Levels in triangulated categories

Srikanth Iyengar

University of Nebraska, Lincoln

Leeds, 18th August 2006

The goal

My aim is to make a case that the invariants that I call levels areuseful and interesting invariants.

Towards this end, I discuss the proof of the following result.

Theorem

Let R be a commutative noetherian local ring.For every finite free complex F the following inequality holds:∑

n∈ZLoewyR Hn(F ) ≥ 1 + conormal free-rank of R .

For now, it is not relevant what “conormal free-rank of R” is.What is relevant is that the statement makes no mention of levels,or triangulated categories, or...

Applied to group algebras of elementary abelian p-groups, thisrecovers results of G. Carlsson, and C. Allday and V. Puppe.

Outline

Thickenings and levels

DG modules over DG algebras

A New Intersection Theorem for DG modules

Homology of perfect complexes

The dimension of the stable derived category of a local ring

Joint work with (subsets of)

L. L. Avramov and C. Miller R.-O. Buchweitz

Based on the following articles:

Class and rank for differential modules (on the arXiv.)Homology of perfect complexes (will be on the arXiv before long.)

Thick subcategories

Let C be a non-empty class of objects in triangulated category T.

Let ThickT(C) be the smallest thick subcategory containing C.Its objects may be thought as being finitely built out of C.

Example

Let R be a ring. Write ThickR(C)− for ThickD(R)(C)−.

ThickR(R) is the category of perfect complexes: complexesquasi-isomorphic to one of the form

0→ Ft → · · · → Fs → 0

where each Fi is a finitely generated projective R-module.

If R is semi-local, with Jacobson radical m, then

ThickR(R/m) = {M ∈ D(R) | lengthR H(M) <∞.}

Thickenings

We consider subcategories {thicknT(C)}n>0 of ThickT(C):

thick0T(C) = {0}.

thick1T(C) = C closed up under shifts, finite direct sums, retracts.

thicknT(C) is the subcategory with objects{

M

∣∣∣∣∣ L→ M → N → ΣL is an exact triangle

with L ∈ thickn−1T (C) and N ∈ thick1

T(C)

}

closed up under retracts.

Note that thicknT(C) is closed under shifts and direct sums, but not

under triangles.

We call thicknT(C) the nth thickening of C in T.

It consists of (n − 1)-fold extensions of objects in thick1T(C).

These subcategories provide a filtration

{0} ⊆ thick1T(C) ⊆ thick2

T(C) ⊆ · · · ⊆⋃n>0

thicknT(C) = ThickT(C).

This filtration appears in the work of

Bondal and Van den Bergh: Generators and representability...

Dan Christensen: Ideals in triangulated categories...

Rouquier:

Dimension of triangulated categories.Representation dimension of exterior algebras.

The focus in these works is on “global” aspects of T.

Here we use the filtration to obtains invariants of objects in T.

Another pertinent reference:

Dwyer, Greenlees, I.: Finiteness in derived categories...

Levels

Let M be an object in T. The C -level of M is the number

levelCT(M) = inf{n ≥ 0 |M ∈ thicknT(C)}.

Evidently, levelCT(M) is finite if and only if M is in ThickT(C).

This invariant has good formal properties. For example:

If L→ M → N → ΣL is an exact triangle, then

levelCT(M) ≤ levelCT(L) + levelCT(N) .

Thus, levels are sub-additive.

If f : T→ S is an exact functor between triangulatedcategories, then

levelCT(M) ≥ levelf(C)f(T)(f(M)) .

Why levels?

This is what this talk is about.

By varying the class C one can model various invariants ofinterest: projective dimension, Loewy length, regularity, e.t.c.

Most “ring-theoretic” invariants, and certainly those in thepreceding list, do not behave well under change of categories.Levels do, and provide a versatile tool for studying them.

I will now discuss the case where T = D(A), the derived categoryof a DG (=Differential Graded) algebra A.

I will focus on level with respect to A.This models classical projective dimension for modules over rings.

It is convenient to write levelAA(−) instead of levelAD(A)(−),

DG modules over DG algebras

Let A be a DG algebra and M a DG A-module.

Theorem

One has levelAA(M) ≤ d if and only if M is a retract of a DGA-module F admitting a filtration 0 ⊆ F 0 ⊆ · · · ⊆ F d−1 = F where

F n/F n−1 is isomorphic to a direct sum of shifts of A.

One direction is clear: A filtration as above induces exact triangles

F n−1 → F n → F n/F n−1 → ΣF n−1 for 0 ≤ n ≤ d − 1 ,

so levelAA(M) ≤ d , by sub-additivity.

Note: when M is in ThickA(A), it is quasi-isomorphic to a DGmodule whose underlying graded module is projective over A\.The converse holds under additional hypotheses on A.

Example

Let R be a ring and F a finite free complex:

F = 0→ Ft → · · · → Fs → 0 .

Then F n = 0→ Fs+n → · · · → Fs → 0 gives a filtration of F , so

levelRR(F ) ≤ card{n | Fn 6= 0} .

Often the inequality is strict: F can be built more efficiently.

Definition

Given elements x1, . . . , xn in a commutative ring R, the complex

(0→ Rx1−→ R → 0)⊗R · · · ⊗R (0→ R

xn−→ R → 0)

is the Koszul complex on x.

Example

Let R = k[x , y ], a polynomial ring with |x | = 0 = |y |. Pick d ≥ 1.

Let K be the Koszul complex on xd , xd−1y , . . . , xyd−1, yd . Thus

Kn∼= R(d+1

n ) therefore levelRR(K ) ≤ d + 2 .

However, levelRR(K ) = 3. (This calls for an explanation!)

Remark

In the last example card{n | Kn 6= 0} − levelRR(K ) = d − 1; inparticular, the difference can be made arbitrarily large.

The next few slides discuss bounds on levelAA(M).

Remark

Upper bounds on levels are easier to obtain than lower bounds.

Let A be a DG algebra with ∂A = 0, left coherent as a graded ring.

Proposition

If the A-module H(M) is finitely presented, then

levelAA(M) ≤ proj dimA H(M) + 1 ≤ gl dim A + 1 .

Remark: To get a better result one should consider levels withrespect to projectives.

One way to prove the proposition is to pick a projective resolution

0← H(M)← P0 ← ΣP1 ← Σ2P2 ← · · ·

and construct an Adams resolution:

M = M0 // ΣM1

+1������

���

// Σ2X 2

+1��~~~~

~~~

// · · · · · ·

P0

aaBBBBBBBB

ΣP1

__???????

Σ2P2

ccHHHHHHHHH· · ·

A New Intersection Theorem for DG modules

Theorem

Let A be a DG algebra with ∂A = 0 and M a DG A-module.When A is a commutative noetherian algebra over a field one has

levelAA(M) ≥ codim H(M) + 1 .

Recall: codim H(M) = height AnnA H(M). Observe that thisnumber depends only the support of H(M).

Corollary

Let R be a commutative noetherian algebra over a field.If F = 0→ Fd → · · ·F0 → 0 be a finite free complex, then

d + 1 ≥ card{n |Fn 6= 0} ≥ levelAA(F ) ≥ codim H(F ) + 1 .

In particular, d ≥ codim H(F ).

This is the classical New Intersection Theorem.

Example

Let R = k[x1, . . . , xn] (or any regular local ring)

If F is a finite free complex with lengthR H(F ) 6= 0,∞, then

n + 1 = gl dim R + 1 ≥ levelRR(F ) ≥ codim H(F ) + 1 = n + 1 .

Therefore, levelRR(F ) = n + 1.

This calculation applies, in particular, when:

R = k[x , y ]

F = K , the Koszul complex on xd , xd−1y , . . . , xyd−1, yd .

Thus, levelRR(K ) = 3.

Remarks

The New Intersection Theorem for algebras over a field wasproved by Peskine and Szpiro, Hochster, and P. Roberts.

Using intersection theory, P. Roberts proved it for allcommutative noetherian rings.

Even for rings, we have not been able to deduce our theoremfrom Roberts’ result. Let me remind you that the inequality

card{n |Fn 6= 0} ≥ levelAA(F )

is typically strict.

Our proof uses local cohomology and “big” Cohen-Macaulaymodules. Hochster has constructed them for algebras overfields, hence the restriction on A.

The result is deduced from an analogous statement fordifferential modules, proved in: Class and rank for differentialmodules (Avramov, Buchweitz, I.).

The idea is to construct a sequence of complexes

X (d+1) θ(d+1)−−−−→ X (d) θ(d)−−→ · · · θ(1)−−→ X (0)

where d = codim H(M), with the following properties:

(a) H(θ(n)) = 0 for each n;(b) H(M ⊗A θ) 6= 0, where θ = θ(1) ◦ · · · ◦ θ(d + 1).

Much of what I have said so far applies to differential modules.This is important for some applications. That is the beginningof a different story, and is work in progress with L. Avramov,R.-O. Buchweitz, Lars Christensen, and Greg Piepmeyer.

General strategy to estimate levels

Suppose C is an object in a triangulated category T.

We wish to estimate levelCT(X ), for some object X in T.

One strategy is as follows:

Find a (commutative noetherian) DG algebra A with ∂A = 0 andan exact functor

f : T→ D(A)

such that f(C ) is a finitely generated projective A-module.

Then f(C ) ∈ thick1A(A), so one obtains an estimate

levelCT(X ) ≥ levelf(C)A (f(X )) ≥ levelAA(f(X )) ≥ codim f(X ) .

Free summands of the conormal module

Let k be a field and R ∼= k[X ]/I where

k[X ] is a polynomial over k in variables X = {x1, . . . , xe};I is a homogeneous ideal in (X )2.

The R-module I/I 2 is the conormal module of R.

It is independent of the presentation R = k[X ]/I as above.

The conormal free rank of R is the number

cf-rankR = sup

{n

∣∣∣∣∣ Rn is a free direct summand

of the conormal module of R

}

It is a measure of the singularity of R.

Example

Let R = k[x1, . . . , xc ]/(xn11 , . . . , xnc

c ), with each ni ≥ 2.

Then the conormal module of R is I/I 2, where I = (xn11 , . . . , xnc

c ).

It is easy to check that I/I 2 ∼= Rc , so cf-rank R = c .

Special case: ni = p, with p ≥ 2, covers group algebras ofelementary abelian groups.

Free summands of conormal modules arise in the following cases:

When R has embedded deformations: if R = Q/xQ, where(Q, q) is a local ring and x a non-zero divisor in q2.

When R is the closed fibre of a flat homomorphismϕ : (P, p)→ (Q, q) such that ϕ(p) ⊆ q2.

Aside: there is a notion of conormal free rank for any(commutative noetherian) local ring.

Homology of perfect complexes

The Loewy length of an R-module M equals the number

LoewyRM = inf{n ≥ 0 | mnM = 0} ,

where m is the maximal ideal of R.

When lengthR M is finite, so is LoewyRM;the converse holds when M is finitely generated.

Loewy length better reflects the structure of M than length does.

Theorem

If F is a finite free complex of R-modules with H(F ) 6= 0, then∑n

LoewyR Hn(F ) ≥ cf-rankR + 1 .

Thus, the singularity of R imposes lower bounds on the “size” ofhomology of finite free complexes.

Group algebras of elementary abelian groups

Let p a prime, and R the group algebra over Fp of a rank celementary abelian p-group.

Thus, R ∼= Fp[x1, . . . , xc ]/(xp1 , . . . , xp

c ) and I/I 2 ∼= Rc .

Therefore cf-rankR = c , and the theorem yields:∑n

LoewyR Hn(F ) ≥ c + 1 .

In this way, the theorem specializes to results of

G. Carlsson, who proved it when p = 2;

C. Allday and V. Puppe, who proved it for odd primes,

which has application to the study of finite group actions.Neither of their methods extends to cover the other case...

Proof of theorem

The first step is convert the problem to one about levels:

Lemma

One has an inequality:∑

n LoewyR Hn(F ) ≥ levelkR(F ).

This inequality follows from general properties of levels.

Thus, it suffices to prove the following inequality:

levelkR(F ) ≥ c + 1 where c = cf-rank R .

Let K be the Koszul complex on a set of generators for m.

This is a DG algebra (an exterior algebra with a differential).

Let Λ be an exterior algebra on c variables of degree 1.

A crucial input in the proof is:

Theorem

As DG algebras K ' A, where A ∼= Λ⊗k B, and under the inducedequivalence a: D(K ) ≡ D(A) of derived categories, one has

a(K ⊗R k) '⊕

n

Σnk(cn) .

This is where the free summand of the conormal module comes in.The proof involves calculations with various DG algebra models forthe Koszul complex. It is akin to the Jacobian criterion.

Here is one consequence of the preceding theorem:

Corollary

levela(K⊗Rk)A (−) = levelkA(−).

Let S be a polynomial ring on c variables of degree −2.

We view it as a DG algebra with zero differential.

One has exact functors between triangulated categories

D(R)t // D(K )

a // D(A)i // D(Λ)

h // D(S)

The functors involved are as follows: t = K ⊗R −

a is the equivalence of categories in the last result.

i is induced by the inclusion Λ ↪→ (Λ⊗k B) = A, and

h is the BGG functor representing RHomΛ(k,−).

In particular, h(Λ) ' Σck and h(k) ' S .

Summing up

We want to prove: If F is a finite free complex of R-modules,that is to say, if levelRR(F ) is finite, then

levelkR(F ) ≥ c + 1 where c = cf-rank R .

We will deduce this from the New Intersection Theorem for S :For any DG S-module M, one has an inequality

levelSS(M) ≥ codim H(M) + 1 .

The path from R to S is

D(R)t // D(K )

a // D(A)i // D(Λ)

h // D(S)

A rappel

Levels are non-increasing under application of exact functors.

levelkR(F ) levelRR(F ) 6= 0,∞

levelt(k)K (t(F ))

\W

levelKK (t(F )) 6= 0,∞��

levelat(k)A (at(F )) levelkA(at(F )) levelAA(at(F )) 6= 0,∞

��

levelkΛ(iat(F ))

\W

levelΛΛ(iat(F )) 6= 0,∞��

levelSS(hiat(F ))

BGG

levelkS(hiat(F )) 6= 0,∞��BGG

dim S + 1

\W

NITck PPPPPPP

PPPPPPPlengthS(H(hiat(F ))) 6= 0,∞

��

The proof is better than the theorem:

1. The result can be formulated (and proved) for all local rings.

2. When R is complete intersection, the same argument yields:

Theorem

If M is a complex of R-modules with H(M) noetherian, then∑n

LoewyR Hn(M) ≥ codimVR(M) + 1 ,

where VR(M) is the cohomological variety of M.

Specialized to group algebras, this recovers a result of Benson andCarlson, which was proved using “shifted subgroups”.

Dimension of stable categories

The dimension of a triangulated category T is the number

dim T = inf

{d ≥ 0

∣∣∣∣∣ there is an object G in T

such that thickd+1T (G ) = T

}This invariant was introduced by Rouquier.

Theorem

Let R be a local ring and set T = Db(R)/ThickR(R). Then

dim T ≥ cf-rankR − 1 .

Thus, embedded deformations of R impose lower bounds on dim T .Note: when R is complete intersection cf-rank R = codim R.

Example

When R = k[x1, . . . , xc ]/(xn11 , . . . , xnc

c ) with ni ≥ 2, then

dim stmod(R) ≥ c − 1 .

A partial list of references

L. L. Avramov, R.-O. Buchweitz, S. Iyengar, C. Miller,Homology of perfect complexes, preprint 2006.

L. L. Avramov, R.-O. Buchweitz, S. Iyengar, Class and rank ofdifferential modules, ArXiv: math.AC/0602344.

A. Bondal, M. Van den Bergh, Generators and representabilityof functors in commutative and non-commutative geometry,Moscow Math. J. 3 (2003), 1-36.

D. J. Christensen, Ideals in triangulated categories: phantoms,ghosts and skeleta, Adv. Math. 136 (1998), 284–339.

W. G. Dwyer, J. P. C. Greenlees, S. Iyengar, Finiteness inderived categories of local rings, Commentarii Math. Helvetici81 (2006), 383–432.

R. Rouquier, Representation dimension of exterior algebras,Invent. Math. 165 (2006), 357–367.

— Dimensions of triangulated categories, math.CT/0310134.