p-adic Hodge theory
Peter Scholze
Algebraic Geometry
Salt Lake City
Fontaine’s construction
Let C be an algebraically closed complete extension of Qp .
Fontaine’s construction
Let C be an algebraically closed complete extension of Qp . Then
Frobenius Φ is surjective on OC /p.
Fontaine’s construction
Let C be an algebraically closed complete extension of Qp . Then
Frobenius Φ is surjective on OC /p. Have Fontaine’s field
C [ = Frac(lim OC /p) , OC[ = lim OC /p .
←
−
←
−
Φ
Φ
Fontaine’s construction
Let C be an algebraically closed complete extension of Qp . Then
Frobenius Φ is surjective on OC /p. Have Fontaine’s field
C [ = Frac(lim OC /p) , OC[ = lim OC /p .
←
−
←
−
Φ
Φ
Then C [ is a complete algebraically closed complete field of
characteristic p,
Fontaine’s construction
Let C be an algebraically closed complete extension of Qp . Then
Frobenius Φ is surjective on OC /p. Have Fontaine’s field
C [ = Frac(lim OC /p) , OC[ = lim OC /p .
←
−
←
−
Φ
Φ
Then C [ is a complete algebraically closed complete field of
characteristic p, with ring of integers OC[ .
Fontaine’s construction
Let C be an algebraically closed complete extension of Qp . Then
Frobenius Φ is surjective on OC /p. Have Fontaine’s field
C [ = Frac(lim OC /p) , OC[ = lim OC /p .
←
−
←
−
Φ
Φ
Then C [ is a complete algebraically closed complete field of
characteristic p, with ring of integers OC[ .
Example
c , then C [ = F\
If C = Q
p
p ((t)),
Fontaine’s construction
Let C be an algebraically closed complete extension of Qp . Then
Frobenius Φ is surjective on OC /p. Have Fontaine’s field
C [ = Frac(lim OC /p) , OC[ = lim OC /p .
←
−
←
−
Φ
Φ
Then C [ is a complete algebraically closed complete field of
characteristic p, with ring of integers OC[ .
Example
c , then C [ = F\
If C = Q
p
p ((t)), where t corresponds to
1/p
(p, p , . . .).
Fontaine’s construction
Fact. There is an identification of multiplicative monoids
C [ = lim C .
←
−p
x7→x
Fontaine’s construction
Fact. There is an identification of multiplicative monoids
C [ = lim C .
←
−p
x7→x
This gives rise to a continuous multiplicative map
C [ → C : x 7→ x ] .
Fontaine’s construction
Fact. There is an identification of multiplicative monoids
C [ = lim C .
←
−p
x7→x
This gives rise to a continuous multiplicative map
C [ → C : x 7→ x ] . For example, t ] = p,
Fontaine’s construction
Fact. There is an identification of multiplicative monoids
C [ = lim C .
←
−p
x7→x
This gives rise to a continuous multiplicative map
C [ → C : x 7→ x ] . For example, t ] = p, and
n
n
(1 + t)] = lim (1 + p 1/p )p .
n→∞
Fontaine’s construction
Fact. There is an identification of multiplicative monoids
C [ = lim C .
←
−p
x7→x
This gives rise to a continuous multiplicative map
C [ → C : x 7→ x ] . For example, t ] = p, and
n
n
(1 + t)] = lim (1 + p 1/p )p .
n→∞
The map C [ → C : x 7→ x ] is analytic, highly non-algebraic.
Geometry over C vs. geometry over C [
Geometry over C vs. geometry over C [
Example (The affine line A1 with coordinate T .)
Geometry over C vs. geometry over C [
Example (The affine line A1 with coordinate T .)
Claim:
A1C [ ≈ lim A1C .
←
−p
T 7→T
Geometry over C vs. geometry over C [
Example (The affine line A1 with coordinate T .)
Claim:
A1C [ ≈ lim A1C .
←
−p
T 7→T
On points this is the identification
C [ = lim C .
←
−p
x7→x
Geometry over C vs. geometry over C [
Example (The affine line A1 with coordinate T .)
Claim:
A1C [ ≈ lim A1C .
←
−p
T 7→T
On points this is the identification
C [ = lim C .
←
−p
x7→x
As x 7→ x ] is non-algebraic, need to formalize this in an analytic
world.
Perfectoid Algebras
In algebraic geometry, spaces defined by their sheaves of functions.
Perfectoid Algebras
In algebraic geometry, spaces defined by their sheaves of functions.
In analytic geometry, these rings are Banach algebras.
Perfectoid Algebras
In algebraic geometry, spaces defined by their sheaves of functions.
In analytic geometry, these rings are Banach algebras.
Definition
Let L = C or L = C [ (or any perfectoid field). A perfectoid
L-algebra is a Banach L-algebra R such that the subring R ◦ ⊂ R of
powerbounded elements is bounded,
Perfectoid Algebras
In algebraic geometry, spaces defined by their sheaves of functions.
In analytic geometry, these rings are Banach algebras.
Definition
Let L = C or L = C [ (or any perfectoid field). A perfectoid
L-algebra is a Banach L-algebra R such that the subring R ◦ ⊂ R of
powerbounded elements is bounded, and
Φ : R ◦ /p → R ◦ /p : x 7→ x p is surjective.
Perfectoid Algebras
In algebraic geometry, spaces defined by their sheaves of functions.
In analytic geometry, these rings are Banach algebras.
Definition
Let L = C or L = C [ (or any perfectoid field). A perfectoid
L-algebra is a Banach L-algebra R such that the subring R ◦ ⊂ R of
powerbounded elements is bounded, and
Φ : R ◦ /p → R ◦ /p : x 7→ x p is surjective.
Example
∞
R = C hT 1/p i, convergent power series in variable T and all its
p-power roots.
Perfectoid Algebras
In algebraic geometry, spaces defined by their sheaves of functions.
In analytic geometry, these rings are Banach algebras.
Definition
Let L = C or L = C [ (or any perfectoid field). A perfectoid
L-algebra is a Banach L-algebra R such that the subring R ◦ ⊂ R of
powerbounded elements is bounded, and
Φ : R ◦ /p → R ◦ /p : x 7→ x p is surjective.
Example
∞
R = C hT 1/p i, convergent power series in variable T and all its
p-power roots. This R gives functions on (an open subset of)
limT 7→T p A1C .
←
−
Perfectoid Algebras
In algebraic geometry, spaces defined by their sheaves of functions.
In analytic geometry, these rings are Banach algebras.
Definition
Let L = C or L = C [ (or any perfectoid field). A perfectoid
L-algebra is a Banach L-algebra R such that the subring R ◦ ⊂ R of
powerbounded elements is bounded, and
Φ : R ◦ /p → R ◦ /p : x 7→ x p is surjective.
Example
∞
R = C hT 1/p i, convergent power series in variable T and all its
p-power roots. This R gives functions on (an open subset of)
limT 7→T p A1C .
←
−
Note: If L = C [ is of characteristic p, last condition is equivalent
to requiring R perfect.
Tilting for perfectoid algebras
Tilting for perfectoid algebras
Let R a perfectoid C -algebra. Define
'
R [◦ = lim R ◦ /p ← lim R ◦ ,
←
−
←
−p
Φ
x7→x
Tilting for perfectoid algebras
Let R a perfectoid C -algebra. Define
'
R [◦ = lim R ◦ /p ← lim R ◦ ,
←
−
←
−p
Φ
and R [ = R [◦ ⊗OC [ C [ .
x7→x
Tilting for perfectoid algebras
Let R a perfectoid C -algebra. Define
'
R [◦ = lim R ◦ /p ← lim R ◦ ,
←
−
←
−p
Φ
x7→x
and R [ = R [◦ ⊗OC [ C [ .
Example
∞
∞
If R = C hT 1/p i, then R [ = C [ hT 1/p i.
Tilting for perfectoid algebras
Let R a perfectoid C -algebra. Define
'
R [◦ = lim R ◦ /p ← lim R ◦ ,
←
−
←
−p
Φ
x7→x
and R [ = R [◦ ⊗OC [ C [ .
Example
∞
∞
If R = C hT 1/p i, then R [ = C [ hT 1/p i.
Theorem (S., 2011)
The functor R 7→ R [ is an equivalence between the category of
perfectoid C -algebras and the category of perfectoid C [ -algebras.
Perfectoid Spaces
Perfectoid Spaces
p-adic analytic geometry:
Perfectoid Spaces
p-adic analytic geometry:
I
Tate’s rigid-analytic varieties (late 60’s)
Perfectoid Spaces
p-adic analytic geometry:
I
Tate’s rigid-analytic varieties (late 60’s)
I
Berkovich’s analytic spaces (late 80’s)
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)
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)
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 ’affinoid
perfectoid space’ X = Spa(R) of continuous valuations, equipped
with a structure sheaf OX .
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 ’affinoid
perfectoid space’ X = Spa(R) of continuous valuations, equipped
with a structure sheaf OX .
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R [ ).
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 ’affinoid
perfectoid space’ X = Spa(R) of continuous valuations, equipped
with a structure sheaf OX .
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R [ ). The underlying topological
spaces |X | ∼
= |X [ | are homeomorphic,
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 ’affinoid
perfectoid space’ X = Spa(R) of continuous valuations, equipped
with a structure sheaf OX .
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R [ ). The underlying topological
spaces |X | ∼
= |X [ | are homeomorphic, OX is a sheaf 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 ’affinoid
perfectoid space’ X = Spa(R) of continuous valuations, equipped
with a structure sheaf OX .
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R [ ). The underlying topological
spaces |X | ∼
= |X [ | are homeomorphic, OX is a sheaf of perfectoid
C -algebras, with tilt OX [ .
Perfectoid Spaces
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R [ ). The underlying topological
spaces |X | ∼
= |X [ | are homeomorphic, OX is a sheaf of perfectoid
C -algebras, with tilt OX [ .
Perfectoid Spaces
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R [ ). The underlying topological
spaces |X | ∼
= |X [ | are homeomorphic, OX is a sheaf of perfectoid
C -algebras, with tilt OX [ .
Define general perfectoid spaces by gluing affinoid perfectoid
spaces.
Perfectoid Spaces
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R [ ). The underlying topological
spaces |X | ∼
= |X [ | are homeomorphic, OX is a sheaf of perfectoid
C -algebras, with tilt OX [ .
Define general perfectoid spaces by gluing affinoid perfectoid
spaces.
Corollary
The categories of perfectoid spaces over C and C [ are equivalent.
Example
Example
The inverse limit
X = lim A1C has tilt X [ = lim A1C [ .
←
−p
←
−p
T 7→T
T 7→T
Example
The inverse limit
X = lim A1C has tilt X [ = lim A1C [ .
←
−p
←
−p
T 7→T
X[
..
.
T 7→T
≈
X
..
.
T 7→T p
A1C [
T 7→T p
A1C [
T 7→T p
A1C
T 7→T p
A1C
Example
The inverse limit
X = lim A1C has tilt X [ = lim A1C [ .
←
−p
←
−p
T 7→T
|X [ |
..
.
T 7→T
∼
=
|X |
..
.
∼
= T 7→T p
|A1C [ |
∼
= T 7→T p
|A1C [ |
T 7→T p
|A1C |
T 7→T p
|A1C |
Example
|X [ |
..
.
∼
=
|X |
..
.
∼
= T 7→T p
|A1C [ |
∼
= T 7→T p
|A1C [ |
T 7→T p
|A1C |
T 7→T p
|A1C |
Example
|X [ |
..
.
∼
=
|X |
..
.
∼
= T 7→T p
T 7→T p
|A1C [ |
|A1C |
∼
= T 7→T p
|A1C [ |
T 7→T p
|A1C |
Thus, homeomorphism of topological spaces (underlying adic
spaces)
|A1C [ | ∼
|A1 | .
= lim
←
−p C
T 7→T
Example
|X [ |
..
.
∼
=
|X |
..
.
∼
= T 7→T p
T 7→T p
|A1C [ |
|A1C |
∼
= T 7→T p
|A1C [ |
T 7→T p
|A1C |
Thus, homeomorphism of topological spaces (underlying adic
spaces)
|A1C [ | ∼
|A1 | .
= lim
←
−p C
T 7→T
char p geometry as infinite covering of char 0 geometry.
The almost purity theorem
Theorem
Let X be a perfectoid space over C .
The almost purity theorem
Theorem
Let X be a perfectoid space over C . There is an étale site Xét ,
The almost purity theorem
Theorem
Let X be a perfectoid space over C . There is an étale site Xét ,
[ under tilting.
such that Xét ∼
= Xét
The almost purity theorem
Theorem
Let X be a perfectoid space over C . There is an étale site Xét ,
[ under tilting.
such that Xét ∼
= Xét
There is the sheaf OX+ ⊂ OX of functions bounded by 1.
The almost purity theorem
Theorem
Let X be a perfectoid space over C . There is an étale site Xét ,
[ under tilting.
such that Xét ∼
= Xét
There is the sheaf OX+ ⊂ OX of functions bounded by 1.
Theorem
The global sections H 0 (Xét , OX+ ) = R + ,
The almost purity theorem
Theorem
Let X be a perfectoid space over C . There is an étale site Xét ,
[ under tilting.
such that Xét ∼
= Xét
There is the sheaf OX+ ⊂ OX of functions bounded by 1.
Theorem
The global sections H 0 (Xét , OX+ ) = R + , and H i (Xét , OX+ ) is almost
zero for i > 0,
The almost purity theorem
Theorem
Let X be a perfectoid space over C . There is an étale site Xét ,
[ under tilting.
such that Xét ∼
= Xét
There is the sheaf OX+ ⊂ OX of functions bounded by 1.
Theorem
The global sections H 0 (Xét , OX+ ) = R + , and H i (Xét , OX+ ) is almost
zero for i > 0, i.e., killed by p 1/n ∈ OC for all n > 0.
The almost purity theorem
Theorem
Let X be a perfectoid space over C . There is an étale site Xét ,
[ under tilting.
such that Xét ∼
= Xét
There is the sheaf OX+ ⊂ OX of functions bounded by 1.
Theorem
The global sections H 0 (Xét , OX+ ) = R + , and H i (Xét , OX+ ) is almost
zero for i > 0, i.e., killed by p 1/n ∈ OC for all n > 0.
This is closely related to Faltings’s celebrated “almost purity
theorem”.
The key computation
Let R = OC hT ±1 i,
The key computation
Let R = OC hT ±1 i, and X = Spa(R[1/p], R).
The key computation
Let R = OC hT ±1 i, and X = Spa(R[1/p], R). Then X has an
affinoid perfectoid cover by X̃ = Spa(R̃[1/p], R̃),
The key computation
Let R = OC hT ±1 i, and X = Spa(R[1/p], R). Then X has an
affinoid perfectoid cover by X̃ = Spa(R̃[1/p], R̃), where
∞
R̃ = OC hT ±1/p i .
The key computation
Let R = OC hT ±1 i, and X = Spa(R[1/p], R). Then X has an
affinoid perfectoid cover by X̃ = Spa(R̃[1/p], R̃), where
∞
R̃ = OC hT ±1/p i .
Proposition
H 0 (Xproét , ÔX+ ) = R ,
H 1 (Xproét , ÔX+ ) = Ω1R/OC ⊕ (p 1/(p−1) −torsion) .
The key computation
Let R = OC hT ±1 i, and X = Spa(R[1/p], R). Then X has an
affinoid perfectoid cover by X̃ = Spa(R̃[1/p], R̃), where
∞
R̃ = OC hT ±1/p i .
Proposition
H 0 (Xproét , ÔX+ ) = R ,
H 1 (Xproét , ÔX+ ) = Ω1R/OC ⊕ (p 1/(p−1) −torsion) .
Here, Xproét is the pro-étale site,
The key computation
Let R = OC hT ±1 i, and X = Spa(R[1/p], R). Then X has an
affinoid perfectoid cover by X̃ = Spa(R̃[1/p], R̃), where
∞
R̃ = OC hT ±1/p i .
Proposition
H 0 (Xproét , ÔX+ ) = R ,
H 1 (Xproét , ÔX+ ) = Ω1R/OC ⊕ (p 1/(p−1) −torsion) .
Here, Xproét is the pro-étale site, and ÔX+ = lim OX+ /p n .
←
−
The key computation
Let R = OC hT ±1 i, and X = Spa(R[1/p], R). Then X has an
affinoid perfectoid cover by X̃ = Spa(R̃[1/p], R̃), where
∞
R̃ = OC hT ±1/p i .
Proposition
H 0 (Xproét , ÔX+ ) = R ,
H 1 (Xproét , ÔX+ ) = Ω1R/OC ⊕ (p 1/(p−1) −torsion) .
Here, Xproét is the pro-étale site, and ÔX+ = lim OX+ /p n . One has
←
−
H i (Xproét , ÔX+ ) = lim H i (Xét , OX+ /p n ) .
←
−
The key computation
The p 1/(p−1) -torsion is sometimes called “junk torsion”.
The key computation
The p 1/(p−1) -torsion is sometimes called “junk torsion”. In the last
talk, we will see how to get rid of it.
The key computation
The p 1/(p−1) -torsion is sometimes called “junk torsion”. In the last
talk, we will see how to get rid of it.
Step 1. The Zp -cover X̃ → X induces a map
i
Hcont
(Zp , R̃) → H i (Xproét , ÔX+ ) .
The key computation
The p 1/(p−1) -torsion is sometimes called “junk torsion”. In the last
talk, we will see how to get rid of it.
Step 1. The Zp -cover X̃ → X induces a map
i
Hcont
(Zp , R̃) → H i (Xproét , ÔX+ ) .
This map is an almost isomorphism as H i (X̃proét , ÔX+ ) is almost
zero for i > 0.
The key computation
The p 1/(p−1) -torsion is sometimes called “junk torsion”. In the last
talk, we will see how to get rid of it.
Step 1. The Zp -cover X̃ → X induces a map
i
Hcont
(Zp , R̃) → H i (Xproét , ÔX+ ) .
This map is an almost isomorphism as H i (X̃proét , ÔX+ ) is almost
zero for i > 0.
Remains to compute
∞
i
(Zp , OC hT ±1/p i) =
Hcont
M
d
j∈Z[1/p]
i
(Zp , OC · T j ) .
Hcont
The key computation
Step 2. Computation of
i
Hcont
(Zp , OC · T j ) .
The key computation
Step 2. Computation of
i
Hcont
(Zp , OC · T j ) .
Fix a compatible system of p-power roots of unity
ζp , ζp2 , . . . ∈ OC .
The key computation
Step 2. Computation of
i
Hcont
(Zp , OC · T j ) .
Fix a compatible system of p-power roots of unity
ζp , ζp2 , . . . ∈ OC . Then the generator γ = 1 ∈ Zp acts via
γ · T j = ζpnm T j , j = n/p m , n, m ∈ Z .
The key computation
Step 2. Computation of
i
Hcont
(Zp , OC · T j ) .
Fix a compatible system of p-power roots of unity
ζp , ζp2 , . . . ∈ OC . Then the generator γ = 1 ∈ Zp acts via
γ · T j = ζpnm T j , j = n/p m , n, m ∈ Z .
i
Then Hcont
(Zp , OC · T j ) computed by the complex
γ−1
ζpnm −1
(OC · T j −→ OC · T j ) ∼
= (OC −→ OC ) .
The key computation
ζpnm −1
OC −→ OC .
The key computation
ζpnm −1
OC −→ OC .
Case 1. j ∈ Z, i.e., m = 0.
The key computation
ζpnm −1
OC −→ OC .
Case 1. j ∈ Z, i.e., m = 0. Then ζpnm − 1 = 0, and so one gets OC
in degrees 0, 1.
The key computation
ζpnm −1
OC −→ OC .
Case 1. j ∈ Z, i.e., m = 0. Then ζpnm − 1 = 0, and so one gets OC
in degrees 0, 1. This contributes OC · T j in degrees i = 0, 1 to the
final result.
The key computation
ζpnm −1
OC −→ OC .
Case 1. j ∈ Z, i.e., m = 0. Then ζpnm − 1 = 0, and so one gets OC
in degrees 0, 1. This contributes OC · T j in degrees i = 0, 1 to the
final result.
Case 2. j 6∈ Z.
The key computation
ζpnm −1
OC −→ OC .
Case 1. j ∈ Z, i.e., m = 0. Then ζpnm − 1 = 0, and so one gets OC
in degrees 0, 1. This contributes OC · T j in degrees i = 0, 1 to the
final result.
Case 2. j 6∈ Z. Then ζpnm − 1 divides p 1/(p−1) , and one gets no
H 0 , and p 1/(p−1) -torsion in H 1 .
The key computation
ζpnm −1
OC −→ OC .
Case 1. j ∈ Z, i.e., m = 0. Then ζpnm − 1 = 0, and so one gets OC
in degrees 0, 1. This contributes OC · T j in degrees i = 0, 1 to the
final result.
Case 2. j 6∈ Z. Then ζpnm − 1 divides p 1/(p−1) , and one gets no
H 0 , and p 1/(p−1) -torsion in H 1 .
End result.
H 0 (Xproét , ÔX+ ) =
M
d
H 1 (Xproét , ÔX+ ) =
M
d
j∈Z
j∈Z
OC T j = R ,
OC T j ⊕
M
(OC /(ζpnm − 1))T j
j=n/p m ∈Z[1/p]\Z
= Ω1R/OC ⊕ (p 1/(p−1) −torsion) .
Almost finite generation: Local case
Definition
Let R be an OC -algebra.
Almost finite generation: Local case
Definition
Let R be an OC -algebra. An R-module M is called almost finitely
generated
Almost finite generation: Local case
Definition
Let R be an OC -algebra. An R-module M is called almost finitely
generated if for any n ≥ 1, there is a finitely generated submodule
Mn ⊂ M
Almost finite generation: Local case
Definition
Let R be an OC -algebra. An R-module M is called almost finitely
generated if for any n ≥ 1, there is a finitely generated submodule
Mn ⊂ M such that M/Mn is killed by p 1/n .
Almost finite generation: Local case
Definition
Let R be an OC -algebra. An R-module M is called almost finitely
generated if for any n ≥ 1, there is a finitely generated submodule
Mn ⊂ M such that M/Mn is killed by p 1/n .
Corollary
Let R = OC hT1±1 , . . . , Td±1 i, X = Spa(R[1/p], R).
Almost finite generation: Local case
Definition
Let R be an OC -algebra. An R-module M is called almost finitely
generated if for any n ≥ 1, there is a finitely generated submodule
Mn ⊂ M such that M/Mn is killed by p 1/n .
Corollary
Let R = OC hT1±1 , . . . , Td±1 i, X = Spa(R[1/p], R). Then
H i (Xproét , ÔX+ ) is an almost finitely generated R-module for all
i ≥ 0,
Almost finite generation: Local case
Definition
Let R be an OC -algebra. An R-module M is called almost finitely
generated if for any n ≥ 1, there is a finitely generated submodule
Mn ⊂ M such that M/Mn is killed by p 1/n .
Corollary
Let R = OC hT1±1 , . . . , Td±1 i, X = Spa(R[1/p], R). Then
H i (Xproét , ÔX+ ) is an almost finitely generated R-module for all
i ≥ 0, is almost zero for i > d,
Almost finite generation: Local case
Definition
Let R be an OC -algebra. An R-module M is called almost finitely
generated if for any n ≥ 1, there is a finitely generated submodule
Mn ⊂ M such that M/Mn is killed by p 1/n .
Corollary
Let R = OC hT1±1 , . . . , Td±1 i, X = Spa(R[1/p], R). Then
H i (Xproét , ÔX+ ) is an almost finitely generated R-module for all
i ≥ 0, is almost zero for i > d, and
H i (Xproét , ÔX+ ) = ΩiR/OC ⊕ (p i/(p−1) −torsion) .
Almost finite generation: Local case
Definition
Let R be an OC -algebra. An R-module M is called almost finitely
generated if for any n ≥ 1, there is a finitely generated submodule
Mn ⊂ M such that M/Mn is killed by p 1/n .
Corollary
Let R = OC hT1±1 , . . . , Td±1 i, X = Spa(R[1/p], R). Then
H i (Xproét , ÔX+ ) is an almost finitely generated R-module for all
i ≥ 0, is almost zero for i > d, and
H i (Xproét , ÔX+ ) = ΩiR/OC ⊕ (p i/(p−1) −torsion) .
For the proof, redo the computation in any dimension, or use the
Künneth formula.
Almost finite generation: Global case
Theorem
Let X be a proper smooth rigid-analytic variety over C .
Almost finite generation: Global case
Theorem
Let X be a proper smooth rigid-analytic variety over C . Then
H i (Xproét , ÔX+ ) is an almost finitely generated OC -module for all
i ≥ 0,
Almost finite generation: Global case
Theorem
Let X be a proper smooth rigid-analytic variety over C . Then
H i (Xproét , ÔX+ ) is an almost finitely generated OC -module for all
i ≥ 0, and almost zero for i > 2 dim X .
Almost finite generation: Global case
Theorem
Let X be a proper smooth rigid-analytic variety over C . Then
H i (Xproét , ÔX+ ) is an almost finitely generated OC -module for all
i ≥ 0, and almost zero for i > 2 dim X .
The proof is a version of the Cartan–Serre argument:
Almost finite generation: Global case
Theorem
Let X be a proper smooth rigid-analytic variety over C . Then
H i (Xproét , ÔX+ ) is an almost finitely generated OC -module for all
i ≥ 0, and almost zero for i > 2 dim X .
The proof is a version of the
argument: Take two
S Cartan–Serre
S
(nice) affinoid covers X = i∈I Ui = i∈I Vi such that Ui is
strictly contained in Vi .
Almost finite generation: Global case
Theorem
Let X be a proper smooth rigid-analytic variety over C . Then
H i (Xproét , ÔX+ ) is an almost finitely generated OC -module for all
i ≥ 0, and almost zero for i > 2 dim X .
The proof is a version of the
argument: Take two
S Cartan–Serre
S
(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,ét , OX+ /p) → H j (Ui,ét , OX+ /p)
have almost finitely generated image (over OC ).
Finiteness of Zp -cohomology
Corollary
Let X be a proper smooth rigid-analytic variety over C .
Finiteness of Zp -cohomology
Corollary
Let X be a proper smooth rigid-analytic variety over C . The
natural map
H i (Xét , Zp ) ⊗Zp OC → H i (Xproét , ÔX+ )
is an almost isomorphism.
Finiteness of Zp -cohomology
Corollary
Let X be a proper smooth rigid-analytic variety over C . The
natural map
H i (Xét , Zp ) ⊗Zp OC → H i (Xproét , ÔX+ )
is an almost isomorphism. In particular, H i (Xét , Zp ) is finitely
generated, and vanishes for i > 2 dim X .
Finiteness of Zp -cohomology
Corollary
Let X be a proper smooth rigid-analytic variety over C . The
natural map
H i (Xét , Zp ) ⊗Zp OC → H i (Xproét , ÔX+ )
is an almost isomorphism. In particular, H i (Xét , Zp ) is finitely
generated, and vanishes for i > 2 dim X .
Enough to prove similar result for
H i (Xét , Fp ) ⊗Fp OC /p → H i (Xét , OX+ /p) .
Finiteness of Zp -cohomology
Corollary
Let X be a proper smooth rigid-analytic variety over C . The
natural map
H i (Xét , Zp ) ⊗Zp OC → H i (Xproét , ÔX+ )
is an almost isomorphism. In particular, H i (Xét , Zp ) is finitely
generated, and vanishes for i > 2 dim X .
Enough to prove similar result for
H i (Xét , Fp ) ⊗Fp OC /p → H i (Xét , OX+ /p) .
There, use Artin–Schreier sequence
0 → Fp → OX+ /p → OX+ /p → 0 .
The Hodge–Tate decomposition
Recall that we want to prove the Hodge–Tate decomposition, for a
proper smooth rigid-analytic variety X0 over K with X = X0 ⊗K C ,
H i (Xét , Zp ) ⊗Zp C ∼
=
i
M
j=0
H i−j (X , ΩjX )(−j) .
The Hodge–Tate decomposition
Recall that we want to prove the Hodge–Tate decomposition, for a
proper smooth rigid-analytic variety X0 over K with X = X0 ⊗K C ,
H i (Xét , Zp ) ⊗Zp C ∼
=
i
M
H i−j (X , ΩjX )(−j) .
j=0
At this point, we have an isomorphism
H i (Xét , Zp ) ⊗Zp C ∼
= H i (Xproét , ÔX ) ,
where ÔX = ÔX+ [1/p].
The Hodge–Tate decomposition
The local computation of the cohomology of ÔX in terms of
differentials gives the Hodge–Tate spectral sequence
E2ij = H i (X , ΩjX )(−j) ⇒ H i+j (Xproét , ÔX ) .
The Hodge–Tate decomposition
The local computation of the cohomology of ÔX in terms of
differentials gives the Hodge–Tate spectral sequence
E2ij = H i (X , ΩjX )(−j) ⇒ H i+j (Xproét , ÔX ) .
(The Tate twist is needed to make things Galois equivariant;
The Hodge–Tate decomposition
The local computation of the cohomology of ÔX in terms of
differentials gives the Hodge–Tate spectral sequence
E2ij = H i (X , ΩjX )(−j) ⇒ H i+j (Xproét , ÔX ) .
(The Tate twist is needed to make things Galois equivariant; it
appears because we had to choose roots of unity in the
computation.)
The Hodge–Tate decomposition
The local computation of the cohomology of ÔX in terms of
differentials gives the Hodge–Tate spectral sequence
E2ij = H i (X , ΩjX )(−j) ⇒ H i+j (Xproét , ÔX ) .
(The Tate twist is needed to make things Galois equivariant; it
appears because we had to choose roots of unity in the
computation.)
If X = X0 ⊗K C , then Galois equivariance forces all differentials in
the spectral sequence to be 0.
The Hodge–Tate decomposition
The local computation of the cohomology of ÔX in terms of
differentials gives the Hodge–Tate spectral sequence
E2ij = H i (X , ΩjX )(−j) ⇒ H i+j (Xproét , ÔX ) .
(The Tate twist is needed to make things Galois equivariant; it
appears because we had to choose roots of unity in the
computation.)
If X = X0 ⊗K C , then Galois equivariance forces all differentials in
the spectral sequence to be 0. Moreover, there is a unique
Galois-equivariant splitting of the resulting abutment filtration.
The Hodge–Tate decomposition
The local computation of the cohomology of ÔX in terms of
differentials gives the Hodge–Tate spectral sequence
E2ij = H i (X , ΩjX )(−j) ⇒ H i+j (Xproét , ÔX ) .
(The Tate twist is needed to make things Galois equivariant; it
appears because we had to choose roots of unity in the
computation.)
If X = X0 ⊗K C , then Galois equivariance forces all differentials in
the spectral sequence to be 0. Moreover, there is a unique
Galois-equivariant splitting of the resulting abutment filtration.
This proves the Hodge–Tate decomposition.
The Hodge–Tate decomposition
The local computation of the cohomology of ÔX in terms of
differentials gives the Hodge–Tate spectral sequence
E2ij = H i (X , ΩjX )(−j) ⇒ H i+j (Xproét , ÔX ) .
(The Tate twist is needed to make things Galois equivariant; it
appears because we had to choose roots of unity in the
computation.)
If X = X0 ⊗K C , then Galois equivariance forces all differentials in
the spectral sequence to be 0. Moreover, there is a unique
Galois-equivariant splitting of the resulting abutment filtration.
This proves the Hodge–Tate decomposition.
Remark. One can show that the Hodge–Tate spectral sequence
degenerates always, for a proper smooth rigid-analytic variety X
over C . However, it does not canonically split.