p-adic cohomology in characteristic p - Fakultät für … · work in progress by Jean-Marc Fontaine and Uwe Jannsen Regensburg, ... 2prM g, f : Mr!Mr+1 the multiplication by p, v

A new p-adic cohomology theoryin characteristic p

work in progress by Jean-Marc Fontaine and Uwe Jannsen

Regensburg, 17 February 2011

Conference Arithmetic and Motivic Algebraic Geometry

Uwe Jannsen (University of Regensburg) p-adic cohomology in characteristic p 17 February 2011 1 / 27

Principal aims

Development of a cohomology theory in characteristic pwhich refines the crystalline cohomology

–and works well for torsion.

Development of a sheaf theory which generalizes theDieudonné theory

–and works well for torsion.


Gauges and ϕ-gauges over a perfect field

k perfect field of char. p; Wn, W∞ = W Witt vectors of k .

Let Dn = Wn[f , v ]/(fv − p) as a graded commutative ring withdeg f = 1 and deg v = −1.Definition: A Wn-gauge is a graded Dn-module.

explicitly: · · · M rfv

M r+1 . . . where fv = p = vf .

Definition: A ϕ-Wn-gauge is a gauge with a σ-linearisomorphism

ϕ : M∞ := lim−→f

M r ∼=−→ M−∞ := lim−→v

M r .

Morphisms are morphisms of graded Dn-modules respecting ϕ.


Finite type gaugesCall a Wn-gauge of finite type, if it finitely generated asDn-module. For integers a ≤ b call a finite type gaugeconcentrated in the interval [a,b], if f is an isomorphism forn ≥ b and v is an isomorphism for n ≤ a.This category is equivalent to the category of finite diagrams

Mafv

Ma+1 . . .Mb−1fv

Mb .

Here M∞ = Mb and M−∞ = Ma.

Any finite type gauge is concentrated in some interval. Call agauge effective if it is concentrated in [0,∞].Remark: The category of finite type ϕ-W -gauges concentratedin [0,1] is equivalent to the category of Dieudonné modules:

M0fv

M1 ϕ→M0σ ! M0 mit F = ϕf und V = vϕ−1.


The prototype

Let M be an F-crystal, i.e., a finitely generated free W -modulewith an injective σ-linear endomorphism ϕ : M −→ M.We get an associated ϕ-W -gauge as follows:

M r = m ∈ M|ϕ(m) ∈ pr M ,f : M r → M r+1 the multiplication by p,v : M r+1 → M r the inclusion.

There is an N ≥ 0 with pNM ⊂ ϕ(M). Then ϕ(MN) = M, and

M = M0 ∼→ M−∞ and MN ∼→ M∞ .

Define a σ-linear isomorphism ϕ : M∞ −→ M−∞ by the maps

ϕr : M r −→ M , ϕr (m) = p−rϕ(m).


F -crystals and ϕ-gauges

This gives a full embedding of categories

F -crystals → finite type free effective ϕ-W -gauges .

More generally we have a full embedding of categories

virtual F crystals → finite type free ϕ-W -gauges .

Here a virtual F -crystal is an F -isocrystal D (i.e., afinite-dimensional vector space over K0 = Frac(W ) with aσ-linear isomorphism ϕ : D → D) together with a W -latticeM ⊂ D. The associated ϕ-W -gauge is literally defined as above.The above embeddings are not surjective on objects. Thus thecategory of all ϕ-W -gauges is much bigger than the category ofF -crystals.


Some constructions and definitions forϕ-gaugesReduction: If m < n ≤ ∞, and M is a ϕ-Wn-gauge, then we getan induced ϕ-Wm-gauge M ⊗Wn Wm = M/pm.For r ∈ Z define the Tate ϕ-Wn-gauge Wn(r) as the gauge whichis Wn concentrated in the place −r (i.e., the interval [−r ,−r ]):

. . .pid

Wn

pid

Wnidp

Wnidp. . .

with the middle Wn place in degree −r .

Definition Call a Wn-gauge M(a) strict if (fr , vr ) : M r → M∞ ⊕M−∞ is injective for all r ,

(b) quasi-rigid if M r f→M r+1 v→M r f→M r+1 is exact for all r ,(c) rigid if it is strict and quasi-rigid.


Reductions of virtual F -crystals and freeϕ-W -gauges

Theorem1. For a finite type ϕ-W1-gauge M the following are equivalent:(a) M is the reduction of a virtual F -crystal.(b) M is strict.(c) As a gauge, M is the direct sum of gauges W1(ri).(d) M corresponds to an F-zip (see below).2. If M is the reduction of a free ϕ-W-gauge, then M isquasi-rigid.

Hence not all ϕ-W1-gauges come via reduction from virtualF -crystals or free ϕ-W -gauges.


Algebraic propertiesOne has tensor products of Wn-gauges and ϕ-Wn-gauges: Thetensor product of graded modules M and N over a graded ring Ais given by

(M ⊗A N)n = ( ⊕i+j=n

M i ⊗A0 N j)/Un ,

where Un is the sub-A0-module generated by the elementsλx ⊗ y − x ⊗ λy for x ∈ M i , y ∈ N j and λ ∈ Ak with i + j + k = n.One easily checks that for Wn-gauges M and N one hascanonically

(M ⊗ N)∞ ∼= M∞ ⊗Wn N∞ and (M ⊗ N)−∞ ∼= M−∞ ⊗Wn N−∞ .

So for ϕ-gauges M and N the gauge M ⊗N becomes a ϕ-gaugeby ϕM ⊗ ϕN : M∞ ⊗ N∞ → M−∞ ⊗ N−∞.Similarly one has internal Homs, and (Tate) twists M(m) by(M(m))n = Mn+m. In fact, M(m) = M ⊗Wn(m).


The sheaves Ocrisn

A morphism of schemes f : X → Y is called syntomic, if it is flatand locally a complete intersection (factors locally as X i→Z p→Ywhere p is smooth and i is a regular immersion). The syntomictopology is the topology generated by these morphisms.Work of Fontaine and Messing: For n ≥ 1 let Ocris

n be thepresheaf on the site Xsyn of all syntomic schemes over k definedby the crystalline cohomology:

Ocrisn (X ) = H0

cris(X/Wn(k),OX/Wn) .

Then one has canonical isomorphisms for all r ≥ 0

H r (X ,Ocrisn ) := H r (Xsyn,Ocris

n ) ∼= H rcris(X/Wn,OX/Wn) ,

i.e., the sheaves Ocrisn compute the crystalline cohomology.


Explicit description of the sheaves Ocrisn

There is a rather explicit description of the sheaves Ocrisn : For an

affine syntomic scheme Spec(A) let Wn(A) be the ring ofn-truncated Witt vectors, and let W DP

n (A) be the divided powerenvelope of W (A) w.r.t. the ideal I = (a0, . . . ,an)|apn

0 = 0,compatible with the standard divided powers on pWn. Thenthere is a canonical isomorphism of sheaves

W DPn

∼−→Ocrisn ,

W DPn = the sheaf associated to the presheaf A 7→W DP

n (A).From both description it is clear that one has a canonicalFrobenius morphism ϕ : Ocris

n → Ocrisn .


The sheaves I [r ]1

Lemma/Definition (a) There is an epimorphism of ring sheavesOcris

1 → O, given in terms of Witt vectors by (a0) 7→ ap0.

(b) Its kernel, denoted I [1]1 , is a PD-ideal. Its r -th DP-power is

denoted I [r ]1 .(c) There is a natural monomorphism O → Ocris

1 . Thecomposition Ocris

1 O → Ocris1 is ϕ, the Frobenius on Ocris

1 .(d) The composition O → Ocris

1 O is the Frobenius on O.


The universal gauge Gr

Lemma One has a canonical exact sequence for m,n ≥ 0

0→ Ocrism → Ocris

m+n → Ocrisn → 0 ,

where the map Ocrism+n → Ocris

n is the natural projection, andOcris

m → Ocrism+n is induced by the multiplication by pn. Thus we

have a torsion free pro-sheaf

Ocris = (Ocrisn ) ,

with the projections as transition maps. We can now apply theconstruction of the prototype ϕ-gauge and define

Gr = ker(Ocris ϕ→Ocris → Ocrisr = Ocris/prOcris) .

(The sub-pro-sheaf on which the Frobenius is divisible by pr .)Uwe Jannsen (University of Regensburg) p-adic cohomology in characteristic p 17 February 2011 13 / 27

The universal gauges Grn

By reduction modulo pn we obtain a gauge (Grn).

There is also a more direct description of Grn: For any m ≥ n + r

defineGr

m = ker(Ocrism

ϕ→Ocrism → Ocris

r )

and then let Grn = Gr

m/pr .

One shows that this is independent of m ( ≥ n + r ), and that onehas exact sequences

0→ Grm → Gr

m+n → Grn → 0 .


The gauges Grn are ϕ-gauges

One can define a σ-linear Frobenius ϕ : G∞ → G−∞ = G0, byletting “ϕr = p−rϕ” on Gr , similarly for (Gr

n). It follows from workof Fontaine and Messing that ϕ is an isomorphism, so that weget ϕ-gauges:In fact, this is easily reduced to the case of (Gr

1), and here oneargues as follows. Let FrOcris

1 = im(ϕr : Gr1 → Ocris

1 ).Theorem (’Cartier isomorphism’; Fontaine, Messing) ϕr inducesan isomorphism

I [r ]1 /I[r+1]1

∼−→FrOcris1 /Fr−1Ocris

1 .

This implies the surjectivity of ϕ, i.e., Ocris1 = ∪n≥0Fn, and also

the injectivity of ϕ.


Gauge cohomology

Theorem (Fontaine, J.)For any n, there is a cohomology theory with values in ϕ-Wn-gauges, (H i(−,Wn)r , f , v), so that the following holds for smoothvarieties X over k:a. For each i, the ϕ-gauge (H i(X ,Wn)r ) is concentrated in theinterval [0, i ] and vanishes for i > 2d where d = dim(X ).b. H i(X ,Wn)0 = H i

cris(X ,Wn).c. For X connected, smooth and proper of dimension d thegauges are of finite type, and the Poincaré duality for crystallinecohomology extends to a perfect duality of gauges

(H i(X ,Wn)r )× (H2d−i(X ,Wn)r ) −→ (H2d (X ,Wn)r )∼→Wn(−d), .


Comments 1

1. The cohomology theory is simply obtained by definingH i(X ,Wn)r = H i

syn(X ,Grn) and using transport of structure. Then

statement b. is clear.2. The other results are obtained by projecting to the Zariski siteand reducing to the case n = 1. If

α : Xsyn −→ XZar

is the morphism of sites, it follows from results of Berthelot oncrystalline cohomology that one has

Rα∗I[r ]1 = Ω≥r

X

(the upper part, starting with ΩrX , of the de Rham complex).


Comments 2Then one uses induction and the following exact sequences:

0→ FrOcris1 → Gr+1

1v→Gr

1 → FrOcris1 → 0

0→ I [r ]1 → Gr−11

f→Gr1 → I [r ]1 → 0

Gr1

f→Gr+11

v→Gr1

f→Gr+1 .

The last exactness is the quasi-rigidity of (Gr1); in fact, this gauge

is also rigid.These sequences can also be used to prove that thecohomology of the Gr

n can be computed by much coarsertopologies, e.g., the so-called quasi-étale (quiet) topology orsimply the p-th root topology, generated by open immersionsand extraction of p-th roots (conjectured by Fontaine, proved byF. Schnellinger).


Examples 1

1. On H1(X ,Wn) one obtains just a Dieudonné module, see theremark above.2. (H2(X ,Wn)r ) is a ϕ-gauge in the interval [0,2]:

H2,0 H2,1 H2,2 ϕ→H2,0.

This is already a finer structure than the ‘generalized Dieudonnémodule’ H2,0 = H2

cris(X ,Wn) with F (= ϕ f 2) and V (= v2 ϕ−1)satisfying FV = p2 = VF .


Examples 23. For X smooth of dimension 2, the projection (Gr

1 = Rα∗Gr1) of

the universal gauge to the Zariksi site is given by the diagram

G01

fv

G11

fv

G21

of complexes (where C is the Cartier operator and Z Ω1σ = kerd)

Ω20id

Ω20C

Ω2σ

↑ d ↑ dC ↑ d

Ω10C

Z Ω1σ

incl0

Ω1σ

↑ d ↑ d ↑ d

OC−1=F0

Oσid0Oσ


Relation with Fontaine-Laffaille theory 1

We note that, for a W -gauge M, one has

M∞ = M/(f − 1)M and M−∞ = M/(v − 1)M .

Now consider W [v ] = polynomial ring with v in degree −1, andthe category of graded W [v ]-modules with σ-linear isomorphism

ϕ : M/(v − p)M ∼−→M/(v − 1)M .

If M is p-torsion and of finite type, it is concentrated in aninterval [a,b] in a similar way as above. The bijectivity of ϕshows that v is injective, and so one can identify M r , fora ≤ r ≤ b, with a submodule of Ma (via v r−a), and one getssemi-linear maps ϕr : M r → Ma satisfying ϕr = pϕr+1.Moreover, Ma is the sum of the images of the ϕr .


Relation with Fontaine-Laffaille theory 2Thus the considered category is the one of filtered ϕ-modulesconsidered by Fontaine and Laffaille, which is equivalent to thecategory of ‘crystalline’ GK0-Zp-representations, K0 = Frac(W ).Now we have a morphism of graded ringsW [v ]→ D = W [v , f ]/(fv − p) into the ring we considered up tonow, and for an object M as above the D-module

M = D ⊗W [v ] M

has a natural grading. Moreover the morphisms

M/(v − 1)M → M/(v − 1)M and M/(v − p)M → M/(f − 1)M

induced by the map x 7→ 1⊗ x are isomorphisms (note thatv − p = (f − 1)v−1). Hence M becomes a ϕ-W -gauge, which isalso of finite type and concentrated in [a,b].


Relation with Fontaine-Laffaille theory 3

Thus we have a ‘special fiber functor’ (of abelian categories)from finite type crystalline Zp-representations over K0 toϕ-W -gauges.

To be checked: That, for a smooth and proper scheme X overW the functor transports H i(XK0 ⊗K0 K0,Zp) to the ϕ-W -gauge(H i(Xk ,W )r ).


Relation with F -zips and displays 1

Note that ϕ-gauges can be defined over any scheme S ofcharacteristic p, in a more or less obvious way.Definition (Moonen and Wedhorn) Let S be a scheme over Fp.An F -zip over S is a tuple M = (M,C•,D•, ϕ•) where(a) M is a locally free sheaf of OS-modules of finite type.(b) C• is a descending chain in M(σ) such that the quotients arelocally free.(c) D• is an ascending chain in M with locally free quotients.(d) (ϕi)i∈Z is a family of isomorphisms of OS-modules

ϕi : C i/C i+1 −→ Di/Di−1

such that ... (some obvious commutations and one furtherassumption).


Relation with F -zips and displays 2

Proposition (Fontaine/F. Schnellinger) There is a fully faithfulembedding of the category of F -zips into the category ofϕ-OS-gauges. The essential image consist of rigid locally freegauges of finite type.

It seems also that the category of (generalized) displays, asdefined by Langer and Zink, can be embedded into the categoryof ϕ-W (OS)-gauges. This was proved for perfect rings OS (M.Wid). The image should be the gauges which are locally free.


Sheaves and generalized Dieudonné theoryThe idea is to define certain syntomic sheaves which shouldcorrespond to the smooth Z` sheaves for ` 6= p, by usingϕ-gauges.Let M be a ϕ-W -gauge. Then we get an associated syntomicsheaf F(M) on the big site of Spec(k) by defining

F(M) = Hom(1,G• ⊗M) .

Here the Hom is the sheaf Hom of sheaves of gauges.Moreover, 1 is the trivial constant sheaf of ϕ-gauges, M standsfor the constant sheaf of gauges associated to M, and thetensor product is the tensor product of sheaves of ϕ-gauges.More explicitely one has

F(M) = ker((G ⊗M)0 ϕ−v−→(G ⊗M)−∞) .


Outlook

It is hoped that this operation gives a full embedding in thesetting of the derived categories. But one can show that onegets a full embedding without deriving after restricting to gaugesin smaller intervals.

Moreover, there should be a

Projection formula For a ϕ-Wn-gauge M in the interval [−∞,0]and π : X → Spec(k) smooth and proper, one has

Rπ∗π∗F(M) = RF∗(M⊗ Rπ∗Gn,X ) .


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p-adic cohomology in characteristic p - Fakultät für … · work in progress by Jean-Marc Fontaine and Uwe Jannsen Regensburg, ... 2prM g, f : Mr!Mr+1 the multiplication by p, v