition 2.3.18] shows that SpecTan

Tettakes Tet-schemes locally of

finite presentation to Tan-schemes locally of finite presentation. We can thereforeinvoke [11, Corollary 12.22] to conclude that SpecTan

Tettakes derived Deligne-Mumford

stacks locally of finite presentation to derived analytic spaces. In this section, weshow that this analytification functor satisfies all the good properties one wouldexpect. In particular, we will show that if (X,OX) is a classical scheme locallyof finite type over C, then SpecTan

Tet(X,OX) can be canonically identified with the

classical analytification of (X,OX). Moreover, we will show that if (X,OX) isa derived Deligne-Mumford stack locally of finite presentation over C, then thecanonical map

SpecTanTet

(X,OX)alg → (X,OX)

is flat. We will use these results to deduce derived versions of GAGA theorems inthe next section.

6.1. Relative spectrum functor. Let us begin with a couple of general resultsconcerning the relative spectrum functor associated to a morphism of geometries.

Proposition 6.1. Let ϕ : G′ → G be a morphism of geometries and suppose thatboth G′ and G are compatible with n-truncations. Let (X,OX) ∈ Top(G′). Then thecanonical morphism

SpecGG′(X, τ≤nOX)→ SpecGG′(X,OX)

Exhibits SpecGG′(X, τ≤n) as n-truncation of SpecGG′(X,OX). In particular, it inducesan equivalence on the underlying ∞-topos.

Proof. Let G′ → G′≤n and G → G≤n be n-stubs for G′ and G respectively. Theuniversal property defining n-stubs, implies the existence of a commutative squareof morphism of geometries

G′ G

G′≤n G≤n

ϕ

ϕn


Therefore we have

SpecG≤nG SpecGG′ ' SpecG≤nG′≤n SpecG

′≤n

G′

Combining this with Proposition 3.11, we obtain the desired result.

Furthermore we can prove the following result:

Proposition 6.2. Let G,G′ be geometries compatible with n-truncations. Letϕ : G′ → G be a morphism of geometries and let G′ → G′≤n, G→ G≤n be n-stubs forG′ and G, respectively. The diagram

Sch(G′≤n) Sch(G≤n)

Sch(G′) Sch(G)

SpecG≤n

G′≤n

SpecGG′

commutes.

Proof. Since ϕ commutes with finite limits, the induced morphism

ϕ : Ind((G′)op)→ Ind(Gop)

commutes with finite limits as well. In particular, we see that it takes k-truncatedobjects to k-truncated objects. The statement now follows from the following pairof observations:

(1) if A ∈ Ind(Gop) is k-truncated, then, writing SpecG(A) = (XA,OA), OA isk-truncated;

(2) if A ∈ Ind((G′)op), then SpecGG′(SpecG(A)) ' SpecG′(ϕ(A)).As these statements follow directly from the definitions, the proof is complete.

Remark 6.3. We don’t know whether it is possible to extend the above result tothe category of G′-structured topoi. In what follows, we won’t need but the resultwe proved.

6.2. Comparison with the classical analytification. Consider the morphismof pregeometries Tet → Tan. The associated relative spectrum functor is by definitionthe analytification functor. Since it preserves the class of schemes locally of finitepresentations, it defines a functor

Schf.p.(Tet)→ Schf.p.(Tan) ⊂ dAnC

The goal of this section is to show that it coincides with the classical analytificationof Grothendieck.

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Lemma 6.4. Let X = (X,OX) be a derived Deligne-Mumford stack and let Xan =(Xan,OXan) be its analytification. Let Y = (Y,OY) → X be an etale morphism ofDeligne-Mumford stacks. Then the analytification Y an can be explicitly describedas the pair (Z,OZ), where the ∞-topos Z is defined to be the pullback of

Z Xan

Y X

j∗

p∗

i∗

while OZ is defined to be j−1OXan. Suppose furthermore that Y ' X/U and thatU → 1X is an effective epimorphism. Then Z ' Xan

/p−1U and p−1U → 1Z is aneffective epimorphism.

Proof. The first part is just a reformulation of [12, Lemma 2.1.3]. The secondpart follows from the universal property of etale subtopoi (cf. [10, 6.3.5.8]) andfrom the fact that p−1 commutes with truncations and (therefore) with effectiveepimorphisms (see [10, 5.5.6.28]).

Proposition 6.5. Let X ∈ Tet be a smooth (derived) scheme. Then the ana-lytification of X is 0-localic and 0-truncated, and it coincides with the classicalanalytification defined in [9].

Proof. In virtue of Lemma 6.4, we only need to show that this results holds true forX = An

C, the algebraic n-dimensional affine space. If (Y,OY) is any Tet-structuredtopos, we have

HomLTop(Tet)op((Y,OY), SpecTet(AnC)) ' OY(A1)n

Suppose now that (Z,OZ) is a derived C-analytic space. Then, if we denote by EnCthe analytic n-dimensional affine space, we have

HomLTop(Tan)op((Z,OZ), SpecTan(EnC)) ' OZ(E1C)n ' O

algZ (A1

C)n

We conclude that SpecTan(EnC) is the analytification of AnC.

Proposition 6.6. Let X = (X,OX) be a scheme locally of finite type over C, seenas a derived scheme. Let (Xan,OXan) be the analytification in the sense of [11].Then OXan is 0-truncated and moreover the canonical morphism

(Xan,OalgXan)→ (X,OX)

exhibits (Xan,OalgXan) as analytification of X in the sense of [9].


Proof. Choose a geometric envelope Tan → Gan. The universal property defining Ganshows that the morphism Tet → Tan induces a (essentially) unique limit-preservingfunctor Get → Gan. Since the analytification functor of [11] is defined to be therelative spectrum SpecGan

Get, the first statement follows directly from Proposition 6.2.

As for the second statement, we can assume X to be affine and therefore choose aclosed immersion X → An

C. Let J ⊂ C[X1, . . . , Xn] be the ideal defining X. Choosegenerators f1, . . . , fm ∈ J and consider the morphism f : An → Am classified bythese elements. We can therefore describe X as the truncation of the pullback

X AnC

Spec(C) AmC

f

0

computed in dSchC. The analytification functor of [11] is a right adjoint. Inparticular, it commutes with limits. We can therefore identify (Xan,OXan) with thepullback in Top(Tan)

(Xan,OXan) SpecTan(EnC)

SpecTan(∗) SpecTan(EmC )

Observe that the bottom horizontal morphism is a closed immersion. Therefore,this is a pullback in dAnC. In particular, we see that Xan is a closed subtopos of XEnC

,and it is therefore 0-localic, and inspection shows that it is the topos associated tothe topological space Xan, the analytification of [9]. The result now follows from[11, Lemma 12.19].

Corollary 6.7. Let f : X → Y be a closed immersion of derived schemes locally offinite presentation over C. Then f an : Xan → Y an is a closed immersion of derivedC-analytic spaces.

Proof. In virtue of Proposition 6.1, it is enough to prove the statement when bothX and Y are 0-truncated. In this case, the result is an immediate consequence ofProposition 6.6 and of [9, Expose XII, Proposition 3.2].

Let now X = (X,OX) be a 0-truncated higher Deligne-Mumford stack, locally offinite presentation over Spec(C). The functor ψ : Sch(Gder

et (k))→ Sh(dAff, τqet) of[12, Theorem 2.4.1] is fully faithful. It can be shown that if X is n-localic, thenψ(X) is a geometric Deligne-Mumford stack in the sense of [26] (see [18, Theorem1.7] for a proof and a more precise comparison statement). On the other side,

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Xan = (Xan,OXan) defines via the functor φ : dAnC → Sh(StnderC , τ) of Section 3.1

an analytic geometric stack. Since X was 0-truncated, Proposition 6.2 shows thatXan is 0-truncated and therefore Corollary 3.10 shows that we can identify φ(Xan)with a higher Deligne-Mumford analytic stack in the sense of [20]. We can thereforecompare φ(Xan) with ψ(X)an, where the latter is the analytification of ψ(X) in thesense of [20, §6.1]. It is a rather easy task to show that the two notions coincide:

Proposition 6.8. Let (X,OX) be an n-localic 0-truncated higher Deligne-Mumfordstack locally of finite presentation over Spec(C). Keeping the above notations, thereexists a natural isomorphism φ(Xan)→ ψ(X)an.

Proof. To be clearer in this proof, let us explicitly write SpecTanTet

(X) instead ofXan, and let us reserve the notation (−)an for the analytification functor of [20].It follows from Lemma 6.4 that SpecTan

Tet(X) commutes with geometric realization

of etale groupoids. The same is true for φ (see Corollary 3.5) and for ψ (see [12,2.4.13]). Finally, (−)an is defined to be a left Kan extension, and therefore it hasthis property by construction. Therefore, it is sufficient to prove the statementwhen X is an affine scheme of finite presentation over Spec(C), and in this casethe statement follows directly from Proposition 6.6.

6.3. Flatness I. Let us begin with the following definition.

Definition 6.9. Let (X,OX) and (Y,OY) be (connective) E∞-structured topoi (i.e.elements of Top(Tdisc)). We will say that a morphism (f, ϕ) : (X,OX)→ (Y,OY) isflat if the ϕ : f−1OY → OX is a flat morphism of sheaves of E∞-rings.

Remark 6.10. Recall that a morphism ϕ : A→ B of (connective) E∞-sheaves onan ∞-topos X is said to be flat if the induced base change functor

−⊗A B : A-Mod→ B-Mod

is t-exact (with respect to the canonical t-structures of A-Mod and B-Mod). IfX has enough points, this is equivalent to ask that for every geometric pointη−1 : X S : η∗, the induced morphism η−1(ϕ) : η−1A→ η−1B is a flat morphismof (connective) E∞-rings. In other words, the π0(η−1(ϕ)) is flat and one hasequivalences

πi(η−1A)⊗π0(η−1A) π0(η−1B)→ πi(η−1B)(that is, the morphism η−1(f) is strong).

Let (X,OX) be a derived Deligne-Mumford stack locally of finite presentationover C and let us write

SpecTanTet

(X,OX) = (Xan,OXan)


Our goal is to show that the canonical morphism

(Xan,OalgXan)→ (X,OX)

is flat in the sense of Definition 6.9. This statement is local on the Deligne-Mumfordstack (X,OX), and therefore it will be sufficient to prove it for SpecTet(A), whereA is a connective E∞-ring of finite presentation over C. The task will be greatlysimplified if we could replace the Tet-scheme SpecTet(A) (whose underlying∞-toposis 1-localic) with the TZar-scheme SpecTZar(A) (whose underlying ∞-topos is 0-localic). For this, we will need a digression on the relative spectrum associated tothe morphism of pregeometries TZar → Tet.

Let X be an ∞-topos. Recall that the forgetful functor

StrlocTet

(X)→ StrlocTZar

(X)

is fully faithful. If moreover we suppose that the hypercompletion X∧ has enoughpoints, then an hypercomplete object O ∈ Strloc

TZar(X∧) belongs to the essential image

of this functor if and only if all the stalks η−1O are strictly henselian E∞-rings.Therefore, if (X,OX) is a Tet-structure topos, we will denote by again by (X,OX)the associated TZar-structured topos.

Proposition 6.11. Let A ∈ CAlgC be a connective C-algebra. Then:(1) SpecTan

Tet(SpecTZar(A)) ' SpecTet(A).

(2) SpecTanTZar

(SpecTZar(A)) ' SpecTanTet

(SpecTet(A)).(3) the canonical map q : SpecTet(A) → SpecTZar(A) is flat in the sense of

Definition 6.9.

Proof. Statements (1) and (2) follow directly from the universal properties of therelative and absolute spectrum functors. We will prove statement (3). Let usdenote by Oet

A the Tet-structure sheaf of SpecTet(A) and by OA the TZar-structuresheaf of SpecTZar(A). We know that Oet

A is an hypercomplete object of Sh(Aet, τqet)(see [13, Theorem 8.4.2.(3)]). Let A → B be an etale morphism. We can thenfactor it as A→ A′ → B, where A→ A′ is a Zariski open immersion, and one hasq−1(OA)(B) = OA(A′) = A′. From this, we conclude that q−1OA is hypercompleteas well.

It follows from J. Lurie’s version of Deligne’s completeness theorem [13, Theorem4.1] that the hypercompletion Sh(Aet, τqet)∧ has enough points, we can check theflatness of Oet

A → q−1OA on stalks. If η−1 : Sh(Aet, τqet) S : η∗ is a geometricpoint, [14, Proposition 1.1.6] shows that we can find a filtered diagram Aα ofetale A-algebras such that for every F ∈ Sh(Aacuteet, τqet) one has

η−1(F ) = limF (Aα)

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Moreover, η−1 is canonically determined by a morphism x : Spec(k)→ Spec(A) forsome algebraically closed field k. It follows that

η−1q−1OA = Ax

where Ax denotes the Zariski stalk at the point x. On the other side, As.h.x = η−1Oet

A

is a strict henselianization of Ax. It follows that the canonical morphism Ax → As.h.x

is flat. This completes the proof.

Corollary 6.12. Let A be a connective E∞-algebra of finite presentation over C.Let us write SpecTan

TZar(SpecTZar(A)) = (X,OX). Then the following are equivalent:

(1) the morphism (X,OalgX )→ SpecTZar(A) is flat;

(2) the morphism (X,OalgX )→ SpecTet(A) is flat.

6.4. Analytification of germs. From this point on, we will focus on the Zariskianalytification functor SpecTan

TZar.

Let us recall that the analytification functor of [9, Expose XII] is defined onlyfor schemes locally of finite presentation over C. One of the advantages of therelative spectrum functor introduced in [12] is that it can be applied also to germsof schemes, that is to TZar-topoi of the form (S, A), where A is a local E∞-algebra.We will turn our attention to the study of the Tan-structured topos SpecTan

TZar(S, A)

in the special case where A arises as the ring of Zariski germs at one point of aderived (Zariski) scheme locally of finite presentation over C.

Lemma 6.13. Let f : X → Y be a morphism of topological spaces. Let y ∈ Y be aclosed point and suppose that f−1(y) = x. Then the diagram

S S

Sh(X) Sh(Y )

x∗

id

y∗

f∗

is a pullback square in Top.

Proof. The morphism y∗ is a closed immersion of ∞-topoi. Set V := Y \ y.Seeing V as a (−1)-truncated object in Sh(Y ), we can identify y∗ : S → Sh(Y )with the inclusion Sh(Y )/V → Sh(Y ). The pullback along f∗ can therefore beidentified with Sh(X)/f−1(V ). Observe that f−1(V ) can be identified with the(−1)-truncated object in Sh(Y ) represented by the inverse image U of V alongf . By hypothesis X \ U = x. It follows that Sh(X)/f−1(V ) → Sh(X) can beidentified with x∗ : S→ Sh(X). The proof is now complete.


Let (X,OX) be a (Zariski) derived scheme locally of finite presentation overC and let (Xan,OXan) be its analytification in the sense of [11]. It follows fromProposition 6.6 that Xan = Sh(Xan), where Xan is the underlying topological spaceof the analytification of (X,OX) in the sense of [9]. Let x∗ : S → Sh(Xan) be ageometric point. The induced map

x−1p−1OX → x−1OalgXan

can be seen as a morphism

(S, y−1OalgXan)→ (S, x−1OX)

It follows from Lemma 6.13 and [12, Lemma 2.1.3] that we can identify, via theabove morphism, (S, y−1OXan) with the analytification of (S, x−1OX).

We will now give a more explicit characterization of the Tan-structure OXan,y :=y−1OXan in term of OX,x := x−1OX . Introduce the functor

Ψ: StrlocTZar

(S)/C → StrlocTan(S)/H0

which is by definition the left adjoint to

Φ: StrlocTan(S)/H0 → Strloc

TZar(S)/C

We have:

Lemma 6.14. Keeping the above notations, OXan,y = Ψ(OX,x).

Proof. We already argued that (S,OXan,y) is the analytification in the sense of [11]of (S,OX,x). The universal property of the analytification shows therefore that forevery O ∈ Strloc

Tan(S)/H0 we have

MapStrlocTan (S)/H0

(OXan,y,O) ' MapTop(Tan)((S,O), (S,OXan,y))

' MapTop(TZar)((S,Oalg), (S,OX,x))

' MapStrlocTZar

(S)/C(OX,x,Oalg)

Therefore, we conclude that O′ = Ψ(O).

We formulate the following conjecture generalizing Lemma 6.14:

Conjecture 6.15. Let O ∈ StrlocTZar

(S)/C. Introduce the functor

Ψ: StrlocTZar

(S)/C → StrlocTan(S)/H0

left adjoint to the underlying algebra functor. Then the unit O→ Ψ(O)alg exhibits(S,Ψ(O)) as analytification of (S,O).

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6.5. Flatness II. Let us denote by CAlglocC the ∞-category of local C-algebras

(with residue field C). We have a canonical identification CAlglocC ' Strloc

TZar(S)/C.

Recall also from Definition 2.5 that we denote by AnRinglocC the ∞-category

StrlocTan(S)/H0 . The adjunction considered at the end of the previous section can be

rewritten asΨ: CAlgloc

C AnRinglocC : Φ

Accordingly to the notations of [11], we will write (−)alg to denote the functor Φ.There is a forgetful functor

CAlgloc → S

which admits a left adjoint. We will denote it by C[−]. Observe that π0C[∗] canbe identified with the ring of germs C[T ](0).

The universal properties involved show that there is a natural equivalenceH− ' Ψ C[−], where H− is the ∞-functor associated to the functor intro-duced in Section 2.5. To further simplify notations, we will write Aan instead ofΨ(A)alg. If K ∈ S is a space, we will further write CK instead of C[K]an.

Lemma 6.16. Suppose that f : A→ B is a morphism in the∞-category AnRinglocC

which is surjective on π0. Then for every other morphism g : A→ C, the forget-ful functor (−)alg : AnRingloc

C → CAlglocC preserves the homotopy pushout of this

diagram.

Proof. We can see A, B and C as Tan-structures on S. The maps A → B andA → C define morphisms (S, B) → (S, A) and (S, C) → (S, A) in Top(Tan). Theresult follows now from combining [11, Proposition 10.3, Lemma 11.10].

Lemma 6.17. For every n ∈ N, the maps C[∆n] → C∆n and C[∂∆n] →C∂∆n are flat.

Proof. The morphism C[∆0]→ C[∆n] is an acyclic cofibration, and the same goesfor Ψ(C[∆0])→ Ψ(C[∆n]). In particular, C∆0 → C∆n is a weak equivalence.Now we have π0(C[∆0]) = (C[∆0])0 = C[T ], while π0(C∆0) = (C∆0)0 = CT.Since C[T ]→ CT is flat, the first statement follows at once.

Let us turn to the case of ∂∆n. When n = 0, 1, the result is trivial. If n ≥ 2, wecan present ∂∆n as the (homotopy) pushout of a diagram of the form

∐∆n−2 t∆n−2 ∐∆n−1

∐∆n−2 ∂∆n

f


Both the functors C[−] and Ψ(C[−]) preserve this pushout. Moreover, the morphism∐∆n−2 t∆n−2 →

∐∆n−2

is a (degreewise) epimorphism, and the functors C[−] and Ψ commute with degree-wise epimorphisms (being left adjoints). It follows that the induced morphism⊗

Ψ(C[∆n−2])⊗Ψ(C[∆n−2])→⊗

Ψ(C[∆n−2])

is a degreewise epimorphism. It is moreover an epimorphism on π0, and thereforethe image via C− of the above homotopy pushout is still a homotopy pushout.In other words, the diagram

C ∐∆n−2 t∆n−2 C ∐∆n−1

C ∐∆n−2 C∂∆n

is both a homotopy pushout and a strict pushout. Observe moreover that

C [∐∆n−2 t∆n−2] C [∐∆n−2]

C ∐∆n−2 t∆n−2 C ∐∆n−2

is a strict pushout, that this diagram is homotopy equivalent to a (strict) pushoutdiagram of discrete objects and that the left vertical map is flat. Therefore it ishomotopy equivalent to a homotopy pushout and therefore it is a homotopy pushoutitself. As consequence, we see that the map C[∂∆n]→ C∂∆n can be identifiedwith the homotopy pushout of the map C[∐∆n−1]→ C∐∆n−1, which is flat. Itfollows that C[∂∆n]→ C∂∆n is flat as well, thus completing the proof.

Lemma 6.18. The diagram

C[∂∆n] C[t]

C∂∆n Cz

is a pushout in the category of connective E∞-algebras over C.

Proof. Let R = C∂∆n ⊗C[∆n] C[t]. We have a canonical map g : R → Cz.Lemma 6.17 shows that both C[t]→ R and Cz are flat. Therefore R is discrete,

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and it will be sufficient to show that g induces an isomorphism on π0. Thecomputations of Proposition 2.37 and Proposition 2.38 show that

π0(R) = Torπ0(C[∂∆n])0 (π0(C∂∆n, π0(C[t])) ' Cz.

The proof is thus completed.

Theorem 6.19. Let A be a local (connective) E∞-algebra over C. Then themorphism A→ Aan is flat.

Proof. We can obtain A as colimit of a (possibly transfinite) diagram Aαα<λ,depicted as

A0 → A1 → · · · → An → · · ·where A0 = C each map Aα → Aα+1 is the pushout of

C[∂∆m] C[∆0]

Aα Aα+1

We will prove by transfinite induction that Aα → Aanα is flat. When α = 0 there is

nothing to prove. Since the functor (−)alg commutes with sifted colimits (thus inparticular with sequential ones) and since flat morphisms are stable under filteredcolimits, we only need to prove that Aα+1 → Aan

α+1 is flat given that Aα → Aanα is.

Applying Ψ to the above diagram, we obtain a pushout

H∂∆m H∆0

Ψ(An) Ψ(An+1)

Proposition 2.38 shows that H∂∆m → H∆0 is a surjection on π0 and thereforeLemma 6.16 guarantees that (−)alg preserves this pushout. In particular, we obtaina commutative diagram

C[∂∆m] C[∆0]

C∂∆m C∆0

An An+1

Aann Aan

n+1


Lemma 6.18 shows that the top square is a pushout. Therefore, in the rectangle

C[∂∆m] An Aann

C[∆0] An+1 Aann+1

both the left square and the outer one are pushouts. Therefore the same goes forthe one on the right. Since An → Aan

n was flat by hypothesis, we conclude that thesame goes for An+1 → Aan

n+1.

Corollary 6.20. Let (X,OX) be a derived Deligne-Mumford stack locally of finitepresentation over C and write SpecTan

Tet(X,OX) = (Xan,OXan). Then the canonical

map p : (Xan,OalgXan)→ (X,OX) is flat (in the sense of Definition 6.9).

Proof. The question being local on (X,OX), we can assume (X,OX) = SpecTet(A)for a connective E∞-algebra A of finite presentation over C. It follows fromCorollary 6.12 that it is enough to prove that the canonical map

(Xan,OalgXan)→ SpecTZar(A)

is flat. In this case, Proposition 6.2 and Proposition 6.6 show that Xan is a 0-localic∞-topos and Lemma 3.2 shows that Xan is hypercomplete and has enough points.We are therefore reduced to show that for every geometric point x∗ : S→ Xan, theinduced map

x−1p−1OX → x−1OalgXan

is flat. Therefore, we can invoke Lemma 6.14 to reduce to the case of Theorem 6.19.This completes the proof.

6.6. Computing the analytification. A first important application of this flat-ness result is that it allows to give a more concrete description of the analytificationof a derived Deligne-Mumford stack in terms of the analytification of its truncation.

Before giving the details of this, though, we will need to recall some terminology.If A is a simplicial ring and M is a connective A-module one can form the splitsquare-zero extension A⊕M . This is rather an easy task if we are working in themodel category sCRing. However, if we need to generalize it to a less elementary∞-category, things become substantially more complicated. We refer to [15, §7.3.4]for a detailed account of this construction. To stress the non-triviality of thisconstruction, we prefer to suppress the notation A⊕M in virtue of Ω∞A (M), whichis reminiscent of the way the construction goes. The framework developed in loc.cit. applies as well to sheaves of connective E∞-rings on any ∞-topos and to theircategory of modules, and we will be using it precisely in this setting.

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Let (X,OX) be a Deligne-Mumford stack locally of finite presentation over C.Proposition 6.1 shows that for every n ≥ 0 the natural morphism

SpecTanTet

(X, τ≤nOX)→ SpecTanTet

(X,OX)

exhibits SpecTanTet

(X, τ≤nOX) as the n-truncation of SpecTanTet

(X,OX). We can thereforewrite

SpecTanTet

(X, τ≤nOX) = (Xan, τ≤nOXan)Let us further denote by p∗ : Xan X : p−1 the induced geometric morphism.

We know that τ≤nOX → τ≤n−1OX is a square-zero extension. There existstherefore a pullback diagram

τ≤nOX τ≤n−1OX

τ≤n−1OX Ω∞(πn(OX)[n+ 1])

d

Applying the functor p−1 we obtain the pullback diagram

τ≤np−1OX τ≤n−1p

−1OX

τ≤n−1p−1OX Ω∞(πn(p−1OX)[n+ 1])

p−1(d)

Since the morphism p−1OX → OalgXan is flat, we can invoke Lemma A.3 to conclude

thatτ≤nO

algXan τ≤n−1O

algXan

τ≤n−1OalgXan Ω∞(πn(Oalg

Xan)[n+ 1])

p−1(d) ⊗p−1OXO

algXan

is a pullback square. In other words, the analytification of the n-th Postnikovinvariant of OX is the n-th Postnikov invariant of Oalg

Xan .

Corollary 6.21. Let (X,OX) be a Deligne-Mumford stack locally of finite presen-tation over C. Let (Xan,O) be the analytification of (X,Ω∞τ≤n−1OX

(πn(OX[n+ 1]))).Then Oalg = Ω∞

τ≤n−1OalgXan

(πn(OalgXan [n+ 1])).

Proof. Indeed, we know that Oalg is n-truncated and

τ≤n−1O ' τ≤n−1OalgXan ' τ≤n−1Ω∞τ≤n−1OX

(πn(OX)[n+ 1])

Therefore Oalg is determined by its n-th Postnikov invariant. The above discussionallows to identify it with the analytification of the n-th Postnikov invariant of


Ω∞τ≤n−1OX(πn(OX)[n+ 1]), which is the null derivation. Therefore the n-th Postnikov

invariant of Oalg is the null derivation as well, and therefore we see that Oalg can beidentified with the split square-zero extension of τ≤n−1O by πn(O)[n+1] ' πn(Oalg

Xan).The conclusion follows.

Corollary 6.22. Let (X,OX) be a Deligne-Mumford stack locally of finite presen-tation over C. Then the analytification functor SpecTan

Tetpreserves the pushout:

(X,Ω∞(πn(OX)[n+ 1])) (X, τ≤n−1OX)

(X, τ≤n−1OX) (X, τ≤nOX)

Proof. Let us write (Xan,O) = SpecTanTet

(X,Ω∞(πn(OX)[n + 1]). We only need tocheck that the commutative diagram

τ≤nOXan τ≤n−1OXan

τ≤n−1OXan O

is a pullback. Since (−)alg preserves pullbacks and it is conservative, it is enoughto check that

τ≤nOalgXan τ≤n−1O

algXan

τ≤n−1OalgXan Oalg

is a pullback. This follows from the previous corollary and the discussion beforeit.

Remark 6.23. One could ask whether a similar description holds before passingto the underlying Tet-structured topos. The answer is affirmative, but the proofrequires quite a bit of machinery that is beyond the scope of the present article.We will return on this point in [19].

7. GAGA for derived Deligne-Mumford stacks

This section is devoted to the two GAGA theorems. Let us first introducethe notion of proper morphism of derived Deligne-Mumford stacks. As for theanalogous notion for derived C-analytic spaces, we will reduce via the truncationto the definition given in [20]. Let us briefly recall the definitions given there:

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Definition 7.1 ([20, Definition 4.7]). A morphism f : X → Y of algebraic Deligne-Mumford stacks is said to be weakly proper if there exists an atlas Yii∈I of Ysuch that for every i ∈ I, there exists a scheme Pi proper over Yi and a propersurjective Yi-morphism from Pi to X ×Y Yi.

Definition 7.2 ([20, Definition 4.8]). We define by induction on n ≥ 0.(i) An n-representable morphism of algebraic Deligne-Mumford stacks is said to

be separated if its diagonal being an (n− 1)-representable morphism is proper.(ii) An n-representable morphism of algebraic stacks is said to be proper if it is

separated and weakly proper.

We now introduce the following definition:

Definition 7.3. Let f : X → Y be a morphism of derived Deligne-Mumford stacks.We will say that f is separated (resp. proper) if t0(f) is separated (resp. proper) inthe sense introduced above.

Remark 7.4 (Algebraic proper direct image theorem). Using [20, Theorem 5.11]as basis of the induction, the same proof given in Proposition 5.5 yields a versionof the proper direct image theorem statement for proper morphisms of higheralgebraic Deligne-Mumford stacks whose truncation is locally noetherian. This isprobably a folklore result (as it was the aforementioned theorem), but we couldn’tlocate it in the literature.

7.1. GAGA 1. Let f : (X,OX) → (Y,OY ) be a morphism of derived Deligne-Mumford stacks locally of finite presentations over C. We have the followingcommutative diagram in Top(Tet):

(Xan,OalgXan) (X,OX)

(Y an,OalgY an) (Y,OY )

hX

fan f

hY

which in turn induces a commutative diagram of stable ∞-categories

OX-Mod OalgXan-Mod

OY -Mod OalgY an-Mod

Rf∗ Rfan∗

RhX∗

RhY ∗

Given F ∈ OX-Mod, adjoint nonsense produces a canonical map

ϕF : (Rf∗F)an → Rf an∗ (Fan)


where Fan := h∗X(F) (and we don’t write Lh∗X(F) thanks to the flatness resultCorollary 6.20). Similarly, we wrote (Rf∗F)an to denote h∗Y (Rf∗F).

Theorem 7.5. Let f : X → Y be a proper morphism of derived Deligne-Mumfordstacks locally of finite presentation over C. For every F ∈ Coh+(X), the canonicalmap

ϕF : (Rf∗F)an → Rf an∗ (Fan)

is an equivalence.

Proof. Let C be the full subcategory of OX-Mod spanned by those F for which ϕF

is an equivalence. We observe that:

(1) C is stable under loops, suspensions and extensions: indeed, this followsimmediately from the fact that all the functors Rf∗, Rf an

∗ , h∗X and h∗Y areexact functors between stable ∞-categories;

(2) C contains Coh♥(X) ' Coh♥(t0(X)). Indeed, we have the commutativediagram

t0(X) t0(Y )

X Y

f0

i j

f

If F ∈ Coh♥(X), we can write F ' Ri∗(F′) with F′ ∈ Coh♥(t0(X)) andtherefore Rf∗(F) ' Rj∗(Rf0∗F). The GAGA theorem of [20, Theorem 7.3]combined with Proposition 6.8 and Proposition 4.3 shows that the canonicalmap

(Rf0∗(F′))an → Rf an0∗ ((F′)an)

is an equivalence. We are therefore left to prove that the theorem holdsfor the inclusion j : t0(Y )→ Y . This follows combining Corollary 6.20 andProposition 6.2.

At this point, the devissage lemma [20, Lemma 5.10] shows that Cohb(X) ⊂ C. Tocomplete the proof, we only need to show that whenever τ≤nF ∈ Coh+(X) ∩ C forevery n, then F ∈ C.

Form a fiber sequence

τ≤nF → F → τ>nF

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Since Rf∗, Rf an∗ and (−)an are exact functors between stable ∞-categories, we

obtain a morphism of fiber sequences

(Rf∗(τ≤nF))an (Rf∗F)an (Rf∗(τ>nF))an

Rf an∗ (τ≤nF)an Rf an

∗ Fan Rf an∗ (τ>nF)an

Corollary 6.20 shows that (τ>nF)an ' τ>n(Fan) and therefore both Hi(Rf∗(τ>nF)an)and Hi(Rf an

∗ (τ>nF)an) vanish for i ≤ n. Since τ≤nF ∈ Cohb(X) by hypothesis, wesee the morphism (Rf∗F)an → Rf an

∗ Fan induces an isomorphism on the cohomologygroups Hi for every i ≤ n. Letting n vary, we conclude that ϕF is an equivalence,thus completing the proof.

7.2. GAGA 2.

Lemma 7.6. Let X be a derived Deligne-Mumford stack locally of finite presenta-tion over C. Let F ∈ OX-Mod and suppose it can be written as

F ' lim τ≥−nF

Then the analytification (−)an commutes with this limit.

Proof. It follows from Corollary 6.20 that (τ≥nF)an ' τ≥nFan. Therefore we have

Fan ' lim τ≥−nFan ' lim(τ≥−nF)an

completing the proof.

Theorem 7.7. Let X = (X,OX) be a derived Deligne-Mumford stack proper overC. The analytification functor

(−)an : OX-Mod −→ OalgXan-Mod

restricts to an equivalence(−)an : Coh(X) −→ Coh(Xan)

Proof. Let us start by proving fully faithfulness. Since OX-Mod and OalgXan-Mod are

stable and C-linear, they are canonically enriched over D(Ab), see [8, Examples7.4.14, 7.4.15]. We will denote by MapD(Ab)

OXand MapD(Ab)

OalgXan

the enrichedmapping spaces of these two categories, respectively. For every F,G ∈ OX-Modthere is a natural map

ψF,G : MapD(Ab)OX

(F,G)→ MapD(Ab)O

algXan

(Fan,Gan)

and we want to prove that ψF,G is an equivalence. Observe that when F,G ∈Coh♥(X) ' Coh♥(t0(X)) the statement follows from the analogous result for


higher Deligne-Mumford stacks proved in [20, Proposition 7.2]. The same extensionargument given in loc. cit. shows that ψF,G is an equivalence whenever F,G ∈Cohb(X).

Let us now turn to the general case. Since D(Ab) is both left and right t-complete,it is enough to prove that for every integer n ∈ Z, π−nψF,G is an isomorphism ofabelian groups. Recalling that

π−n MapD(Ab)OX

(F,G) = π0 MapD(Ab)OX

(F,G[n])

π−n MapD(Ab)O

algXan

(Fan,Gan) = π0 MapD(Ab)O

algXan

(Fan,Gan[n])

we see that it is enough to treat the case n = 0. Observe that π0 MapD(Ab)OX

(F,G) canbe identified with the global sections of the cohomology sheaf H0(RHomOX

(F,G)).Since F and G are hypercomplete objects (see [14, Proposition 2.3.21]), we canwrite

F ' colimn

τ≤nF, G ' limmτ≥mG

where the limits and colimits are computed in OX-Mod. Using the fact that (−)an

commutes with all colimits and invoking Lemma 7.6, we conclude that

Fan ' colimn

(τ≤nF)an, Gan ' limm

(τ≥mG)an

Moreover, Corollary 6.20 shows that (τ≤nF)an ' τ≤nFan and (τ≥mG)an ' τ≥mG

an.We are therefore reduced to the case where F ∈ Coh−(X) and G ∈ Coh+(X).Suppose more precisely that Hi(F) = 0 for i ≥ n0 and cHj(G) = 0 for j ≤ m0.Then the same bounds hold for Fan and Gan, so that we obtain:

π0 MapD(Ab)OX

(F,G) ' π0 MapD(Ab)OX

(τ≥m0−1F, τ≤n0+1G)

π0 MapD(Ab)O

algXan

(Fan,Gan) ' π0 MapD(Ab)OX

(τ≥m0−1Fan, τ≤n0+1G

an)

Since both τ≥m0−1F and τ≤n0+1G belong to Cohb(X), we already know that thecanonical map

MapD(Ab)OX

(τ≥m0−1F, τ≤n0+1G)→ MapD(Ab)OX

(τ≥m0−1Fan, τ≤n0+1G

an)

is an equivalence. In conclusion, ψF,G is an equivalence for every F,G ∈ Coh(X),completing the first part of the proof.

Let us now turn to the essential surjectivity part. Let C be the full subcategoryof Cohb(Xan) spanned by the essential image of the analytification functor (−)an.Since Coh♥(Xan) ' Coh♥(t0(Xan)), Proposition 6.8 and Proposition 4.3 can becombined together with the GAGA theorem of [20, Theorem 7.3] to conclude thatC contains Coh♥(Xan). Observe that C is clearly stable under loop and suspensions

66 MAURO PORTA

in OX-Mod. We claim that C is a thick subcategory of OX-Mod. Indeed, if we aregiven a fiber sequence

F′ → F → F′′

with F′,F′′ ∈ C, we can rotate it and express F as the fiber of F′′ → F′[1].Let G′,G′′ ∈ Cohb(X) be such that (G′)an ' F′ and (G′′)an ' F′′. By the fullyfaithfulness we already proved, the map F′′ → F′[1] is the analytification of amap G′′ → G′[1]. Let G be the fiber of this map. Since (−)an is an exact functorbetween stable ∞-categories, we see that Gan ' F, thus completing the proof ofthe claim. Therefore, the hypotheses of the devissage lemma [20, Lemma 5.10] aresatisfied and therefore we conclude that C contains the whole Cohb(Xan). If nowF ∈ Coh−(Xan), we can write

F ' lim τ≥nF

Since the analytification functor (−)an is fully faithful and essentially surjectiveon Cohb(Xan), we see that the diagram τ≥nF is the analytification of a towerGn in Cohb(X). Consider the morphism Gn → τ≥nGn. Corollary 6.20 showsthat it becomes an equivalence after passing to the analytification. Since (−)an isconservative, we conclude that Gn ∈ Coh≥n(X) ∩ Cohb(X). Therefore there arecanonical maps τ≥nGn−1 → Gn, which become equivalences after applying (−)an.These remarks show that G := limGn ∈ Coh−(X). At this point, we can applyLemma 7.6 to get

Gan ' limGann ' lim τ≥nF ' F.

At last, this proves that (−)an is essentially surjective also on Coh−(Xan), completingthe proof. Since we already know that (−)an : Coh(X)→ Coh(Xan) is fully faithful,we conclude now that (−)an : Coh−(X)→ Coh−(Xan) is an equivalence.

Finally, let F ∈ Coh(Xan). Repeating the same reasoning as before but using thetruncations τ≤nF and using the fact that (−)an commutes with limits, we finallyconclude that there exists G ∈ Coh(X) such that Gan ' F. The proof is thereforecomplete.

8. Extension to Artin stacks

In this section we outline how it is possible to extend all the results we obtainedso far to the setting of derived Artin stacks.

We begin with a definition of derived Artin analytic stacks.

Definition 8.1. A morphism f : X → Y in StnderC is said to be smooth if it is

strong and its truncation t0(f) is smooth.


Let Psm be the collection of smooth morphisms in StnderC . Then the triple

(StnderC , τ,Psm) is a geometric context in the sense of [20]. We therefore give the

following definition:

Definition 8.2. A derived Artin analytic stack is a geometric stack for the context(Stnder

C , τ,Psm).

With this definition, the proof of the comparison results Corollary 3.10 andProposition 4.3 remain unchanged. The analytification functor for derived Artinanalytic stacks locally of finite presentation over C is obtained as left Kan extensionof the analytification functor

SpecTanTet

: dAff f.p.C → Stnder

C

exactly as it is done in [20, 24]. Therefore the comparison result Proposition 6.8also extends to the setting of derived Artin stacks. We now observe that the proofsof our main theorems 5.5, 7.5 and 7.7 all rely on the analogous results of [20] bydevissage to the heart using the comparison results we cited above. Therefore thesame technique applies to the setting of derived Artin stacks.

Appendix A. Flat morphisms

We collect in this section a couple of basic facts about flat morphisms of E∞-ringswe couldn’t find a reference for.

Lemma A.1. Let f : A → B be a flat morphism of connective E∞-rings. Thenτ≤n(A)⊗A B ' τ≤n(B).

Proof. It follows from [15, 7.2.2.13] that for every A-module M one hasπi(M)⊗π0A π0B ' πi(M ⊗A B)

On the other sideπi(M)⊗A B = πi(M)⊗π0A π0A⊗A B

Since A→ B is flat, we see that π0A⊗A B is discrete and therefore there exists amap

π0B → π0A⊗A Band the spectral sequence of [15, 7.2.1.19] shows that this is indeed an isomorphism.In conclusion, we see that

πi(M)⊗A B ' πi(M ⊗A B)Now, since A → B is flat, we see that τ≤nA ⊗A B is n-truncated, and thereforethere exists a morphism

τ≤nB → τ≤nA⊗A B

68 MAURO PORTA

The above argument shows that this morphism induces isomorphisms on the πi forevery i and therefore it is an equivalence.

For the terminology used in the following lemma we refer back to the discussionat the beginning of Section 6.6.

Lemma A.2. Let f : A→ B be a morphism of E∞-rings. Let M be an A-module.Then Ω∞A (M)⊗A B ' Ω∞B (f ∗(M)).

Proof. The functor−⊗AB : CAlg/A → CAlg/B commutes with finite limits (becausefinite limits are computed in the category Sp/A and Sp/B respectively and −⊗A Bis a functor between stable ∞-categories). Therefore, the conclusion follows.

Lemma A.3. Let f : A → B be a flat morphism of connective E∞-rings. Letd : τ≤n−1A → Ω∞A (πn(A)[n + 1]) be the n-th Postnikov invariant of A. Thend⊗τ≤n−1A τ≤n−1B : τ≤n−1B → Ω∞B (πn(B)[n+ 1]) is the n-th Postnikov invariant ofB.

Proof. The n-th Postnikov invariant of A is characterized by the fact that it makesthe diagram

τ≤nA τ≤n−1A

τ≤n−1A Ω∞A (πn(A)[n+ 1])

into a pullback diagram in the category of connective E∞-rings. In particular, itis a pullback diagram in the category of τ≤nA-modules. Therefore the functor− ⊗τ≤nA τ≤nB preserves this pullback. Since τ≤n−1A ⊗τ≤nA τ≤nB ' τ≤n−1B andΩ∞A (πn(A)[n+ 1])⊗τ≤nA τ≤nB ' Ω∞B (πn(B)[n+ 1]), we conclude that the diagram

τ≤nB τ≤n−1B

τ≤n−1B Ω∞B (πnB[n+ 1])

d⊗τ≤nA τ≤nB

is a pullback square in the category of B-modules and, a posteriori, in the categoryof B-algebras. We conclude that d⊗τ≤nA τ≤nB is the n-th Postnikov invariant ofB.

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70 MAURO PORTA

Mauro PORTA, Institut de Mathematiques de Jussieu, CNRS-UMR 7586, Case7012, Universite Paris Diderot - Paris 7, Batiment Sophie Germain 75205 ParisCedex 13 France

E-mail address: [email protected]

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