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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, . . .).
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, . . .).
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, . . .).
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, . . .).
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, . . .).
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, . . .).
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, . . .).
Fontaine’s construction
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.
Fontaine’s construction
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.
Fontaine’s construction
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.
Fontaine’s construction
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.
Fontaine’s construction
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.
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 [ = lim←−x 7→xp
C .
As x 7→ x ] is non-algebraic, need to formalize this in an analyticworld.
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 [ = lim←−x 7→xp
C .
As x 7→ x ] is non-algebraic, need to formalize this in an analyticworld.
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 [ = lim←−x 7→xp
C .
As x 7→ x ] is non-algebraic, need to formalize this in an analyticworld.
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 [ = lim←−x 7→xp
C .
As x 7→ x ] is non-algebraic, need to formalize this in an analyticworld.
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 [ = lim←−x 7→xp
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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
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
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
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
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
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
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
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
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
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 [ .
Define general perfectoid spaces by gluing affinoid perfectoidspaces.
Corollary
The categories of perfectoid spaces over C and C [ are equivalent.
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 [ .
Define general perfectoid spaces by gluing affinoid perfectoidspaces.
Corollary
The categories of perfectoid spaces over C and C [ are equivalent.
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 [ .
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
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
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
A1C [
T 7→T p
A1C
T 7→T p
A1C [ A1
C
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
|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.
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.
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”.
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”.
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”.
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”.
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”.
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”.
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”.
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”.
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
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
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
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
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
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
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
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
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]
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]
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]
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]
H icont(Zp,OC · T j) .
The key computation
Step 2. Computation of
H icont(Zp,OC · T j) .
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
Step 2. Computation of
H icont(Zp,OC · T j) .
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
Step 2. Computation of
H icont(Zp,OC · T j) .
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
Step 2. Computation of
H icont(Zp,OC · T j) .
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
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
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
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
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
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
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
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) .
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 ).
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 ).
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 ).
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 ).
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 ).
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 ).
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 .
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 .
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 .
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 .
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 .
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 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 Hodge–Tate decomposition
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.
The Hodge–Tate decomposition
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.
The Hodge–Tate decomposition
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.
The Hodge–Tate decomposition
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.
The Hodge–Tate decomposition
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.
The Hodge–Tate decomposition
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.
The Hodge–Tate decomposition
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.