Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their...

p-adic Hodge theory

Peter Scholze

Algebraic GeometrySalt Lake City

Fontaine’s construction

Let C be an algebraically closed complete extension of Qp.

ThenFrobenius Φ is surjective on OC/p. Have Fontaine’s field

C [ = Frac(lim←−Φ

OC/p) , O[C = lim←−Φ

OC/p .

Then C [ is a complete algebraically closed complete field ofcharacteristic p, with ring of integers O[C .

Example

If C = Qp, then C [ = Fp((t)), where t corresponds to(p, p1/p, . . .).


Let C be an algebraically closed complete extension of Qp. ThenFrobenius Φ is surjective on OC/p.

Have Fontaine’s field



OC/p .


Example



Let C be an algebraically closed complete extension of Qp. ThenFrobenius Φ is surjective on OC/p. Have Fontaine’s field



OC/p .


Example






OC/p .

Then C [ is a complete algebraically closed complete field ofcharacteristic p,

with ring of integers O[C .

Example






OC/p .


Example






OC/p .


Example

If C = Qp, then C [ = Fp((t)),

where t corresponds to(p, p1/p, . . .).





OC/p .


Example



Fact. There is an identification of multiplicative monoids

C [ = lim←−x 7→xp

C .

This gives rise to a continuous multiplicative mapC [ → C : x 7→ x ]. For example, t] = p, and

(1 + t)] = limn→∞

(1 + p1/pn)pn.

The map C [ → C : x 7→ x ] is analytic, highly non-algebraic.




C .

This gives rise to a continuous multiplicative mapC [ → C : x 7→ x ].

For example, t] = p, and

(1 + t)] = limn→∞

(1 + p1/pn)pn.





C .

This gives rise to a continuous multiplicative mapC [ → C : x 7→ x ]. For example, t] = p,

and

(1 + t)] = limn→∞

(1 + p1/pn)pn.





C .

This gives rise to a continuous multiplicative mapC [ → C : x 7→ x ]. For example, t] = p, and

(1 + t)] = limn→∞

(1 + p1/pn)pn.


Geometry over C vs. geometry over C [

Example (The affine line A1 with coordinate T .)

Claim:A1C [ ≈ lim←−

T 7→T p

A1C .

On points this is the identification


C .

As x 7→ x ] is non-algebraic, need to formalize this in an analyticworld.

Perfectoid Algebras

In algebraic geometry, spaces defined by their sheaves of functions.

In analytic geometry, these rings are Banach algebras.

DefinitionLet L = C or L = C [ (or any perfectoid field). A perfectoidL-algebra is a Banach L-algebra R such that the subring R ⊂ R ofpowerbounded elements is bounded, andΦ : R/p → R/p : x 7→ xp is surjective.

Example

R = C 〈T 1/p∞〉, convergent power series in variable T and all itsp-power roots. This R gives functions on (an open subset of)lim←−T 7→T p A1

C .

Note: If L = C [ is of characteristic p, last condition is equivalentto requiring R perfect.

Perfectoid Algebras

In algebraic geometry, spaces defined by their sheaves of functions.In analytic geometry, these rings are Banach algebras.


Example


C .


Perfectoid Algebras


DefinitionLet L = C or L = C [ (or any perfectoid field). A perfectoidL-algebra is a Banach L-algebra R such that the subring R ⊂ R ofpowerbounded elements is bounded,

andΦ : R/p → R/p : x 7→ xp is surjective.

Example


C .


Perfectoid Algebras



Example


C .


Perfectoid Algebras



Example

R = C 〈T 1/p∞〉, convergent power series in variable T and all itsp-power roots.

This R gives functions on (an open subset of)lim←−T 7→T p A1

C .


Perfectoid Algebras



Example


C .


Tilting for perfectoid algebras

Let R a perfectoid C -algebra. Define

R[ = lim←−Φ

R/p'← lim←−

x 7→xpR ,

and R[ = R[ ⊗OC[

C [.

Example

If R = C 〈T 1/p∞〉, then R[ = C [〈T 1/p∞〉.

Theorem (S., 2011)

The functor R 7→ R[ is an equivalence between the category ofperfectoid C -algebras and the category of perfectoid C [-algebras.

Perfectoid Spaces

p-adic analytic geometry:

I Tate’s rigid-analytic varieties (late 60’s)

I Berkovich’s analytic spaces (late 80’s)

I Huber’s adic spaces (early 90’s)

To perfectoid C -algebra R (+ R+ ⊂ R) can attach an ’affinoidperfectoid space’ X = Spa(R) of continuous valuations, equippedwith a structure sheaf OX .

Theorem (S., 2011)

Let X = Spa(R) and X [ = Spa(R[). The underlying topologicalspaces |X | ∼= |X [| are homeomorphic, OX is a sheaf of perfectoidC -algebras, with tilt OX [ .

Perfectoid Spaces






Theorem (S., 2011)

Let X = Spa(R) and X [ = Spa(R[).

The underlying topologicalspaces |X | ∼= |X [| are homeomorphic, OX is a sheaf of perfectoidC -algebras, with tilt OX [ .

Perfectoid Spaces






Theorem (S., 2011)

Let X = Spa(R) and X [ = Spa(R[). The underlying topologicalspaces |X | ∼= |X [| are homeomorphic,

OX is a sheaf of perfectoidC -algebras, with tilt OX [ .

Perfectoid Spaces






Theorem (S., 2011)

Let X = Spa(R) and X [ = Spa(R[). The underlying topologicalspaces |X | ∼= |X [| are homeomorphic, OX is a sheaf of perfectoidC -algebras,

with tilt OX [ .

Perfectoid Spaces






Theorem (S., 2011)


Perfectoid Spaces

Theorem (S., 2011)


Define general perfectoid spaces by gluing affinoid perfectoidspaces.

Corollary

The categories of perfectoid spaces over C and C [ are equivalent.

Example

The inverse limit

X = lim←−T 7→T p

A1C has tilt X [ = lim←−

T 7→T p

A1C [ .

|X [|...

T 7→T p∼=

∼= |X |...

T 7→T p

|A1

C [ |

T 7→T p∼=

|A1C |

T 7→T p

|A1

C [ | |A1C |

Example

The inverse limit



T 7→T p

A1C [ .

X [

...

T 7→T p

≈ X...

T 7→T p

A1C [

T 7→T p

A1C

T 7→T p

A1C [ A1

C

Example

The inverse limit



T 7→T p

A1C [ .

|X [|...

T 7→T p∼=

∼= |X |...

T 7→T p

|A1

C [ |

T 7→T p∼=

|A1C |

T 7→T p

|A1

C [ | |A1C |

Example

|X [|...

T 7→T p∼=

∼= |X |...

T 7→T p

|A1

C [ |

T 7→T p∼=

|A1C |

T 7→T p

|A1

C [ | |A1C |

Thus, homeomorphism of topological spaces (underlying adicspaces)

|A1C [ | ∼= lim←−

T 7→T p

|A1C | .

char p geometry as infinite covering of char 0 geometry.

The almost purity theorem

TheoremLet X be a perfectoid space over C.

There is an etale site Xet,such that Xet

∼= X [et under tilting.

There is the sheaf O+X ⊂ OX of functions bounded by 1.

TheoremThe global sections H0(Xet,O+

X ) = R+, and H i (Xet,O+X ) is almost

zero for i > 0, i.e., killed by p1/n ∈ OC for all n > 0.

This is closely related to Faltings’s celebrated “almost puritytheorem”.


TheoremLet X be a perfectoid space over C. There is an etale site Xet,

such that Xet∼= X [

et under tilting.







TheoremLet X be a perfectoid space over C. There is an etale site Xet,such that Xet












X ) = R+,

and H i (Xet,O+X ) is almost









zero for i > 0,

i.e., killed by p1/n ∈ OC for all n > 0.


The key computation

Let R = OC 〈T±1〉,

and X = Spa(R[1/p],R). Then X has anaffinoid perfectoid cover by X = Spa(R[1/p], R), where

R = OC 〈T±1/p∞〉 .

Proposition

H0(Xproet, O+X ) = R ,

H1(Xproet, O+X ) = Ω1

R/OC⊕ (p1/(p−1)−torsion) .

Here, Xproet is the pro-etale site, and O+X = lim←−O

+X /pn. One has

H i (Xproet, O+X ) = lim←−H i (Xet,O+

X /pn) .

The key computation

Let R = OC 〈T±1〉, and X = Spa(R[1/p],R).

Then X has anaffinoid perfectoid cover by X = Spa(R[1/p], R), where

R = OC 〈T±1/p∞〉 .

Proposition





+X /pn. One has


X /pn) .

The key computation

Let R = OC 〈T±1〉, and X = Spa(R[1/p],R). Then X has anaffinoid perfectoid cover by X = Spa(R[1/p], R),

where

R = OC 〈T±1/p∞〉 .

Proposition





+X /pn. One has


X /pn) .

The key computation

Let R = OC 〈T±1〉, and X = Spa(R[1/p],R). Then X has anaffinoid perfectoid cover by X = Spa(R[1/p], R), where

R = OC 〈T±1/p∞〉 .

Proposition





+X /pn. One has


X /pn) .

The key computation


R = OC 〈T±1/p∞〉 .

Proposition




Here, Xproet is the pro-etale site,

and O+X = lim←−O

+X /pn. One has


X /pn) .

The key computation


R = OC 〈T±1/p∞〉 .

Proposition





+X /pn.

One has


X /pn) .

The key computation


R = OC 〈T±1/p∞〉 .

Proposition





+X /pn. One has


X /pn) .

The key computation

The p1/(p−1)-torsion is sometimes called “junk torsion”.

In the lasttalk, we will see how to get rid of it.

Step 1. The Zp-cover X → X induces a map

H icont(Zp, R)→ H i (Xproet, O+

X ) .

This map is an almost isomorphism as H i (Xproet, O+X ) is almost

zero for i > 0.

Remains to compute

H icont(Zp,OC 〈T±1/p∞〉) =

⊕j∈Z[1/p]

H icont(Zp,OC · T j) .

The key computation

The p1/(p−1)-torsion is sometimes called “junk torsion”. In the lasttalk, we will see how to get rid of it.

Step 1. The Zp-cover X → X induces a map

H icont(Zp, R)→ H i (Xproet, O+

X ) .

This map is an almost isomorphism as H i (Xproet, O+X ) is almost

zero for i > 0.

Remains to compute

H icont(Zp,OC 〈T±1/p∞〉) =

⊕j∈Z[1/p]


The key computation

Step 2. Computation of


Fix a compatible system of p-power roots of unityζp, ζp2 , . . . ∈ OC . Then the generator γ = 1 ∈ Zp acts via

γ · T j = ζnpmT j , j = n/pm , n,m ∈ Z .

Then H icont(Zp,OC · T j) computed by the complex

(OC · T j γ−1−→ OC · T j) ∼= (OC

ζnpm−1−→ OC ) .

The key computation



Fix a compatible system of p-power roots of unityζp, ζp2 , . . . ∈ OC .

Then the generator γ = 1 ∈ Zp acts via



(OC · T j γ−1−→ OC · T j) ∼= (OC


The key computation



Fix a compatible system of p-power roots of unityζp, ζp2 , . . . ∈ OC . Then the generator γ = 1 ∈ Zp acts via



(OC · T j γ−1−→ OC · T j) ∼= (OC


The key computation

OC

ζnpm−1−→ OC .

Case 1. j ∈ Z, i.e., m = 0. Then ζnpm − 1 = 0, and so one gets OC

in degrees 0, 1. This contributes OC · T j in degrees i = 0, 1 to thefinal result.Case 2. j 6∈ Z. Then ζnpm − 1 divides p1/(p−1), and one gets no

H0, and p1/(p−1)-torsion in H1.End result.

H0(Xproet, O+X ) =

⊕j∈ZOCT j = R ,

H1(Xproet, O+X ) =

⊕j∈ZOCT j ⊕

⊕j=n/pm∈Z[1/p]\Z

(OC/(ζnpm − 1))T j

= Ω1R/OC

⊕ (p1/(p−1)−torsion) .

The key computation

OC


Case 1. j ∈ Z, i.e., m = 0.

Then ζnpm − 1 = 0, and so one gets OC



H0(Xproet, O+X ) =

⊕j∈ZOCT j = R ,

H1(Xproet, O+X ) =

⊕j∈ZOCT j ⊕



= Ω1R/OC


The key computation

OC



in degrees 0, 1.

This contributes OC · T j in degrees i = 0, 1 to thefinal result.Case 2. j 6∈ Z. Then ζnpm − 1 divides p1/(p−1), and one gets no


H0(Xproet, O+X ) =

⊕j∈ZOCT j = R ,

H1(Xproet, O+X ) =

⊕j∈ZOCT j ⊕



= Ω1R/OC


The key computation

OC



in degrees 0, 1. This contributes OC · T j in degrees i = 0, 1 to thefinal result.

Case 2. j 6∈ Z. Then ζnpm − 1 divides p1/(p−1), and one gets no


H0(Xproet, O+X ) =

⊕j∈ZOCT j = R ,

H1(Xproet, O+X ) =

⊕j∈ZOCT j ⊕



= Ω1R/OC


The key computation

OC



in degrees 0, 1. This contributes OC · T j in degrees i = 0, 1 to thefinal result.Case 2. j 6∈ Z.

Then ζnpm − 1 divides p1/(p−1), and one gets no


H0(Xproet, O+X ) =

⊕j∈ZOCT j = R ,

H1(Xproet, O+X ) =

⊕j∈ZOCT j ⊕



= Ω1R/OC


The key computation

OC




H0, and p1/(p−1)-torsion in H1.

End result.

H0(Xproet, O+X ) =

⊕j∈ZOCT j = R ,

H1(Xproet, O+X ) =

⊕j∈ZOCT j ⊕



= Ω1R/OC


The key computation

OC





H0(Xproet, O+X ) =

⊕j∈ZOCT j = R ,

H1(Xproet, O+X ) =

⊕j∈ZOCT j ⊕



= Ω1R/OC


Almost finite generation: Local case

DefinitionLet R be an OC -algebra.

An R-module M is called almost finitelygenerated if for any n ≥ 1, there is a finitely generated submoduleMn ⊂ M such that M/Mn is killed by p1/n.

Corollary

Let R = OC 〈T±11 , . . . ,T±1

d 〉, X = Spa(R[1/p],R). Then

H i (Xproet, O+X ) is an almost finitely generated R-module for all

i ≥ 0, is almost zero for i > d, and

H i (Xproet, O+X ) = Ωi

R/OC⊕ (pi/(p−1)−torsion) .

For the proof, redo the computation in any dimension, or use theKunneth formula.


DefinitionLet R be an OC -algebra. An R-module M is called almost finitelygenerated

if for any n ≥ 1, there is a finitely generated submoduleMn ⊂ M such that M/Mn is killed by p1/n.

Corollary

Let R = OC 〈T±11 , . . . ,T±1








DefinitionLet R be an OC -algebra. An R-module M is called almost finitelygenerated if for any n ≥ 1, there is a finitely generated submoduleMn ⊂ M

such that M/Mn is killed by p1/n.

Corollary

Let R = OC 〈T±11 , . . . ,T±1








DefinitionLet R be an OC -algebra. An R-module M is called almost finitelygenerated if for any n ≥ 1, there is a finitely generated submoduleMn ⊂ M such that M/Mn is killed by p1/n.

Corollary

Let R = OC 〈T±11 , . . . ,T±1









Corollary

Let R = OC 〈T±11 , . . . ,T±1

d 〉, X = Spa(R[1/p],R).

Then








Corollary

Let R = OC 〈T±11 , . . . ,T±1



i ≥ 0,

is almost zero for i > d, and






Corollary

Let R = OC 〈T±11 , . . . ,T±1



i ≥ 0, is almost zero for i > d,

and






Corollary

Let R = OC 〈T±11 , . . . ,T±1







Almost finite generation: Global case

TheoremLet X be a proper smooth rigid-analytic variety over C.

ThenH i (Xproet, O+

X ) is an almost finitely generated OC -module for alli ≥ 0, and almost zero for i > 2 dim X .

The proof is a version of the Cartan–Serre argument: Take two(nice) affinoid covers X =

⋃i∈I Ui =

⋃i∈I Vi such that Ui is

strictly contained in Vi .

The key point is that the transition maps

H j(Vi ,et,O+X /p)→ H j(Ui ,et,O+

X /p)

have almost finitely generated image (over OC ).


TheoremLet X be a proper smooth rigid-analytic variety over C. ThenH i (Xproet, O+

X ) is an almost finitely generated OC -module for alli ≥ 0,

and almost zero for i > 2 dim X .


⋃i∈I Ui =





X /p)






⋃i∈I Ui =





X /p)





The proof is a version of the Cartan–Serre argument:

Take two(nice) affinoid covers X =

⋃i∈I Ui =





X /p)






⋃i∈I Ui =





X /p)


Finiteness of Zp-cohomology

Corollary

Let X be a proper smooth rigid-analytic variety over C.

Thenatural map

H i (Xet,Zp)⊗Zp OC → H i (Xproet, O+X )

is an almost isomorphism. In particular, H i (Xet,Zp) is finitelygenerated, and vanishes for i > 2 dim X .

Enough to prove similar result for

H i (Xet,Fp)⊗Fp OC/p → H i (Xet,O+X /p) .

There, use Artin–Schreier sequence

0→ Fp → O+X /p → O+

X /p → 0 .


Corollary

Let X be a proper smooth rigid-analytic variety over C. Thenatural map


is an almost isomorphism.

In particular, H i (Xet,Zp) is finitelygenerated, and vanishes for i > 2 dim X .




0→ Fp → O+X /p → O+

X /p → 0 .


Corollary

Let X be a proper smooth rigid-analytic variety over C. Thenatural map


is an almost isomorphism. In particular, H i (Xet,Zp) is finitelygenerated, and vanishes for i > 2 dim X .




0→ Fp → O+X /p → O+

X /p → 0 .

The Hodge–Tate decomposition

Recall that we want to prove the Hodge–Tate decomposition, for aproper smooth rigid-analytic variety X0 over K with X = X0 ⊗K C ,

H i (Xet,Zp)⊗Zp C ∼=i⊕

j=0

H i−j(X ,ΩjX )(−j) .

At this point, we have an isomorphism

H i (Xet,Zp)⊗Zp C ∼= H i (Xproet, OX ) ,

where OX = O+X [1/p].


The local computation of the cohomology of OX in terms ofdifferentials gives the Hodge–Tate spectral sequence

E ij2 = H i (X ,Ωj

X )(−j)⇒ H i+j(Xproet, OX ) .

(The Tate twist is needed to make things Galois equivariant; itappears because we had to choose roots of unity in thecomputation.)If X = X0 ⊗K C , then Galois equivariance forces all differentials inthe spectral sequence to be 0. Moreover, there is a uniqueGalois-equivariant splitting of the resulting abutment filtration.This proves the Hodge–Tate decomposition.Remark. One can show that the Hodge–Tate spectral sequencedegenerates always, for a proper smooth rigid-analytic variety Xover C . However, it does not canonically split.



E ij2 = H i (X ,Ωj


(The Tate twist is needed to make things Galois equivariant;

itappears because we had to choose roots of unity in thecomputation.)If X = X0 ⊗K C , then Galois equivariance forces all differentials inthe spectral sequence to be 0. Moreover, there is a uniqueGalois-equivariant splitting of the resulting abutment filtration.This proves the Hodge–Tate decomposition.Remark. One can show that the Hodge–Tate spectral sequencedegenerates always, for a proper smooth rigid-analytic variety Xover C . However, it does not canonically split.



E ij2 = H i (X ,Ωj


(The Tate twist is needed to make things Galois equivariant; itappears because we had to choose roots of unity in thecomputation.)

If X = X0 ⊗K C , then Galois equivariance forces all differentials inthe spectral sequence to be 0. Moreover, there is a uniqueGalois-equivariant splitting of the resulting abutment filtration.This proves the Hodge–Tate decomposition.Remark. One can show that the Hodge–Tate spectral sequencedegenerates always, for a proper smooth rigid-analytic variety Xover C . However, it does not canonically split.



E ij2 = H i (X ,Ωj


(The Tate twist is needed to make things Galois equivariant; itappears because we had to choose roots of unity in thecomputation.)If X = X0 ⊗K C , then Galois equivariance forces all differentials inthe spectral sequence to be 0.

Moreover, there is a uniqueGalois-equivariant splitting of the resulting abutment filtration.This proves the Hodge–Tate decomposition.Remark. One can show that the Hodge–Tate spectral sequencedegenerates always, for a proper smooth rigid-analytic variety Xover C . However, it does not canonically split.



E ij2 = H i (X ,Ωj


(The Tate twist is needed to make things Galois equivariant; itappears because we had to choose roots of unity in thecomputation.)If X = X0 ⊗K C , then Galois equivariance forces all differentials inthe spectral sequence to be 0. Moreover, there is a uniqueGalois-equivariant splitting of the resulting abutment filtration.

This proves the Hodge–Tate decomposition.Remark. One can show that the Hodge–Tate spectral sequencedegenerates always, for a proper smooth rigid-analytic variety Xover C . However, it does not canonically split.



E ij2 = H i (X ,Ωj


(The Tate twist is needed to make things Galois equivariant; itappears because we had to choose roots of unity in thecomputation.)If X = X0 ⊗K C , then Galois equivariance forces all differentials inthe spectral sequence to be 0. Moreover, there is a uniqueGalois-equivariant splitting of the resulting abutment filtration.This proves the Hodge–Tate decomposition.

Remark. One can show that the Hodge–Tate spectral sequencedegenerates always, for a proper smooth rigid-analytic variety Xover C . However, it does not canonically split.



E ij2 = H i (X ,Ωj


(The Tate twist is needed to make things Galois equivariant; itappears because we had to choose roots of unity in thecomputation.)If X = X0 ⊗K C , then Galois equivariance forces all differentials inthe spectral sequence to be 0. Moreover, there is a uniqueGalois-equivariant splitting of the resulting abutment filtration.This proves the Hodge–Tate decomposition.Remark. One can show that the Hodge–Tate spectral sequencedegenerates always, for a proper smooth rigid-analytic variety Xover C . However, it does not canonically split.

Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their...

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