Peter Scholze Algebraic Geometry Salt Lake CityAlgebraic Geometry Salt Lake City. Classical Hodge...

p-adic Hodge theory

Peter Scholze

Algebraic GeometrySalt Lake City

Classical Hodge theory

Let X be a compact complex manifold.

(i) If X is Kahler, there is a natural Hodge decomposition

H i (X ,Z)⊗Z C ∼=i⊕

H i−j(X ,ΩjX ) .

(ii) There is a natural isomorphism

H i (X ,Z)⊗Z C ∼= H idR(X ) .

H i (X ,Z)⊗Z C ∼=i⊕

H i−j(X ,ΩjX ) .

H i (X ,Z)⊗Z C ∼= H idR(X ) .

H i (X ,Z)⊗Z C ∼=i⊕

H i−j(X ,ΩjX ) .

H i (X ,Z)⊗Z C ∼= H idR(X ) .

(iii) If X is Kahler, then the Hodge-de Rham spectral sequence

E ij1 = H j(X ,Ωi

X )⇒ H i+jdR (X )

degenerates at E1.

Examples

All projective smooth algebraic varieties over C give rise to Kahlercomplex manifolds.

Examples of non-Kahler complex manifolds:

The Hopf surface: Let q ∈ C∗, |q| < 1. Then

X := (C2 \ (0, 0))/qZ .

One has H1(X ,OX ) = C while H0(X ,Ω1X ) = 0, so Hodge

symmetry fails. However, the Hodge-de Rham spectral sequencedegenerates at E1.

Examples

X := (C2 \ (0, 0))/qZ .

Examples

The Hopf surface:

Let q ∈ C∗, |q| < 1. Then

X := (C2 \ (0, 0))/qZ .

Examples

X := (C2 \ (0, 0))/qZ .

Examples

X := (C2 \ (0, 0))/qZ .

symmetry fails.

However, the Hodge-de Rham spectral sequencedegenerates at E1.

Examples

X := (C2 \ (0, 0))/qZ .

Examples

The Iwasawa threefold:

1 ∗ ∗0 1 ∗0 0 1

be the unipotent subgroup of GL3. Then

X := N(C)/N(Z[i ]) .

The Hodge-de Rham spectral does not degenerate at E1; inparticular X is non-Kahler.

Examples

The Iwasawa threefold: Let

1 ∗ ∗0 1 ∗0 0 1

be the unipotent subgroup of GL3.

X := N(C)/N(Z[i ]) .

Examples

The Iwasawa threefold: Let

1 ∗ ∗0 1 ∗0 0 1

be the unipotent subgroup of GL3. Then

X := N(C)/N(Z[i ]) .

p-adic Hodge theory

Setup: Let K be a p-adic field, i.e. a discretely valued completenonarchimedean extension of Qp with perfect residue field k . (I.e.,K is a finite extension of W (k)[p−1].)

Let C = ˆK be the completed algebraic closure of K .

Theorem (Tate, 1967)

Let A/OK be an abelian variety. Then there is a naturalGal(K/K )-equivariant isomorphism

H1et(AC ,Zp)⊗Zp C ∼= H1(A,OA)⊗K C ⊕ H0(A,Ω1

A)(−1)⊗K C .

p-adic Hodge theory

Setup:

Let K be a p-adic field, i.e. a discretely valued completenonarchimedean extension of Qp with perfect residue field k . (I.e.,K is a finite extension of W (k)[p−1].)

A)(−1)⊗K C .

p-adic Hodge theory

Setup: Let K be a p-adic field, i.e. a discretely valued completenonarchimedean extension of Qp with perfect residue field k .

(I.e.,K is a finite extension of W (k)[p−1].)

A)(−1)⊗K C .

p-adic Hodge theory

A)(−1)⊗K C .

p-adic Hodge theory

Let A/OK be an abelian variety.

Then there is a naturalGal(K/K )-equivariant isomorphism

A)(−1)⊗K C .

p-adic Hodge theory

A)(−1)⊗K C .

p-adic Hodge theory: A conjecture

Question (Tate, 1967)

One can ask whether a similar Hodge-like decomposition exists forthe etale cohomology with values in C in all dimensions, for ascheme XC coming from a scheme X projective and smooth overOK , or perhaps even over K , or for suitable rigid-analytic spaces.

Such “Hodge-like” decompositions are now called Hodge-Tatedecompositions.

One can ask whether a similar Hodge-like decomposition exists forthe etale cohomology with values in C in all dimensions,

for ascheme XC coming from a scheme X projective and smooth overOK , or perhaps even over K , or for suitable rigid-analytic spaces.

One can ask whether a similar Hodge-like decomposition exists forthe etale cohomology with values in C in all dimensions, for ascheme XC coming from a scheme X projective and smooth overOK ,

or perhaps even over K , or for suitable rigid-analytic spaces.

One can ask whether a similar Hodge-like decomposition exists forthe etale cohomology with values in C in all dimensions, for ascheme XC coming from a scheme X projective and smooth overOK , or perhaps even over K ,

or for suitable rigid-analytic spaces.

p-adic Hodge theory: Results

Fontaine-Messing (1985): X projective smooth scheme overOK = W (k), p > dim X .

Faltings (1990): X proper smooth scheme over K .

Other proofs given by Tsuji, Niziol, Beilinson.

Rigid-analytic case remained open. Not even finiteness ofH i

et(XC ,Zp) was known!

Rigid-analytic varieties: Examples

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

Examples: analytification of proper smooth scheme over K , genericfibre of non-algebraic deformation of abelian variety or K3 surface(or other proper smooth scheme over k),

Hopf surface: If q ∈ K ∗, |q| < 1, let

X := (C 2 \ (0, 0))/qZ .

Thus, there exist “non-Kahler” rigid-analytic varieties. However,there is no p-adic analogue of the Iwasawa manifold. Namely, Chas no cocompact discrete subgroups like Z[i ] ⊂ C.

Examples: analytification of proper smooth scheme over K ,

genericfibre of non-algebraic deformation of abelian variety or K3 surface(or other proper smooth scheme over k),

X := (C 2 \ (0, 0))/qZ .

Thus, there exist “non-Kahler” rigid-analytic varieties.

However,there is no p-adic analogue of the Iwasawa manifold. Namely, Chas no cocompact discrete subgroups like Z[i ] ⊂ C.

X := (C 2 \ (0, 0))/qZ .

Thus, there exist “non-Kahler” rigid-analytic varieties. However,there is no p-adic analogue of the Iwasawa manifold.

Namely, Chas no cocompact discrete subgroups like Z[i ] ⊂ C.

X := (C 2 \ (0, 0))/qZ .

p-adic Hodge theory

Theorem (S., 2012)

(i) For all i ≥ 0, H iet(XC ,Zp) is a finitely generated Zp-module,

which is zero for i > 2 dim X .

(ii) There is a natural Gal(K/K )-equivariant Hodge–Tatedecomposition

H iet(XC ,Zp)⊗Zp C ∼=

i⊕j=0

H i−j(X ,ΩjX )(−j)⊗K C .

p-adic Hodge theory

Theorem (S., 2012)

i⊕j=0

p-adic Hodge theory

Theorem (S., 2012)

i⊕j=0

p-adic Hodge theory

Theorem (S., 2012)

(iii) There is a natural Gal(K/K )-equivariant de Rhamcomparison isomorphism

H iet(XC ,Zp)⊗Zp BdR

∼= H idR(X )⊗K BdR

preserving filtrations; in particular, H iet(XC ,Qp) is de Rham in

the sense of Fontaine.

(iv) The Hodge-de Rham spectral sequence

E ij1 = H j(X ,Ωi

X )⇒ H i+jdR (X )

degenerates at E1.

p-adic Hodge theory

Theorem (S., 2012)

E ij1 = H j(X ,Ωi

X )⇒ H i+jdR (X )

degenerates at E1.

p-adic Hodge theory

Theorem (S., 2012)

E ij1 = H j(X ,Ωi

X )⇒ H i+jdR (X )

degenerates at E1.

On finiteness for Zp-cohomology

Finiteness for Z`-cohomology, ` 6= p, known.

Idea: Choose formal model and use finiteness of nearby cycles.Works for any quasicompact separated smooth rigid-analyticvariety over K . (I.e., without properness.)

Method does not work for ` = p.In fact, if X closed unit disc over K , then H1

et(XC ,Fp)infinite-dimensional.Related to non-finiteness of H1

et(A1k,Fp), i.e. Artin-Schreier covers.

Need global argument, using finiteness of coherent cohomology tocontrol Artin-Schreier type covers.

Finiteness for Z`-cohomology, ` 6= p, known.Idea: Choose formal model and use finiteness of nearby cycles.

Works for any quasicompact separated smooth rigid-analyticvariety over K . (I.e., without properness.)

Finiteness for Z`-cohomology, ` 6= p, known.Idea: Choose formal model and use finiteness of nearby cycles.Works for any quasicompact separated smooth rigid-analyticvariety over K .

(I.e., without properness.)

Finiteness for Z`-cohomology, ` 6= p, known.Idea: Choose formal model and use finiteness of nearby cycles.Works for any quasicompact separated smooth rigid-analyticvariety over K . (I.e., without properness.)

Method does not work for ` = p.

In fact, if X closed unit disc over K , then H1et(XC ,Fp)

infinite-dimensional.Related to non-finiteness of H1

et(XC ,Fp)infinite-dimensional.

Related to non-finiteness of H1et(A1

k,Fp), i.e. Artin-Schreier covers.

Local structure of rigid-analytic varieties

Complex manifolds are locally contractible. In contrast,rigid-analytic varieties have large etale fundamental group, evenlocally. But at least no higher homotopy groups:

Theorem (S., 2012)

Let X be a connected affinoid rigid-analytic variety over C . ThenX is a K (π, 1) for p-torsion coefficients, i.e. for all p-torsion localsystems L on X ,

H iet(X ,L) ∼= H i

cont(π1(X , x),Lx) ,

where x ∈ X is a geometric base point, and π1(X , x) is theprofinite etale fundamental group.

Complex manifolds are locally contractible.

In contrast,rigid-analytic varieties have large etale fundamental group, evenlocally. But at least no higher homotopy groups:

Theorem (S., 2012)

Complex manifolds are locally contractible. In contrast,rigid-analytic varieties have large etale fundamental group, evenlocally.

But at least no higher homotopy groups:

Theorem (S., 2012)

Let X be a connected affinoid rigid-analytic variety over C .

ThenX is a K (π, 1) for p-torsion coefficients, i.e. for all p-torsion localsystems L on X ,

Theorem (S., 2012)

Let X be a connected affinoid rigid-analytic variety over C . ThenX is a K (π, 1) for p-torsion coefficients,

i.e. for all p-torsion localsystems L on X ,

Theorem (S., 2012)

where x ∈ X is a geometric base point,

and π1(X , x) is theprofinite etale fundamental group.

Theorem (S., 2012)

K (π, 1)’s, equal characteristic case

Also true for X over equal characteristic field k((t)).

Enough to consider L = Fp. Consider Artin-Schreier sequence

0→ Fp → OX → OX → 0 ,

exact sequence of sheaves on Xet. As X is affinoid,H i

et(X ,OX ) = 0 for i ≥ 1, get

H iet(X ,Fp) = 0

for i ≥ 2.

Also true for X over equal characteristic field k((t)). Easy:

0→ Fp → OX → OX → 0 ,

H iet(X ,Fp) = 0

for i ≥ 2.

Enough to consider L = Fp.

Consider Artin-Schreier sequence

0→ Fp → OX → OX → 0 ,

H iet(X ,Fp) = 0

for i ≥ 2.

0→ Fp → OX → OX → 0 ,

H iet(X ,Fp) = 0

for i ≥ 2.

0→ Fp → OX → OX → 0 ,

exact sequence of sheaves on Xet.

As X is affinoid,H i

H iet(X ,Fp) = 0

for i ≥ 2.

0→ Fp → OX → OX → 0 ,

et(X ,OX ) = 0 for i ≥ 1,

H iet(X ,Fp) = 0

for i ≥ 2.

0→ Fp → OX → OX → 0 ,

H iet(X ,Fp) = 0

for i ≥ 2.

Moreover, we get a long exact sequence

0→ H0et(X ,Fp)→ R → R → H1

et(X ,Fp)→ 0 ,

where R = H0(X ,OX ).

As exactness of Artin-Schreier sequenceneeds only finite etale covers, get same result for

H icont(π1(X , x),Lx) .

Moreover, we get a long exact sequence

0→ H0et(X ,Fp)→ R → R → H1

et(X ,Fp)→ 0 ,

where R = H0(X ,OX ). As exactness of Artin-Schreier sequenceneeds only finite etale covers, get same result for

H icont(π1(X , x),Lx) .

K (π, 1)’s, mixed characteristic case

Reduce to case of equal characteristic:

Use perfectoid spaces.

DefinitionA perfectoid C -algebra is a Banach C -algebra R such that thesubring of powerbounded elements R ⊂ R is bounded, and theFrobenius Φ : R/p → R/p is surjective.

Recall Fontaine’s field

C [ = R(C ) = Frac (lim←−Φ

OC/p) ∼= k(($)) .

Reduce to case of equal characteristic: Use perfectoid spaces.

OC/p) ∼= k(($)) .

DefinitionA perfectoid C -algebra is a Banach C -algebra R

such that thesubring of powerbounded elements R ⊂ R is bounded, and theFrobenius Φ : R/p → R/p is surjective.

OC/p) ∼= k(($)) .

DefinitionA perfectoid C -algebra is a Banach C -algebra R such that thesubring of powerbounded elements R ⊂ R is bounded,

and theFrobenius Φ : R/p → R/p is surjective.

OC/p) ∼= k(($)) .

Theorem (S., 2011)

Perfectoid C-algebras are equivalent to perfectoid C [-algebras.

There exist ’rigid-analytic varieties’ (more precisely: adic spaces)associated to perfectoid C-, resp. C [-, algebras, called affinoidperfectoid spaces over C , resp. C [. These categories areequivalent, X 7→ X [. In this situation, Xet

∼= X [et.

The functor from perfectoid C -algebras R to perfectoidC [-algebras R[ is given by Fontaine’s construction:

R[ = (lim←−φ

R/p)⊗lim←−φOC/p C [ .

Theorem (S., 2011)

Perfectoid C-algebras are equivalent to perfectoid C [-algebras.There exist ’rigid-analytic varieties’

(more precisely: adic spaces)associated to perfectoid C-, resp. C [-, algebras, called affinoidperfectoid spaces over C , resp. C [. These categories areequivalent, X 7→ X [. In this situation, Xet

∼= X [et.

R[ = (lim←−φ

Theorem (S., 2011)

Perfectoid C-algebras are equivalent to perfectoid C [-algebras.There exist ’rigid-analytic varieties’ (more precisely: adic spaces)

associated to perfectoid C-, resp. C [-, algebras, called affinoidperfectoid spaces over C , resp. C [. These categories areequivalent, X 7→ X [. In this situation, Xet

∼= X [et.

R[ = (lim←−φ

Theorem (S., 2011)

Perfectoid C-algebras are equivalent to perfectoid C [-algebras.There exist ’rigid-analytic varieties’ (more precisely: adic spaces)associated to perfectoid C-, resp. C [-, algebras,

called affinoidperfectoid spaces over C , resp. C [. These categories areequivalent, X 7→ X [. In this situation, Xet

∼= X [et.

R[ = (lim←−φ

Theorem (S., 2011)

Perfectoid C-algebras are equivalent to perfectoid C [-algebras.There exist ’rigid-analytic varieties’ (more precisely: adic spaces)associated to perfectoid C-, resp. C [-, algebras, called affinoidperfectoid spaces over C , resp. C [.

These categories areequivalent, X 7→ X [. In this situation, Xet

∼= X [et.

R[ = (lim←−φ

Theorem (S., 2011)

Perfectoid C-algebras are equivalent to perfectoid C [-algebras.There exist ’rigid-analytic varieties’ (more precisely: adic spaces)associated to perfectoid C-, resp. C [-, algebras, called affinoidperfectoid spaces over C , resp. C [. These categories areequivalent,

X 7→ X [. In this situation, Xet∼= X [

R[ = (lim←−φ

Theorem (S., 2011)

Perfectoid C-algebras are equivalent to perfectoid C [-algebras.There exist ’rigid-analytic varieties’ (more precisely: adic spaces)associated to perfectoid C-, resp. C [-, algebras, called affinoidperfectoid spaces over C , resp. C [. These categories areequivalent, X 7→ X [.

In this situation, Xet∼= X [

R[ = (lim←−φ

Theorem (S., 2011)

Perfectoid C-algebras are equivalent to perfectoid C [-algebras.There exist ’rigid-analytic varieties’ (more precisely: adic spaces)associated to perfectoid C-, resp. C [-, algebras, called affinoidperfectoid spaces over C , resp. C [. These categories areequivalent, X 7→ X [. In this situation, Xet

∼= X [et.

R[ = (lim←−φ

Theorem (S., 2011)

∼= X [et.

R[ = (lim←−φ

Theorem (S., 2011)

∼= X [et.

R[ = (lim←−φ

Want to show:

Any connected affinoid rigid-analytic variety X overC is a K (π, 1).

Proof.

I Find affinoid perfectoid space X over C , X → X an inverselimit of finite etale covers.(Iterate adjoining p-power roots of units.)

I Its tilt X [ is affinoid and lives in equal characteristic, hence isa K (π, 1) for p-torsion coefficients.

I As Xet∼= X [

et, also X is a K (π, 1) for p-torsion coefficients.

I As X → X is pro-finite etale, also X is a K (π, 1) for p-torsioncoefficients.

Want to show: Any connected affinoid rigid-analytic variety X overC is a K (π, 1).

Proof.

I As Xet∼= X [

Proof.

I Find affinoid perfectoid space X over C , X → X an inverselimit of finite etale covers.

(Iterate adjoining p-power roots of units.)

I As Xet∼= X [

Proof.

I Find affinoid perfectoid space X over C , X → X an inverselimit of finite etale covers.

(Iterate adjoining p-power roots of units.)

I As Xet∼= X [

Proof.

I As Xet∼= X [

Proof.

I As Xet∼= X [

Proof.

I As Xet∼= X [

Proof.

I As Xet∼= X [

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