an introduction to rigid cohomology (special week ... · introduction rigidcohomology...

An introduction to rigid cohomology(Special week – Strasbourg – 2012)

Bernard Le Stum

Université de Rennes 1

Version of June 4, 2012

Contents

Introduction

Rigid cohomology

Algebraic and analytic geometry

Topos and derived category

The overconvergent site

Conclusion

Counting pointsLet Fq be a finite field with q elements (q a power of a prime p)and X an algebraic variety over Fq. We want to do the following:

I Compute the number of rational points of X

We will denote it by N(X ) := |X (Fq)|.

Example (always assuming q is odd)

We may consider the affine curve X defined by

y2 = x3 + x , y 6= 0

inside A2Fq, or its projective closure X defined by

y2z = x3 + xz2

inside P2Fq.

Example (continuing)

Of course, we have

N(X ) = N(X ) + |a ∈ Fq, a3 + a = 0|+ |a ∈ Fq, a3 = 0|

N(X ) + 2 if i 6∈ Fq (q ≡ −1 mod 4)N(X ) + 4 if i ∈ Fq (q ≡ 1 mod 4).

Recall that, in general, we know from the exact sequence

1→ ±1 → F×q → (F×q )2 → 1

that there are exactly q−12 squares in F×q .

We consider now the first case which concerns q = 3, 7, 27, . . . andcan easily be done in a very general way. Since −1 is not a squarein Fq, we see that, given any a ∈ F×q , then

either a or −a is a square but not both.

For the same reason,

either a3 + a or (−a)3 + (−a) = −(a3 + a) is a square but not both.

Thus we see that it happens exactly q−12 times that a3 + a has the

form b2. And when this happens, we get exactly 2 possibilities forb. It follows that N(X ) = q − 1 and therefore N(X ) = q + 1.

The second case which concerns q = 5, 9, 25, 49, . . . is a lot morecomplicated. For example, if q = 5, we may draw the followingtable

a −2 −1 1 2a2 −1 1 1 −1a3 2 −1 1 −2

a3 + a 0 −2 2 0

It follows that no element of the form a3 + a can be a non zerosquare and therefore N(X ) = 0 so that N(X ) = 4.

We can also work out the case of F9 := F3[i ]. On easily computes

a3 + a = a + a = 2Re(a) = −Re(a) ∈ F3

and see that it is a non zero square in F9 if and only if Re(a) 6= 0.Thus we obtain 6 possibilities for a and therefore N(X ) = 12 sothat N(X ) = 16.

The Zeta functionIf Fqr /Fq is a finite extension, and X is any algebraic variety overFq, we will write

Nr (X ) := |X (Fqr )|(

= N(X ⊗Fq Fqr )).

And we define the Zeta function of X as

Z (X , t) = exp( ∞∑

1Nr (X )

If we can compute it, we will recover

N(X ) =

(d lnZ (X , t)

and more generally, all other Nr (X ) by looking at the coefficientsof lnZ (X , t).

Thus, what we want to do now is the following:

I Compute the Zeta function of X

Example

Let us first verify that if X is defined over Fq by

y2 = x3 + x , y 6= 0

and X denotes its projective closure as before, then we have

Z (X , t) =

Z(X ,t)

(1−t)2(1−t2)if q ≡ −1 mod 4

Z(X ,t)(1−t)4 if q ≡ 1 mod 4.

Since Z (X , t) = Z (X , t)× Z (X \ X , t), we simply have to identifythe numerator with Z (X \ X , t).

First of all, an equation x = a has exactly 1 solution in Fqr foreach r and therefore, the Zeta function of a rational point is

exp( ∞∑

)= exp (− ln(1− t)) =

11− t .

This gets rid of the second case where there are 4 rational points.

However, when q ≡ −1 mod 4, then x2 + 1 has no solution in Fqr

for r odd and exactly 2 solutions for r even. Thus thecorresponding Zeta function is

exp( ∞∑

11− t2 .

And the second case is settled as well.

When q = 3, we can deduce the first terms of the Zeta functionsof our affine and projective curves above from our previouscomputations.

More precisely, we have N1(X ) = 3− 1 = 2 andN3(X ) = 27− 1 = 26 and we did directly N2(X ) = 12 so that

Z (X , t) = exp(2t + 12 t2

2 + 26 t3

3 ) = 1 + 2t + 8t2 + 34t3 mod t4.

Also, we have N1(X ) = 3 + 1 = 4 and N3(X ) = 27 + 1 = 30 andN2(X ) = 12 + 4 = 16 so that

Z (X , t) = exp(4t + 16 t2

2 + 28 t3

3 ) = 1+ 4t + 16t2 + 52t3 mod t4.

Alternatively, one can derive this by dividing out the previous oneby (1− t)2(1− t2) (exercise !).

Using cohomologyWe can use étale ([8]) or rigid ([3]) cohomology in order tocompute a Zeta function. We will do rigid cohomology here.

Theorem (Etesse-LS)

If X is a smooth algebraic variety of pure dimension d over Fq,then

Z (X , t) =2d∏i=0

det(1− tqd (F ∗)−1

|H irig(X)

)(−1)i+1

Therefore, what we want to do is the following:

I Compute the rigid cohomology of XI Compute the action of Frobenius

Example

Assume for a while that we know the standard properties of rigidcohomology. Then, one easily sees that the Zeta function of anelliptic curve such as X above has the form

Z (X , t) =1− at + qt2

(1− t)(1− qt).

We will also see below that we can actually do the explicitcomputations. In particular, the Zeta function is completelydetermined once we know N(X ) = −a + 1 + q (use logarithmicderivative).

When q ≡ −1 mod 4, we get a = 0 and an easy computationshows that

Z (X , t) = 1+(q+1)t+(q2+2q+1)t2+(q3+2q2+2q+1)t3 mod t4

which is a generalization of the above formula (case q = 3).

As an application, we may take q = 7 and get

lnZ (X , t) = 8t + 32t2 mod t3.

We recover N(X ) = 8 and discover N2(X ) = 64 so thatN2(X ) = 60. Thus, the equation y2 = x3 + x has 60 solutionswith y 6= 0 over F7[i ].

We can also do the case q = 5. We know that N(X ) = 4 so that4 = −a + 1 + 5 and thus a = 2. In other words, we have

Z (X , t) =1− 2t + 5t2

(1− t)(1− 5t).

It follows that

lnZ (X , t) = 4t + 16t2 mod t3

form which we recover N(X ) = 4 but we also discover N2(X ) = 32from which we derive N2(X ) = 28. Thus, the equationy2 = x3 + x has 28 solutions with y 6= 0 over F5[

√2].

Defining cohomologyWe may define rigid cohomology whenever we are in a suitablegeometric situation and show afterwards that this is well defined.

Assume for example that there exists a scheme X overZq := W (Fq) such that

X = X ⊗Zq Fq

and a smooth proper formal scheme X over Zq such that X is thecomplement of a relative normal crossing divisor with smoothcomponents. Then, one can define

H∗rig(X ) := H∗dR(X ⊗Zq Qq).

Recall that de Rham cohomology is obtained by differentiatingfunctions. We will work out an example right now.

Example

We consider again the affine curve y2 = x3 + x , y 6= 0. We willhave

H∗rig(X ) := H∗(A d−→ Adx)

(meaning H0rig(X ) = ker d : A→ Adx and H1

rig(X ) = Adx/dA)with

A := Qq[x , y , 1y ]/(y2 − x3 − x) and dy =3x2 + 1

2y dx .

Actually, it is convenient to set B := Qq[x , 1x3+x ], so that

A = B ⊕ By and dy =3x2 + 12(x3 + x)

We may then split the computation in two parts:

H∗rig(X ) := H∗(B d−→ Bdx)⊕ H∗(By d−→ Bydx).

Any element of B can be written in a unique way as a finite sum

f (x) =∑

Pk(x)(x3 + x)k

with degPk ≤ 2. All terms can be integrated unless k = −1 andwe obtain

H1(B d−→ Bdx) ' Qqdxy2 ⊕Qqx

dxy2 ⊕Qqx2 dx

The second part requires some more work but one finds

H1(By d−→ Bydx) ' Qqdxy ⊕Qqx

Using standard properties of rigid cohomology, one can show thatthis last vector space is actually identical to H1

rig(X ).

Frobenius actionThe Frobenius map on an Fq-variety X is the identity on theunderlying space and raises functions to the q-th power.

Unfortunately, the map f 7→ f q on X does not lift to X .

Example

The endomorphism

F : (x , y) 7→ (xq, yq)

of the affine plane over Zq does not leave X stable in the exampleabove:

(xq)3 − xq = x3q − xq 6= (x3 + x)q = (yq)2

There seems to be a solution: one may replace X with its p-adiccompletion X . In other words, we will replace polynomials withseries that converge on the closed p-adic ball of radius one.

Example

In the case of the curve y2 = x3 + x , y 6= 0, we will replace A by

A := Qqx , y , 1/y/(y2 − x3 − x)

whereQqx , y , 1/y =

∑i∈N,j∈Z

ai ,jx iy j , ai ,j → 0.

We may then define

F : (x , y) 7→(xq, yq

√x3q + xq

(x3 + x)q

in order to get a lifting of Frobenius on A. We need to give ameaning to this square root.

Since(x3 + x)q ≡ x3q + xq mod p,

we can writex3q + xq

(x3 + x)q = 1 + pz

and use √1 + pz =

∑n≥0

)pnzn.

This series converges for |z | < 1|p| , and in particular on the closed

disc of radius one.

Unfortunately, unless X is proper, we have

H∗dR(X ⊗Zq Qq) 6= H∗dR(X ⊗Zq Qq).

Example

A1Zq⊗Zq Qq = DQq (0, 1+)

andH∗dR(DQp (0, 1+)) = H∗(Qqt

d−→ Qqtdt).

One easily sees that the series∑k

pktpk ∈ Qqt,

for example, is not integrable and it follows that

H1dR(DQp (0, 1+)) 6= 0 = H1

dR(A1Qp ).

Actually, there exists an object X † that lies between X and Xcalled the weak completion of X such that

H∗dR(X † ⊗Zq Qq) = H∗dR(X ⊗Zq Qq).

The idea is to consider functions that converge on a bigger ball.

Example

In the case of the curve y2 = x3 + x , y 6= 0, we will replace A with

A† := Qq[x , y , 1/y ]†/(y2 − x3 − x)

Qq[x , y , 1/y ]† = ∑

i∈N,j∈Zai ,jx iy j ,∃λ > 1, |ai ,j |λi+|j| → 0.

(overconvergent series) and the above Frobenius is actually definedon A†.

This technique works as well for all hyperelliptic curves ([5]) andleads to efficient algorithms.

Rigid cohomologyNow, we want to define rigid cohomology in general.

Let K be a non trivial complete ultrametric field of characteristic 0with residue field k (and also valuation ring V and maximal idealm). Any finite extension of Qp would satisfy these assumptions.

We have to define a functor

X 7→ H∗rig(X/K )

from algebraic varieties over k to vector spaces over K .

More generally, we want to define a category of coefficientsIsoc†(X/K ) and a functorial assignment

E 7→ H∗rig(X/K ,E ).

Some geometryWe need to consider several categories:

• Sch/k is the category of schemes over k that have a locallyfinite covering of the form

Spec k[t1, . . . , tn]/a

(we recall that Spec A denotes the set of prime ideals of A).

• Sch/K is the analogous category over K .

• Sch/V is the category of schemes over V that have a locallyfinite covering of the form

Spec V[t1, . . . , tn]/a

where a is of finite type (condition always satisfied when thevaluation is discrete).

There are functors

Sch/V // Sch/k

X // Xk

and Sch/V // Sch/K

X // XK

called special fiber and generic fiber respectively. Locally, they aregiven by reduction mod m (same as k ⊗V −)

V[t1, . . . , tn]/a 7→ k[t1, . . . , tn]/a,

and scalar extension (same as K ⊗V −)

V[t1, . . . , tn]/a 7→ K [t1, . . . , tn]/(a).

Note also that there is a pair of complementary closed and openimmersions

Xk → X ← XK

locally obtained by pulling back prime ideals.

• FSch/V is the category of formal schemes over the ring V ofintegers of K that have a locally finite covering of the form

Spf Vt1, . . . , tn/a

where a is of finite type. Recall that Spf A denotes the set of openprimes (i.e. that contain mA) and that

Vt1, . . . , tn =

∞∑

i1=0,...,in=0ai1,...,int

i11 · · · t

inn , V 3 ai1,...,in → 0

.There exists a completion functor

Sch/V → FSch/V , X 7→ X

locally given by (same as completion)

V[t1, . . . , tn]/a 7→ Vt1, . . . , tn/(a).

Moreover, there is a canonical morphism X → X locally obtainedby pulling back open primes.

There exists also a special fiber functor

FSch/V // Sch/k

X // Xk

Locally, it is given by (same as k ⊗V −)

Vt1, . . . , tn/a 7→ k[t1, . . . , tn]/a.

Note that when X is a formal scheme, the canonical map (locallyobtained by pulling back prime ideals) Xk → X is ahomeomorphism.

In order to define the generic fibre, we need to introduce analyticvarieties.

• An/K is the category of analytic varieties over K inBerkovich theory. Locally (in a sense that will be made preciselatter), they have the form

M(Kt1, . . . , tn/a)

Kt1, . . . , tn =

∞∑

i1=0,...,in=0ai1,...,int

i11 · · · t

inn , K 3 ai1,...,in → 0

andM(A) denotes the set of continuous multiplicative semi-normson the K -algebra A. There exists an analytification functor

Sch/K → An/K , V 7→ V an

locally given bySpec A 7→ Malg(A)

whereMalg(A) denotes the set of multiplicative semi-norms on theK -algebra A.

Moreover, there is a canonical surjective map V an → V locallygiven by

x 7→ ker x := f ∈ A, x(f ) = 0.There exists also a generic fibre functor

FSch/V → An/K , X 7→ XK

locally given by (same as K ⊗V −)

Vt1, . . . , tn/a 7→ Kt1, . . . , tn/(a).

Also, there exists a specialization map sp : XK → X , which is notcontinuous (for Berkovich topology), locally given by

x 7→ p = f ∈ A, x(f ) < 1.

Finally, if we start with a scheme X over V, there exists a naturalmap XK → X an

K locally given by the embedding

M(Kt1, . . . , tn/(a)) →Malg(K [t1, . . . , tn]/(a)).

Example (K algebraically closed)

As sets, we may write

A1k = k ∪ ξ, A1

K = K ∪ Ξ and A1V = k ∪ ξ ∪ K ∪ Ξ

where ξ and Ξ denote the generic points of the lines.

Of course, we may also consider A1V = k ∪ ξ (as sets) whose

generic fibre is

D(0, 1+) :=M(Kt) =⋃

a∈V mod m

D(a, 1−) ∪ ‖ − ‖

with the Gauss norm defined as ‖∑

ai t i‖ = max |ai |.Specialization sends the residue disc D(a, 1−) to a and the Gaussnorm to ξ.

We will also describe A1anK , but we want to do the projective line

before.

As sets again, we have

P1k = k ∪ ∞k ∪ ξ, P1

K = K ∪ ∞K ∪ Ξ

andP1V = k ∪ ∞k ∪ ξ ∪ K ∪ ∞K ∪ Ξ.

Of course, we may also write P1V = k ∪ ∞k ∪ ξ. But now,

(P1V)K = P1an

K = D(0, 1+) ∪ D(∞, 1−).

Specialization sends the rational point (a; b) ∈ P1anK with

max(|a|, |b|) = 1 to (a; b) ∈ P1k .

Finally, we haveA1an

K = P1anK \ ∞

and the canonical map XK → X anK is the inclusion

D(0, 1+) → A1anK .

The smooth proper caseWe can first define rigid cohomology for a liftable smooth propervariety (or more generally for the special fibre of the complementof a relative normal crossing divisor with smooth components):

DefinitionLet P be a smooth proper (formal) scheme over V with specialfibre X and generic fibre V , then

H∗rig(X ) := H∗dR(V ).

We must show that this definition is independent of the choice ofP and functorial in X . This is not too difficult if one is ready touse crystalline cohomology ([1]) and the following result ([2]):

H∗cris(X/W (k))⊗W (k) K ' H∗dR(P)⊗V K

(at least when k is perfect and the valuation is discrete).

TubesWe will generalize a little bit the previous case.

Let X → P be a locally closed embedding of an algebraic k-varietyinto a formal V-scheme. Then, the tube of X in P is

]X [P := x ∈ PK , sp(x) ∈ X.

Note thatx ∈]X [P ⇔ sp(x) ∈ X .

If P is affine and

X = a ∈ Pk , ∀i , f i (a) = 0 and ∃j , g j(a) 6= 0,

one easily checks that

]X [P= x ∈ V , ∀i , x(fi ) < 1 and ∃j , x(gj) = 1.

Let’s do it: one may assume that

X = a ∈ Pk , f (a) = 0.

Then, we can simply recall that, by definition,

f (sp(x)) = 0⇔ x(f ) < 1.

Example

We have]0[An

V= Bn(0, 1−).

When X is open in Pk so that X = Xk with X open in P, the tubeof X in P is exactly the generic fibre of X : we have ]X [P= XK . Inorder to verify this assertion, we may assume P = Spf A andX = Spf A1/f so that

]X [P= x ∈M(AK ), x(f ) = 1 ' M(AK1/f ) = XK .

We can now define rigid cohomology for proper (possibly singular)algebraic varieties:

DefinitionLet X → P be a closed embedding of a proper algebraic varietyinto a smooth (around X ) formal scheme, then

H∗rig(X ) := H∗dR(]X [P).

Note that we recover the previous case when X = Pk . Note alsothat this definition also makes sense when X is not proper in whichcase, it should be called convergent cohomology.

We must show that this definition is independent of P andfunctorial in X . In order to do that, we can use Ogus convergentcohomology and a comparison theorem. Alternatively, we can givea direct proof based on the Poincaré lemma and the weak fibrationtheorem that we describe now.

Theorem (Local Poincaré lemma)

Let V be an affinoid variety. If F is a coherent OV -module, wehave a short exact sequence

0→ Γ(V ,F)→ Γ(V×D(0, 1−), p∗F)∂/∂t→ Γ(V×D(0, 1−), p∗F)→ 0.

Proof.The point is to consider a series

∑n sntn that converges on the

open disc. One first checks that we can integrate it with respect tot which means that the series∑

snn + 1 t

also converges on the open disc. This gives exactness on the right.Next, one has to show that the derivative of the series

∑n sntn

with respect to t is zero if and only if the series is constant. Butthe condition means that nsntn−1 = 0 for n > 0. This givesexactness in the middle. Exactness on the left is clear.

Theorem (Weak fibration theorem)

Let X be an algebraic variety, X → P (resp. X → P ′) be a locallyclosed embedding of X into a formal scheme, and P ′ → P asmooth (around X) morphism. Then, locally on X, we have

]X [P′']X [P×Bn(0, 1−).

Proof.One easily reduces to the case of an étale map of affine formalschemes (so that n = 0). It is therefore sufficient to show that ifan étale map A→ A′ induces an isomorphism

A/(f1, . . . , fn) ' A′/(f1, . . . , fn)

then, it induces an isomorphism

Af1/r , . . . , fn/r ' A′f1/r , . . . , fn/r

when r < 1 is close to 1. This is formal.

Towards the general caseIn order to go further, we will consider the following situation:

X → P sp←− PKλ←− V

where the first map is a locally closed embedding of a k-algebraicvariety X into a formal V-scheme P and λ is a morphism from ananalytic variety V to the generic fibre of P. Most of the times, λwill be the inclusion of a neighborhood V of ]X [P in PK .

DefinitionThe tube of X in V is

]X [V := x ∈ V , sp(λ(x)) ∈ X.

Note that

x ∈]X [V ⇔ λ(x) ∈]X [P ⇔ sp(λ(x)) ∈ X .

Example (Monsky-Washnitzer)

Let A be a finitely presented V-algebra,

X := Spec Ak and V := (Spec AK )an.

The choice of a presentation

A := V[t1, . . . , tr ]/a

defines an embedding

Spec A → AnV → Pn

We will then consider:

X = Spec Ak // Pn

V Pn,anK

spoo (Spec AK )an = V .? _oo

Alternatively, if we denote by Q the projective closure of SpecA inPnV , we may consider

X = Spec Ak // Q Qan

Kspoo (Spec AK )an = V .? _oo

If we write X = Spf A, we have

]X [V =]X [Q= XK = V ∩ Bn(0, 1+).

Actually, we can draw the following commutative diagram

// QanK _

XK // _

QanK _

K // Pn,an

Bn(0, 1+) //

Pn,anK

Working with neighborhoodsBack to

X → P sp←− PKλ←− V ,

we denote byiX :]X [V → V

the inclusion map.

DefinitionThe sheaf i−1

X OV is the sheaf of overconvergent functions on thetube.

Note that Berkovich iX∗i−1X “=” rigid j†X . More generally, we will

consider the sheaves i−1X Ωr

V of overconvergent forms on the tube.

Example

If X → P is a closed embedding, then ]X [V is (Berkovich) open inV and i−1

X OV = O]X [P .

Example (MW)

In the Monsky-Washnitzer situation, one easily sees that thesubsets

Vλ := V ∩ B(0, λ+)

with λ > 1 form a fundamental system of affinoid neighborhoods of

]X [V = V ∩ B(0, 1+)

in V (use the fact that ]X [V is compact to reduce to the affinespace).

Actually, ifA = Spec V [t1, . . . , tn]/a,

then Vλ =M(AλK ) with

Aλ := Vt1/λ, . . . , tn/λ/(a).

Example (Continuing)

It follows that

Γ(]X [P , i−1X OV ) = lim−→AλK = A†K

withA† := V [t1, . . . , tn]†/(a).

V[t1, . . . , tn]† =

∞∑

i=0ai t i , ∃λ > 1,V 3 |ai |λ|i | → 0

The general caseWe may now give a general definition of rigid cohomology:

DefinitionLet X → P be an embedding of an algebraic variety into a smoothproper formal scheme (around X ). Then,

H∗rig(X ) := H∗(]X [P , i−1X Ω•PK ).

In order to show that this definition is independent of P andfunctorial in X , we will need the strong fibration theorem (see [6])and the overconvergent Poincaré lemma (see also [6]).

Example (MW)

In the Monsky-Washnitzer situation, we get

H∗rig(X ) = H∗dR(A†K ) =: H∗MW(X ).

Diagonal embeddingFor more generality, we work in a relative situation

oo Voo

C // S SKoo Ooo

where the lower line is of the same type as the upper one andvertical arrows are morphisms that make the diagramcommutative. The absolute case correspond to the caseC = Speck, S = SpfV and O = SK =M(K ).

If we embed diagonally X in P2 = P ×S P, we also have a map

V 2 = V ×O V → PK ×SK PK = P2K

and we may consider the following diagram:

oo V 2oo

]X [V 2? _oo

oo ]X [V? _oo?

Example (Affine line)

In the simplest case of the affine V-line, we get a quite interestingpicture

(x , x + τ) (x , τ)oo

B2(0, 1+)oo

D(0, 1+)×D(0, 1−)? _oo

(( A1V

D(0, 1+)oo?

D(0, 1+)?

OverconvergenceWe also want to work with general coefficients and we introducethe following:

DefinitionAn overconvergent stratification on an i−1

X OV -module E is anisomorphism of i−1

X OV 2-modules

ε : p†2E := i−1X p∗2 iX∗E ' i−1

X p∗1 iX∗E =: p†1E

satisfying

p†13(ε) = p†12(ε)p†23(ε) on ]X [V 3 and ∆†(ε) = Id on ]X [V .

These are called the cocycle and normalization conditions.

In case of the affine V-line, an overconvergent stratification on afree module of rank one is multiplication by a function

F (x , τ) = 1 +∞∑

k=1fk(x)τk on D(0, 1+)×D(0, 1−)

satisfyingF (x , τ1 + τ2) = F (x , τ1)F (x + τ1, τ2).

ConnectionsIf V is defined by the ideal I in V 2, the first infinitesimalneighborhood of V is the subvariety defined by I2:

V (1) q

I2 ""V

homeo==

There exists an exact sequence

0→ Ω1V → OV (1) → OV → 0

and the derivation may be defined as df = 1⊗ f − f ⊗ 1. We willdenote by p(1)

1 , p(1)2 : V (1) → V the maps induced by the

projections.

We may always pull back an overconvergent stratification to V (1)

and get

E // p(1)†

2 E ' // p(1)†1 E E ⊗i−1

X OVi−1X OV (1)

m // m ⊗ 1 +∇(m)

where∇ : E → E ⊗i−1

X OVi−1X Ω1

happens to be an integrable connection on E that will be said tobe overconvergent.

One easily checks that ∇ is well defined using the normalizationcondition and integrability comes from the cocycle condition. It isa connection by linearity: more precisely,

(1⊗ f )(m ⊗ 1 +∇(m)) = fm ⊗ 1 +∇(fm)

gives∇(fm) = f∇(m) + (1⊗ f − f ⊗ 1)(m ⊗ 1) = f∇(m) + m ⊗ df .

An integrable connection on a free module on one generator e onthe closed disk is determined by

∂x (e) = f (x)e

with some f ∈ Kx. Using the above cocycle condition for F ,one shows by induction that, necessarily,

F (x , τ) =∞∑

k=0fk(x)τk with fk(x)e =

1k!∂k

x (e).

The (over-) convergence condition reads:

∀η < 1, ‖ 1k!∂k

x (e)‖ηk → 0.

This applies to Dwork’s exponential ∂x (e) = −πe with πp−1 = −pwhich is overconvergent and non trivial with F (x , τ) = exp(πτ).

Main independence theoremWe come back to our diagram

oo Voo

]X [V? _oo

C // S SKoo Ooo ]C [O? _oo

that we will denote by (X ,V )/O.

And we define relative rigid cohomology with coefficients (usingderived functors for sake of clarity):

DefinitionAn overconvergent isocrystal on (X ,V )/O is a coherenti−1X OV -module E with an overconvergent integrable connection.Then,

RprigE := Rp∗(E ⊗i−1X OV

i−1Ω•V ).

Overconvergent isocrystals on (X ,V )/O form a categoryIsoc†(X ,V /O) and we have:

Theorem (Berthelot)

Assume

1. P is smooth over S in the neighborhood of X,2. the Zariski closure of X in P is proper over (the Zariski

closure of C in) S,3. V = PK ×SK O.

Then, we have

1. Isoc†(X ,V /O) only depends on X and not on P or V .2. RprigE is functorial and only depends on X and not on P or

We will give a proof of this theorem later on. Before, we need tohave a closer look at the geometrical aspect of the question. Wewill also introduce the necessary cohomological formalism.

RemarksIf we remove the second assumption, we obtain that everythingdepends on the Zariski closure on X in P but not on P or V . Thereis a variant with support: If Z is a closed subset of X , the inclusion

β :]Z [V →]X [V

is an open immersion. And we define

Rprig,Z (E ) := Rp∗(]X [V , β!β−1E ⊗i−1

X OVi−1Ω•V ).

Then the conclusions of the theorem also hold for Rprig,Z (E ).There is also a dual theory. One can replace V with aneighborhood of ]X [V in V such that the inclusion α :]X [V → V isclosed and E = α−1E where E is a coherent module with anintegrable connection on V . Then

Rprig,c(E ) := Rp∗(]X [V , α!(E ⊗OV Ω•V )).

Again, the conclusions of the theorem also hold for Rprig,c(E ).

Algebraic varietiesWe will quickly review the basics of algebraic geometry in a settingsuitable for generalization to formal and analytic geometry.

A k-algebra of finite type is a k-algebra of the formA := k[t1, . . . , tn]/a

and an affine variety over k is a topological space X together witha fixed homeomorphism

X ' Spec A(set of prime ideals with the Zariski topology). The ring offunctions on X is O(X ) := A.

If x ∈ X corresponds to p, k(x) := Frac(A/p) and if f ∈ A, thenf (x) will denote the image of f in k(x). We have a bijection(Nullstellensatz):

X0 := x ∈ X , [k(x) : k] <∞ ' Spm A.

Example (k algebraically closed)

We have

k ∪ ξ ' // Spec k[t] : A1k

a oo // (t − a) closed pointsξ oo // 0 generic point

A morphism of affine varieties is a pair made of a continuous mapϕ : Y → X and a homomorphism ϕ∗ : O(X )→ O(Y ) such that

f (ϕ(y)) = 0⇔ ϕ∗(f )(y) = 0.

Note that ϕ is determined by ϕ∗ but the converse is false.

A subset U of X is an affine domain if the inclusion of U in Xextends to a a morphism U → X which is universal for allmorphisms of affine varieties Y → X that factors through U.Then, this affine structure on U is unique and U is open in X .Moreover, the affine domains of X form a basis for the topology ofX .

The non empty affine domains of A1k are all the subsets

U := A1k \ a1, . . . , an,

where a1, . . . an ∈ k, with affine structure given by

k[t, 1t − a1

, . . . ,1

t − an].

An affine net on a topological space X is a covering U of X suchthat

1. Any finite intersection of elements of U has a covering byelements of U .

2. Any element of U has an affine structure and any inclusion ofelements of U is the inclusion of an affine domain.

An algebraic variety is a topological space X with a maximal affinenet.

A morphism of algebraic varieties Y → X is, up to refinement, acompatible family of morphisms of affine varieties V → U for all Vin some open affine net V of Y and some U in an open affine netU of X .

Of course, this works over any field and we will as well consider thecategory Sch/K of algebraic varieties over K .

Note that this definition of algebraic varieties does not requiresheaf theory that will be introduced later on.

We may consider

k ∪ ∞ ∪ ξ ' // Proj k[t, s] =: P1k

a oo // (t − as)

∞ oo // (s)

ξ oo // (0)

where Proj k[t, s] denote the set of homogeneous prime idealsp ⊂ k[t, s] other than (t, s).

An open affine net may be given by

A1k = Speck[t], P1

k \ 0 ' Speck[s], A1k \ 0 = Speck[t, 1t ].

The maximal affine net is given by ∅ and all P1k \ S where S is any

non-empty finite set of closed points.

ValuationsA valuation on a field K is a map v : K 7→ Γ ∪ ∞ where Γ is atotally ordered (additive) abelian group such that

1. ∀a ∈ K , a = 0⇔ v(a) =∞2. ∀a, b ∈ K , v(a + b) ≥ min(v(a), v(b))

3. ∀a, b ∈ K , v(ab) = v(a) + v(b)

ThenV := a ∈ K , v(a) ≥ 0

is a local ring with maximal ideal

m := a ∈ K , v(a) > 0

and fraction field K . Actually, V is a valuation ring (an integraldomain satisfying

∀a ∈ Frac(V), a ∈ V or 1a ∈ V).

Conversely, given a subring V of a field K , we may orderΓ := K×/V× by setting Γ+ := (V \ 0)/V× and consider thecanonical map

K v // 0 ∪ Γ

0 6= a // a0 // 0

If V is a valuation ring, then Γ is totally ordered (multiplicative)and v is a valuation.

Assume for a while that v is surjective (or replace Γ with v(K×)).One can show that dimV = rank Γ (prime ideals of V correspondto convex subgroups of Γ). Actually, rank Γ = 1 if and only if Γ isisomorphic to an ordered subgroup of R. The valuation is said tobe discrete if Γ is isomorphic to Z.

Example

If p is a non-zero prime, the p-adic valuation on Q is the uniquevaluation such that v(p) = 1. It is discrete.

Absolute valuesAn ultrametric absolute value is a map | − | : K → R≥0 such that

1. ∀a ∈ K , a = 0⇔ |a| = 02. ∀a, b ∈ K , |a + b| ≤ max(|a|, |b|)3. ∀a, b ∈ K , |ab| = |a|+ |b|

Using an isomorphism

(R,+,≤) ∪ ∞ ' // (R>0,×,≥) ∪ 0x // ε−x

− logε y yoo

with ε > 1, we see that an ultrametric absolute value is, up tonotation, a rank one valuation. For example, the p-adic absolutevalue is defined by |a| = p−v(a) on Q.

An absolute value | − | defines a metric on K by d(a, b) = |b − a|and the completion K of K is naturally a field with an absolutevalue. If 0 < |π| < 1, then the valuation ring of K is

V := lim←−V/(πr+1).

For example, the completion of Q for the p-adic absolute value isthe field Qp of p-adic numbers whose valuation ring is

Zp := lim←−Z/(pr+1).

The absolute value extends uniquely to the algebraic closure Qalgp

of Qp whose completion Cp is also algebraically closed (but thevaluation is not discrete anymore).

Formal schemesLet K be a field which is complete for a non trivial ultrametricabsolute value and V its valuation ring. Recall that

Vt1, . . . , tn :=

∞∑

i=0ai t i , V 3 ai → 0

.Alternatively, if π ∈ K with 0 < |π| < 1, we have

Vt1, . . . , tn = lim←−V/(πr+1)[t1, · · · , tn].

A V-algebra is said to be topologically of finite presentation if ithas the form

A := Vt1, . . . , tn/a

where a is finitely generated.

We denote by Spf A the set of open primes of A (primes idealsthat contain mA). Note that the reduction map A 7→ Ak := A/mAinduces a homeomorphism

SpecAk ' Spf A.

An affine formal scheme over V is a topological space X togetherwith a fixed homeomorphism

X ' Spf A

The ring of functions on X is O(X ) := A.

Morphisms of affine formal schemes are defined as before (anyhomomorphism is continuous).

Affine domains are defined as before. They form a basis of opensubsets as before.

A formal scheme over V is a topological space X with a maximalaffine net. Morphisms are defined as before.

Semi-normsLet K be a non-trivial complete ultrametric field.

A semi-norm on a K -algebra A is a map x : A 7→ R≥0 such that

1. ∀a ∈ K , x(a) = |a|.2. ∀f , g ∈ A, x(f + g) ≤ max(x(f ), x(g))

3. ∀f , g ∈ A, x(fg) ≤ x(f )x(g)

It is called multiplicative if we have an equality in the lastcondition. It is a said to be a norm if

x(f ) = 0⇒ f = 0.

Example

We may consider the multiplicative semi-norm defined on K [t] by

a(f ) := |f (a)| if a ∈ K .

Example (Continuing)

We may also consider the Gauss norm

ξ(f ) = max|ai | for f :=∑

ai t i ∈ K [t].

More generally (the previous cases are r = 0 and r = 1respectively),

ξa,r (f ) = max|ai |r i for f :=∑

ai (t − a)i ∈ K [t].

A (semi-) norm x defines a (semi-) metric on A byd(f , g) = x(g − f ) and the completion of A is naturally a normedring.

Example

The completion of K [t1, . . . , tn] for the Gauss normξ(f ) = max|ai | is

Kt1, . . . , tn =

∞∑

i=0ai t i , K 3 ai → 0

The kernelp := f ∈ A, x(f ) = 0

of a multiplicative semi-norm x is a prime ideal in A. Amultiplicative semi-norm x on A induces a multiplicative norm onA/p that extends uniquely to the fraction field and then to itscompletion H(x) (which is therefore a complete ultrametric field):

x**// // A/p

// H(x)|−|// R≥0

f // f (x)

Thus, if we denote by f (x) ∈ H(x) the image of f ∈ A, we have

x(f ) = |f (x)|.

Example

In the above example, H(a) = K and H(ξ) is the completion ofK (t) for the Gauss norm (which is also the fraction field of Kt).

Affinoid varietiesAn affinoid algebra is a quotient of a Tate algebra

Kt1, . . . , tn :=

∞∑

i=0ai t i , K 3 ai → 0

.The Gelfand spectrum of A is the setM(A) of all continuous (justmeans |f (x)| ≤ ‖f ‖) multiplicative semi-norms on A. It is compactHausdorff for the topology of simple convergence (induced by theinclusion

M(A) ⊂∏f ∈A

[0, ‖f ‖] ).

Example

In the above example, a multiplicative semi-norm ξa,r extends(uniquely) to a continuous multiplicative semi-norms on Kt ifand only if |a| ≤ 1 and r ≤ 1.

An affinoid variety over K is a topological space V together with afixed homeomorphism

V 'M(A).

The ring of functions on V is O(V ) := A and we can define H(x)for x ∈ X as well as f (x) if f ∈ A. We have a bijection(Nullstellensatz):

V0 := x ∈ V , [H(x) : K ] <∞ ' Spm A.

Example

If D := D(a, r) is a closed disc in K then ξD := ξa,r only dependson D and we get a map

closed discs ⊂ V //M(Kt) : D(0, 1+).

D // ξD

It is bijective when K algebraically closed and maximally complete1.1|ai+1 − ai | ≤ ri ⇒ ∃a, ∀i , |a − ai | ≤ ri

Affinoid domainsA morphism of affinoid varieties is a pair made of a continuous mapϕ : V ′ → V and a homomorphism ϕ∗ : O(V )→ O(V ′) such that

|f (ϕ(x ′))| = |ϕ∗(f )(x ′)|.

Note that, here again, ϕ is determined by ϕ∗.

A subset W of V is an affinoid domain if the inclusion extends toa morphism W → V which is universal for all morphisms V ′ → Vthat factors through W . Then, the affinoid structure on W isunique and W is closed in V . Moreover, affinoid domains form abasis of compact neighborhoods for V .

Example (K algebraically closed)

The connected affinoid domains of the closed unit disk are thesubsets

W = x ∈ D(0, 1+), |x − a| ≤ r and |x − ai | ≥ ri

with r , r1, . . . , rn ∈ |K |∩ ]0, 1] and a, a1, . . . , an ∈ V. The affinoidstructure is given by

K tr ,r1

t − a1, . . . ,

rnt − an

In particular, we will consider

C(0, 1) = x ∈ D(0, 1+), |x | = 1

andD(0, r+) = x ∈ D(0, 1+), |x | ≤ r

for r ∈ |K |∩ ]0, 1].

Analytic varietiesA quasi-net on a topological space V is a covering W of V suchthat

∀x ∈ V , ∃W1, . . . ,Wn ∈ W,

x ∈W1 ∩ · · · ∩Wn∃U open, x ∈ U ⊂W1 ∪ · · · ∪Wn.

A subset V ′ of V is said to be admissible with respect to W if theset

W ∈ W, W ⊂ V ′

is a quasi-net on V ′.

The quasi-net W is said to be a net if any finite intersection ofelements of W is admissible. A covering of an admissible subset byadmissible subsets is called admissible if it is a quasi-net.

Example

1. The open subsets of a topological space form a net with opensubsets as admissible subsets and open coverings as admissiblecoverings.

2. An affine net on an algebraic variety is a net.

Example

1. The set of all affinoid domains of an affinoid variety X is a netand any finite affinoid covering of an affinoid domain isadmissible. We will always endow X with this net.

2. The infinite affinoid covering of D(0, 1+) by C(0, 1) andD(0, r+) for r ∈ |K |∩ ]0, 1[ is not admissible because ξ hasno neighborhood of any of these types.

3. The open disc

D(0, 1−) = x ∈ D(0, 1+), |x | < 1

is admissible with admissible affinoid covering by the D(0, r+)for r < 1.

An affinoid net on a locally Hausdorff topological space X is a netW such that any element of W has an affinoid structure and anyinclusion of elements of W is the inclusion of an affinoid domain.

An analytic variety (in Berkovich sense) is a topological space Xwith a maximal affinoid net. An admissible subset is called ananalytic domain. Any open subset is an analytic domain.

Example

The analytic affine line is made of all multiplicative (not necessarilycontinuous) semi-norms on K [t]:

A1anK =Malg(K [t]) = ∪R∈|K×|M(Kt/R).

As above, we have when K is algebraically closed and maximallycomplete,

closed discs ⊂ K ' // A1anK .

D // ξD

A morphism of analytic varieties is, up to refinement, a compatiblefamily of morphisms of affinoid varieties V → U for all V in someaffinoid net V of Y and some U in an affinoid net U of X .

Note that if V is a Hausdorff analytic variety, there exists a uniquestructure of rigid analytic variety on

V0 := x ∈ V , [H(x) : K ] <∞such that affinoid domains and admissible affinoid coveringscorrespond bijectively to affinoid open subsets and admissibleaffinoid coverings. This is functorial and fully faithful.

Finally, an analytic variety is said to be good if any point has anaffinoid neighborhood.

Example

An affinoid variety is good. More generally, a proper analyticvariety over an affinoid variety is good. Also, if V is an algebraicvariety, then V an is good. But the generic fibre of A2 \ (0, 0) is notgood.

PresheavesWe introduce the vocabulary that will be necessary to give aformal approach to rigid cohomology.

A presheaf (of sets) on a category C is a contravariant functor

T : C // Sets.X // T (X )(

Y f−→ X) //

(T (X )

T (f )−→ T (Y )

)In other words, we require

T (f g) = T (g) T (f ) and T (IdX ) = IdT (X).

A morphism of presheaves α : T ′ → T is a natural transformation:a family of maps αX : T ′(X )→ T (X ) making commutative all

T ′(X )αX //

T ′(f )

T (X )

T (f )

T ′(Y )αY // T (Y ).

We get a category C of all presheaves and an embedding (Yoneda)

C // C

X //(Y X7−→ Hom(Y ,X )

Actually, we will have for all X in C and T in C,

T (X ) ' Hom(X ,T ).

Be careful that the Yoneda functor does not preserve direct limits(∅ or ∪ for example).

Example (Topological spaces)

If X is a topological space and Open(X ) denotes the category ofopen subsets of X and inclusion maps, a presheaf on X (meaningon Open(X )) is given by a set T (U) for U ⊂ X open, andcompatible restriction maps

T (U)→ T (U ′), s 7→ s|U′

whenever U ′ ( U. Yoneda’s embedding identifies an open subsetU with the presheaf:

U : U ′ 7→⊂ if U ′ ⊂ U∅ otherwise .

If X = ∪Ui is an open covering, we can also consider the subobjectof X :

R := ∪Ui : U ′ 7→⊂ if ∃i ∈ I,U ′ ⊂ Ui∅ otherwise

which is different from X .

If f : Y → X is any continuous map, it induces a functor

f −1 : Open(X )→ Open(Y )

and by composition, a functor

f∗ : Open(Y )→ Open(X ).

in other words, we have

f∗(T )(U) = T (f −1(U)).

One can check that giving a presheaf on the category Top of alltopological spaces is equivalent to giving a presheaf TX (itsrealization) on each topological space X and a compatible familyof morphisms TX 7→ f∗TY for all continuous map f : Y → X .

TopologyLet C be any category (think of Top for example).

A sieve of X ∈ C is a subobject R ⊂ X . If f : Y → X is amorphism, then

f −1(R)(Z ) := s ∈ Y (Z ), f s ∈ R(Z )

defines a sieve in Y .

A topology on C is a set of sieves of the objects of C, calledcovering sieves such that

1. X is a covering sieve of X .2. if R is a covering sieve of X and f : Y → X any map, then

f −1(R) is a covering sieve of Y .3. Let R be a sieve of X . Assume that there exists a covering

sieve S of X such that whenever f : Y → X belongs to S(X ),then f −1(R) is a covering sieve of Y . Then R is a coveringsieve of X .

A category endowed with a topology is a site.

The various topologies on a category C are ordered from coarse(only X covers X ) to discrete (any R covers X ).

We turn Open(X ) into a site by calling a sieve R ⊂ U a coveringif the family

U ′ ∈ Open(X ), R(U ′) 6= ∅

is an open covering of U.

In the same way, R ⊂ X will be a covering sieve in Top if

U ∈ Open(X ), R(U) 6= ∅

is an open covering of X .

PretopologyA pretopology on a category C is a set of families Xi → X calledcovering families such that

1. IdX : X → X is a covering family2. If Xi → X is a covering family and f : Y → X any

morphism, then Xi ×X Y → Y (exists and) is a coveringfamily

3. If Xi → X is a covering family and for each i , Xij → Xi isalso a covering family, then Xij → X is a covering family.

A sieve R ⊂ X will be called a covering sieve for this pretopology ifthere exists a covering family fi : Xi → Xi∈I such that for alli ∈ I, Imfi ⊂ R. This way, we get a topology on C. Conversely, ifC is a site with fibred products, the set of all familiesfi : Xi → Xi∈I such that ∪ Imfi is a covering sieve of X , is apretopology that induces the topology of C.

Usual open coverings define a pretopology giving the abovetopology.

SheavesA sheaf on a site C is a presheaf F : C → Sets such that, for allcovering sieve R of X , we have

(F(X ) =) Hom(X ,F) ' Hom(R,F).

If the topology comes from a pretopology, this is equivalent to

F(X ) // ∏i F(Xi )////∏

i ,j F(Xi ×X Xj)

being exact for any covering family (Exercise : show thatF(∅) = 0).

The category of all sheaves on the site C is the topos C ⊂ C.

Example

The constant presheaf U 7→ Z on a topological space is not a sheafin general but the presheaf U 7→ Zπ0(U) is a sheaf.

1. A sheaf on a topological space is a presheaf that satisfies:given an open covering U = ∪Ui and a family of si ∈ F(Ui )such that (si )|Uj = (sj)|Ui , there exists a unique s ∈ F(U)such that s|UUi

= si .2. A presheaf on Top is a sheaf if and only for any X , the

realization FX is a sheaf on X .

Example

1. For the coarse topology, any presheaf is a sheaf and thecorresponding topos is C.

2. The only sheaf for the discrete topology is the constant sheaf0 and the corresponding topos is the category 0.

3. The canonical topology on a category C is the finest topologysuch that any object of C is a sheaf (meaning that X is asheaf).

If we endow a topos C with its canonical topology, we getC =

If C is a site, the inclusion functor C → C has a left adjointT 7→ T (which is a actually section) given by applying twice

T ′(X ) = lim−→T (R)

with R running through all covering sieves of X .

The sheaf T is called the sheaf associated to the presheaf T .When F is a sheaf, there is a canonical bijection

Hom(T ,F) ' Hom(T ,F).

Example

The sheaf associated to the constant presheaf U 7→ Z on atopological space X is given by U 7→ Zπ0(U).

Morphisms of sitesExample (Topological spaces)

Recall that if f : Y → X is any continuous map, there is anobvious functor

f −1 : Open(X )→ Open(Y )

that gives, by composition, a functor

f∗ : Open(Y )→ Open(X ).

The functor f∗ has left and right adjoints given by

f −1(T )(V ) = lim−→V⊂f −1(U)

and f !(T )(V ) = lim←−f −1(U)⊂V

T (U).

If g : C → C′ is any functor (think of g = f −1 in the previousexample), composition induces a functor

g−1 : C′ // CT ′ // T ′ g

that has left and right adjoints: we have

Hom(g!T ,T ′) = Hom(T , g−1T ′))

and Hom(g−1T ′,T ) = Hom(T , g∗T ′).

By analogy with topological spaces, we often write g =: f −1, andwe have a sequence of adjointness:

g! =: f −1, g−1 =: f∗, g∗ =: f !.

Note that g(X ) = g!(X ) (or with the other notationf −1(X ) = f −1(X )). The point now is to extend this to sheavesand not merely presheaves.

If f : Y → X is any continuous map, the functor f∗ induces afunctor on sheaves, f∗ : ˜Open(Y )→ ˜Open(X ) that has an exactleft adjoint f −1: we have

Hom(f −1F ,G) = Hom(F , f∗G).

If C and C′ are two sites, a functor f −1 : C → C′ is said to becontinuous if f∗ preserves sheaves. By composition, the inducedfunctor f∗ : C′ → C has a left adjoint

f −1 : C // C′

F // ˆf −1(F)

If f −1 is exact, we then say that f : C′ → C is a morphism of sites.

Note that when f −1 is left exact (and we have fibred products inC), then f −1 defines a morphism of sites if and only if it preservescovering families for some pretopologies.

Morphisms of toposA morphism of topos

(f −1, f∗) : C −→ C′

is a couple of adjoint functors (f −1, f∗) with f −1 exact.

Thus, we see that any morphism of site f : C′ → C defines amorphism of topos

(f −1, f∗) : C −→ C′

The converse is true: a functor f −1 : C′ −→ C extends (uniquely)to a morphism of topos if and only if it defines a morphism of sitesf : C −→ C′ for the canonical topologies. In practice, we willtherefore often identify the morphism of sites f and the morphismof topos (f −1, f∗).

A functor g : C′ → C between two sites is said to be cocontinuousif g∗ preserves sheaves. Then, the induced functor g∗ : C′ → Cextends uniquely to a morphism of toposes g := (g−1, g∗).

Restricted (or comma) categoryIf X is an object in a category C, we may consider the categoryC/X of objects of C above X . An object is a pair made of an objectY ∈ C and a structural morphism g : Y → X . If Z has structuralmorphism h : Z → X , a morphism Z → Y over X is simply amorphism f : Z → Y such that h = g f .

More generally, if T ∈ C, then C/T is the category whose objectsare pairs made of an object Y ∈ C and a section s ∈ T (Y ). Amorphism Z → Y over T is a morphism compatible with sections:T (f )(s) = t.

We consider now the forgetful functor : jT : C/T → C. If C is asite, we endow C/T with the induced topology (finest making jTcontinuous). Unfortunately, jT ! is not left exact in general and wedo not get a morphism of sites. However, jT is cocontinuous andwe obtain a morphism of toposes

jT : C/T → C.

Note also that any morphism T ′ → T will induce a morphism oftoposes

C/T ′ // C/T

F/T ′ F .oo

If X is a topological space, we may consider the big site Top/X ofall topological spaces over X . The inclusion map

Open(X ) → Top/X

is continuous, cocontinuous and left exact giving rise to twomorphisms of toposes

Top/XϕX // ˜Open(X )ψXoo

with ϕX ψX = Id and ϕX∗ = ψ−1X .

Let T be any presheaf on Top (or a topological space identifiedwith the corresponding presheaf). If F is a sheaf on Top/T and Xa topological space over T , the realization of F on X is

FX := ϕX∗F/X .

Any morphism f : Y → X over T will induce a morphism

αf : f −1FX → FY

between the realizations.

One can show that, giving a sheaf F on Top/T is equivalent togiving the collection of all FX and compatible morphisms αf . Wemay call F a crystal if all the maps αf are isomorphisms.

Note that if X is a topological space, then, ϕX induces anequivalence between crystals on Top/X and sheaves on X .

Example (from geometry)

We consider the forgetful functor Sch/k → Top that sends analgebraic k-variety to its underlying topological space. We endowSch/k with the coarsest topology making this functor cocontinuous(it will also be continuous but not left exact).

This topology is induced by the pretopology made of coveringsXi → Xi∈I of X by open domains.

Careful: the topology induced by the forgetful functor (finertopology making it continuous) is too fine (an absolute Frobeniusfor example would be a covering).

If T is a presheaf on Sch/k , we can also consider the site Sch/T ofall algebraic varieties over T . As above, giving a (pre) sheaf onSch/T is equivalent to giving its realizations on X for all varietiesX over T with the transition morphisms.

We can do exactly the same thing with Sch/K , Sch/V , FSch/V orAn/K for example. There are morphisms of toposes

Sch/k → FSch/V → Sch/V ← Sch/K ← An/K .

However, the generic fibre functor

FSch/V → An/K

is not continuous (for Berkovich topology).

Nevertheless, admissible coverings define a new pretopology ofAn/K which is finer than the usual topology and makesspecialization continuous. We call it the admissible topology, theGrothendieck topology, the rigid topology or the Tate topology.

Rings and modulesGeometrical objects usually come with a sheaf of functions. Wewant to formalize this now.

If A and C are two categories, a presheaf on C with values in A isa contravariant functor C → A. If A is concrete (endowed with afaithful functor A → Sets) composition gives a functor “underlyingpresheaf of sets”. If moreover, C is a site, a presheaf with values inA is said to be a sheaf if the underlying presheaf of sets is a sheaf.We get the category A(C) of sheaves with values in A.

We will apply this to the case of abelian groups Ab or rings Rngfor example and call the corresponding sheaves just abelian groupsor rings on C. If F and G are two abelian groups on C, theirinternal hom is defined by

HomAb(F ,G)(X ) := HomAb(F/X ,G/X ).

Note that, if F is an abelian group on C, then

EndAb(F) := HomAb(F ,F)

has a natural structure of sheaf of rings.

A ringed site is a pair (C,O) made of a site and a commutativering on this site. We say that (X ,O) is a ringed space whenC = Open(X ).

An O-module is an abelian group on C endowed with a morphismof rings

O → EndAb(F).

A morphism of O-modules u : F → G is a morphism of abeliangroups making commutative the diagram

EndAb(F)

EndAb(G)−u // HomAb(F ,G).

The category O−Mod of all O-modules is an abelian categorywith an internal hom (defined as above) that has a right adjoint ⊗called tensor product.

An O-module is said to be (locally) finitely presented if for any Xin C, the the set R(Y ) of all f : Y → X such that there exists aright exact sequence

OM/Y → O

N/Y → F/Y → 0

defines a covering sieve of X . They form an abelian subcategoryO−Modfp.

A morphism of ringed sites

(C′,O′)→ (C,O)

is a pair made of a morphism of sites f : C′ → C and a morphismof rings f −1O → O′ (or equivalently, O → f∗O′).

If F ′ is an O′-module, then f∗F ′ will have a natural structure ofO-module. The induced functor has a left adjoint given by

F 7→ f ∗F := O′ ⊗f −1O f −1F .

Example (from geometry)

The assignment X 7→ O(X ) defined on affine k-varieties extendsuniquely to a sheaf of rings O on Sch/k called the structural sheaf.

For any presheaf T on Sch/k , we obtain a sheaf of rings on Sch/Tby restriction. We may also consider the realization OX of O on agiven algebraic variety X .

Any morphism of algebraic varieties f : Y → X over k will inducea morphism of ringed spaces (Y ,OY )→ (X ,OX ). But this functoris not fully faithful.

Let T be a presheaf on Sch/k . Giving an O/T -module isequivalent to giving a family of OX -modules for all X over T witha compatible family of transition morphisms f ∗FX → FY . We maycall it a crystal if all transition maps are bijective.

If X is an algebraic variety over k, then realization induces anequivalence

Crys(X ) ' OX−Mod and O/X−Modfp ' Coh(X ).

Again, we can do the same thing with Sch/K , Sch/V , FSch/V orAn/K for example. In the last case, we first define O for the Tatetopology and then restrict to the usual topology.

We will use later on these considerations in order to give a naturaltreatment to the notion of overconvergent isocrystal. Our next aimis to introduce the correct formalism in order to deal with thecohomological aspect.

Derived functorsThe idea behind the notion of triangulated categories and derivedfunctors is to put altogether the cohomology groups in a naturalway.

If C is an additive category, we may consider the category C+(C)of all complexes K • with Kn = 0 for n << 0. Any additive functorF : C → C′ extends trivally to a functor

F : C+(C)→ C+(C′).Assume C is an abelian category with enough injectives (someO−Mod category for example). Then any complex K • has aninjective resolution I• (a quasi-isomorphism K • ' I• with all Ininjective).

When F is left exact, we want to define the right derived functorof F by the formula

RFK • := FI•

in order to getHn(RFK •) = Hn(FI•) = RnFK •.

HomotopyTwo morphisms of complexes f , g : K • → L• are said to behomotopic if there exists a collection of maps sn : Kn → Ln−1 suchthat for all n, we have

gn − f n := sn+1 dn + dn−1 sn.

The category K+(C) is the category of complexes up to homotopy:objects are the same as in C+(C) and morphisms are morphisms inC+(C) modulo homotopy.

Any additive functor F : C → C′ will induce a functor

F : K+(C)→ K+(C′).

Derived categoryIf K • is a complex in a abelian category C, then

Hn(K •) := ker dn/im dn−1.

Any morphism of complexes f : K • → L• induces maps

Hn(K •)→ Hn(L•).

If all these maps are isomorphisms, then f is called aquasi-isomorphism.

The derived category D+(C) is the category of complexes up toquasi-isomorphisms: objects are as in K+(C) and morphisms aregiven by

K ′•

''q−is

K • // L•.

If C has enough injectives and I is the full subcategory of allinjective objects, then

K+(I) ' D+(C)

(see Proposition 1.7.10 of [4]). We may then define the derivedfunctor of F by making commutative the following diagram:

D+(C)RF // D+(C′)

F // K+(C′).

We will writeRnFK • := Hn(RFK •).

Note that any object E of C may be seen as a complexconcentrated in degree 0 and RFE will be a well defined object ofD+(C′).

Example

If C is a site, we may consider the global section functor on X ∈ C:

Ab(C) // AbF // Γ(X ,F) := F(X )

If F• is a complex of abelian group on C, we write

RnΓ(X ,F•) =: Hn(X ,F•).

Assume X is a locally contractile metric space. If ZX denotes thesheaf associated to the constant presheaf Z on X , we have

Hn(X ,ZX ) ' Hnsing(X ).

Now, we have the necessary background in order to formally defineoverconvergent isocrystals and their cohomology.

overconvergent varietiesWe let K be a non trivial complete ultrametric field with valuationring V, maximal ideal m and residue field k.

We want to define rigid cohomology as the cohomology of a topos,generalizing in some sense the crystalline topos of Berthelot andthe convergent topos of Ogus.

An overconvergent variety over V is a diagram

X → P sp←− PKλ←− V

where the first map is a locally closed embedding of a k-algebraicvariety into formal V-scheme and λ is a morphism from a goodanalytic variety to PK .

We will then consider the tube ]X [V which is an analytic domaininside V and the embedding iX :]X [V → V . We recall the followingfundamental example:

Example (MW)

Built out of a V-algebra A of finite presentation, we have

X = Spec Ak // Pn

V Pn,anK

oo V = (Spec AK )an? _oo

and]X [V = (Spf A)K =M(AK ) = V ∩ Bn

K (0, 1+).

Formal morphismsA formal morphism of overconvergent varieties is a commutativediagram

X ′ //

P ′KvK

oo V ′λ′oo

X // P PKoo V .λoo

It will induce a map

]f [u:]X ′[V ′→]X [V

on the tubes.

Overconvergent varieties and formal morphisms form a categoryAn/V .

Example

A very simple but important example of formal morphism is givenby the zero section:

X // An

(0, 1+)

oo BnV (0, 1+)oo

X // P PKoo Voo

that will induce

]X [V×K BnPK (0, 1−)→]X [V

on the tubes.

Strict neighborhoodsA strict neighborhood is a formal morphism of overconvergentvarieties

X // Q

oo Woo_

X // P PKoo Voo

where the first map is an equality and the last map is a (Berkovich)open immersion inducing an equality ]X [W =]X [V on the tubes.

Example (MW)

In the Monsky-Washnitzer situation, we may replace PnV with the

completion of the Zariski closure Q of Spec A in PnV and/or V with

Vλ := V ∩ Bn(0, λ+)

and obtain a strict neighborhood

X // Q

Vλ? _oo_

X // Pn

V Pn,anK

oo V .? _oo

Recall that the link with Monsky-Washnitzer theory is given by

A†K = lim−→Aλ with Vλ =M(Aλ).

Morphisms of overconvergent varietiesA morphism of overconvergent varieties over V is a formalmorphism up to a strict neighborhood:

// Q′

Q′Koo

W ′oo q

]X ′[W ′? _oo

X ′ //

P ′Koo

V ′oo

]X ′[V ′? _oo

X // P PKoo Voo ]X [V? _oo

More precisely, we may consider the category An/V ofoverconvergent varieties and formal morphisms, and then defineAn†/V as the quotient category on the right with respect to thefamily of all strict neighborhoods.

Any morphism in An†/V will provide a pair of morphisms

(f : X ′ → X , u : W ′ → V )

where W ′ is a neighborhood of ]X ′[V ′ in V ′. Moreover, they willsatisfy

sp(λ(u(x ′))) = f (sp(λ′(x ′))

for x ′ ∈]X ′[V ′ . And the same property holds after any isometricextension of K . One can show that, conversely, any such datadetermines a unique morphism of overconvergent varieties:

// P × P ′

PK × P ′Koo

W ′oo o

X ′ //

P ′Koo

V ′oo

X // P PKoo Voo

Taking this into account, we will usually denote by (X ,V ) andobject of An†/V (neglecting P unless it plays a particular role).

Example (MW)

Assume that we are given two V-algebras of finite type A and Band a morphism ϕ : A† → B†. On one hand, we may consider thereduction

ϕk : Ak → Bk

of ϕ mod m and the corresponding morphism

f : Y = Spec Bk → X = Spec Ak .

But we can also work on the generic fibre side.

We consider the weak completion

ϕ†K : A†K = lim−→Aλ → B†K = lim−→Bµ

that will induce Aλ → Bµ for suitable λ, µ and therefore

u : Wµ → Vλ → V

And we get a morphism

Y = Spec Bk //

PmV Pm,an

Koo W = (Spec BK )an? _oo

X = Spec Ak // Pn

V Pn,anK

oo V = (Spec AK )an? _oo

which is not a formal morphism.

Theorem (Strong fibration theorem)

LetP ′

P ′K

oo V ′oo

##P PKoo Voo

be a formal morphism such that

1. v is smooth in the neighborhood of X,2. v induces a proper morphism from the Zariski closure of X in

P ′ to (the Zariski closure of X in) P,3. V ′ = P ′K ×PK V .

Then, locally for the Zariski topology on X and the Tate topologyon V , we have in An†(V),

(X ,V ′) ' (X ,BnV (0, 1+)) over (X ,V ).

Proof.We assume first that the result is known in the étale case and worklocally on X and V .

Since we use Tate topology, the question is also local on P.

Blowing up outside X will induce an isomorphism in An†(V).Thus, we may assume that v is projective (Chow’s lemma).

We embed the closure Y of X in P ′ into some PnP and lift the

regular section s as follows

X // x

++PnX //

X // P

Proof. (the end)

Then, the map Q → P is étale, and we may therefore assume thatQ = P, and in particular that Y is also the closure of X in P. Thequestion is now local on Y . We may therefore assume that P ′ isaffine.

Since P ′ is smooth over P around X, we may assume that theconormal sheaf of X into P ′X is free and lift a basis to sections ofP ′ defining an étale map P ′ → An

V . And we are done.

It still remains to do the étale case which says that(X ,V ′) ' (X ,V ) (global result).

Using the weak fibration theorem, it is sufficient to prove that themorphism V ′ → V is (Berkovich) étale in the neighborhood of thetube. Formally étale follows from hypothesis 1 and boundarylessfollows from hypothesis 2.

The overconvergent siteWe endow An/V with the coarsest topology making the forgetfulfunctor

An/V // An/K

(X ⊂ P ← V ) // V

cocontinuous. Then, we endow An†V with the finest topologymaking the canonical functor An/V → An†/V continuous.

Thus, the topology of An†V is induced by the pretopology whosecoverings are all formal morphisms

(X ⊂ Pi ← Vi )i∈I → (X ⊂ P ← V )

where Vii∈I is a (Berkovich) open covering of a neighborhood of]X [V in V .

Restricted siteAn overconvergent presheaf on V is a presheaf T on An†V . We willbe interested in the restricted site An†/T of objects over T . Inparticular, if (X ,V ) is any overconvergent variety on V, we mayconsider the site

An†/X ,V

of overconvergent varieties over (X ,V ).

There exists a pair of morphisms of toposes

An†/X ,VϕX ,V // ˜Open(]X [V )ψX ,V

withϕX ,V ψX ,V = Id and ϕX ,V∗ = ψ−1

X ,V .

It is characterized by

ϕ−1X ,V (]X [V ′) = (X ,V ′)

if V ′ is an open domain in V .

If F is an overconvergent sheaf over T and (X ,V ) anoverconvergent variety over T , then the realization of F on (X ,V )is

FX ,V := ϕX ,V∗F/X ,V on ]X [V .

Realizations induce an equivalence between the category of sheaveson An†/T and compatible families of sheaves on ]X [V for all (X ,V )over T . it is given by

F(X ,V ) = Γ(]X [V ,FX ,V ).

Overconvergent site over a varietyIf (C ,O) is an overconvergent variety over V, the forgetful functor

An†/C ,O// Sch/C

(X ,V ) // X

is left exact and continuous giving rise to a morphism of toposes

I : Sch/C → An†/C ,O.

If X is a variety over C , we will write X/O := I∗X and consider theoverconvergent site An†/X/O over X/O.

It can be described as follows:

An object of An†/X/O is just an overconvergent variety (U,V ) over(C ,O) together with a fixed factorization U → X over C :

P PKoo Voo

XC // S SKoo O.oo

A morphism in An†/X/O is a morphism of overconvergent varieties(U ′,V ′)→ (U,V ) over (C ,O) where U ′ → U is defined over X :

U // P PKoo Voo

U ′ //

P ′ P ′Koo V ′oo

XC // S SKoo Ooo

Overconvergent modulesThe presheaf

O†V : (X ,V ) 7→ Γ(]X [V , i−1X OV )

is a sheaf of rings on An†V . Its realization on (X ,V ) is i−1X OV .

More generally, if T is an overconvergent presheaf on V, we mayconsider the restriction O†/T of the structural sheaf.

Now, any morphism (f , u) : (X ′,V ′)→ (X ,V ) of overconvergentvarieties over V will induce a morphism of ringed spaces

(u∗, u†) : (]X ′[V ′ , i−1X ′ OV ′)→ (]X [V , i−1

X OV )

withu†F := i−1

X ′ u∗iX∗F .

Giving an O†/T -module E is equivalent to giving the family ofi−1X OV -module EX ,V on ]X [V , for each (X ,V ) over T andcompatible transition maps

u†EX ,V → EX ′,V ′

for all morphism (f , u) : (X ′,V ′)→ (X ,V ).

We will call E a crystal if all these maps are isomorphisms. Notethat this is automatic if E is (locally) finitely presented.

We are interested into deriving the composed functor

pX/O : An†/X/O → An†/C ,O → ˜Open(]C [O).

and apply it to crystals.

Overconvergent isocrystals againAssume that we are given a morphism of overconvergent varieties(X ,V )→ (C ,O). We may consider the diagonal embedding

∆ : (X ,V )→ (X ,V 2).

as well as the projections

p1, p2 : (X ,V 2)→ (X ,V ).

If E is an overconvergent module on X/O, we may consider thetransition maps

p†2EX ,V → EX ,V 2 ← p†1EX ,V

If E is a crystal all these maps are isomorphisms.

Actually, if E is finitely presented, we obtain isomorphisms ofcoherent modules. In other words, EX ,V becomes anoverconvergent isocrystal on (X ,V )/O.

Theorem ([7])

LetX //

oo Voo

]X [V? _oo

C // S SKoo Ooo ]C [O? _oo

1. P is smooth over S in the neighborhood of X,2. the Zariski closure of X in P is proper over (the Zariski

closure of C in) S,3. V = PK ×SK O.

Then, we have

1. O/X/O−Modfp ' Isoc†(X ,V /O)

2. RpX/O∗E ' RprigEX ,V .

Note that the left hand side does not depend on V . In particular,we obtain as corollary, the independence theorem of Berthelot.

The arguments used to prove the second assertion (strong fibrationtheorem and overconvergent Poincaré lemma) may be used toprove the first one. But these arguments require to work locally forthe Tate topology of V (and the Zariski topology of X ).Alternatively, one can use the following result.

Theorem (Covering)

In the situation of the above theorem, (X ,V ) is a covering of X/O.

Proof.The idea is to build local sections.

Essential surjectivity in assertion 1 follows directly from thistheorem. And so does Full faithfulness as we shall see right now.

Proof (full faithfulness).

It follows from the covering theorem that, if E is finitely presented,we have

pX/O∗E ' p∗E∇=0X ,V .

Also, one easily checks that

Hom(E ,F )X ,V = Homi−1X OV

(EX ,V ,FX ,V )

for finitely presented modules.

Thus, we get

pX/O∗Hom(E ,F ) = p∗Hom∇(EX ,V ,FX ,V )

and taking global sections gives

Hom(E ,F ) = Hom∇(EX ,V ,FX ,V ).

Proof. (Assertion 2)

We consider the de Rham complex of EX ,V

F• = EX ,V ⊗i−1X OV

i−1X Ω•V

and the derived linearization operator

RLF• = RjX ,V∗ϕ∗X ,VF•.

with jX ,V : An†/X ,V → An†X/O. It is completely formal to checkthat

RpX/O∗RLF• ' Rp∗F• (= RprigEX ,V ).

In order to finish the proof of the theorem, it is therefore sufficientto verify that

E ' RLF•.

This can be checked on any (Y ,W ) over X/O.

Proof. (continuiing)

We use the diagonal embedding

Y // Q QKoo Woo

Q × Pp2

QK × PK

W × Voo

X // P PKoo Voo

One easily computes

(RLF•)Y ,W = Rp1∗p†2F•.

Thus, we have to show that EY ,W = Rp1∗p†2F•.

Proof. (the end)

We havep†2F

• = p†2EX ,V ⊗i−1Y OW×V

i−1X Ω•W×V/W

= p†1EY ,W ⊗i−1Y OW×V

i−1X Ω•W×V/W

because E is a crystal.

Therefore, we just have to show that if G is a coherenti−1Y OW -module, the canonical map

G → Rp1∗p†1G ⊗i−1Y OW×V

i−1X Ω•W×V/W

is a quasi-isomorphism and apply it to the case G = EY ,W .

Our theorem will therefore follow from the overconvergent Poincarélemma below. More precisely, we will apply it to the embedding ofY into the product Q × P and to the first projection.

Theorem (Overconvergent Poincaré Lemma)

LetP ′

P ′K

oo V ′oo

P PKoo Voo

1. v is smooth in the neighborhood of X,2. v induces a proper morphism from the Zariski closure of X in

P ′ to (the Zariski closure of X in) P,3. V ′ = P ′K ×PK V .

If F is a coherent i−1X OV -module, there is a quasi-isomorphism

F ' Ru∗u†F ⊗i−1X OV ′

i−1X Ω•V ′/V .

Proof.The question is local for the Zariski topology on X and the Tatetopology on V . Therefore, using the strong fibration theorem, wemay assume that

(X ,V ′) = (X ,BnV (0, 1+)).

Using an induction argument, we may actually reduce to the casen = 1.

The theorem therefore follows from its local one dimensionalversion below.

Theorem (Local Overconvergent Poincaré lemma)

LetX → P ← PK ← V

be an overconvergent variety. Denote by X the Zariski closure of Xin P. Assume that

1. P is affine2. The complement of X in X is a hypersurface3. V is affinoid4. We have ]X [V = V

Then, if F is a coherent i−1X OV -module, we have

Γ(]X [V ,F) ' RΓ(]X [V×D(0, 1−), p†F ∂/∂t→ p†F).

Proof.Shrinking V if necessary, we may assume that F = i−1

V M withMcoherent on V . If

X = x ∈ X , g(x) 6= 0

we will write for λ < 1,

V λ = x ∈ V , |g(x)| ≥ λ.

Then, the LHS of the equality is

M = lim−→Mλ with Mλ = Γ(V λ,M).

Now, we set for ηk<→ 1,

Mk := Γ(]X [V×D(0, η+k ), p∗M).

We have Mk = lim−→Mλk and Mλ

k := Mλ⊗Kt/ηk.

Proof. (continuing)

Now, one can check that the RHS of the equality is thecohomology of the bicomplex

∂ //

Mk∂ // ∏Mk

withd(sk) = sk+1 − sk with ∂(sk) = ∂/∂t(sk)

Then, we consider the integration map∫: Mλ

k// Mλ

t i // t i+1

and extend it to∫

Mk →∏

Proof. (the end)

One easily checks that

∫ d and ∂

∫ ∂ + constant = Id + d .

It follows that the bicomplex has no cohomology except in degree0 where we get M. And we are done.

We can check the middle part for example: given s, s ′ ∈∏

Mksuch that ∂(s) = d(s ′), we have to find s ′′ ∈

∏Mk such that

d(s ′′) = s and ∂(s ′′) = s ′. If s has no constant term, we may justchoose s ′′ =

∫s ′ − s. When s is constant and s ′ = 0, we may just

set s ′′ = h(s) where h is the section defined by

0 // M // MN d // MN // 0.(−∑k−1

0 si ) (sk)hoo

ConclusionWe can finally give a definition of rigid cohomology andoverconvergent isocrystals that is completely independent of thechoices.

DefinitionLet X be an algebraic variety over C , C → S a formal embeddingand O → SK any morphism of analytic varieties with O good.

1. An overconvergent isocrystal on X/O is a finitely presentedmodule on the ringed site An†/X/O.

2. If E is an overconvergent isocrystal on X/O, its rigidcohomology is RprigE := RpX/O∗E .

In other words, these objects (overconvergent isocrystals and rigidcohomology) do exist and we can compute them the way we want !

– Thank you –

Pierre Berthelot and Arthur Ogus.Notes on crystalline cohomology.Princeton University Press, Princeton, N.J., 1978.

Pierre Berthelot and Arthur Ogus.F -isocrystals and de Rham cohomology. I.Invent. Math., 72(2):159–199, 1983.

Jean-Yves Étesse and Bernard Le Stum.Fonctions L associées aux F -isocristaux surconvergents. I.Interprétation cohomologique.Math. Ann., 296(3):557–576, 1993.

Masaki Kashiwara and Pierre Schapira.Sheaves on manifolds, volume 292 of Grundlehren derMathematischen Wissenschaften [Fundamental Principles ofMathematical Sciences].Springer-Verlag, Berlin, 1990.With a chapter in French by Christian Houzel.

Kiran S. Kedlaya.Counting points on hyperelliptic curves usingMonsky-Washnitzer cohomology.J. Ramanujan Math. Soc., 16(4):323–338, 2001.

Bernard Le Stum.Rigid cohomology, volume 172 of Cambridge Tracts inMathematics.Cambridge University Press, Cambridge, 2007.

Bernard Le Stum.The overconvergent site.Mémoires de la SMF, 127, 2011.James S. Milne.Étale cohomology, volume 33 of Princeton MathematicalSeries.Princeton University Press, Princeton, N.J., 1980.

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