crystalline cohomology

Tags
- #arithmetic-geometry #arithmetic-geometry/p-adic-hodge-theory
Refs:
Links:
- Galois representations
- arithmetic geometry MOC
- divided power
- semilinear action
- formal disk
- period ring
- admissible representation
- inertia group
- cyclotomic character
- perfect ring
- Galois descent
- Tate twist
- de Rham-Witt complex
- etale comparison theorem
- Types of representations:
  - crystalline representation
  - semistable represenation
  - potentially semistable representation
  - de Rham representation
  - Hodge-Tate representation
- See p-adic monodromy theorem
- de Rham crystalline comparison
- crystalline representation

crystalline cohomology

Motivation

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Relation to Weil cohomology for smooth and proper schemes, algebraic de Rham cohomology, and Witt vectors attachments/Pasted%20image%2020220318193616.png

attachments/Pasted%20image%2020220502145248.png # Definition

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See closed immersion, special fiber, generic fiber, good reduction, semistable reduction.

Frobenius

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Comparisons

Several naturally occurring varieties in number theory do not possess such a well-behaved reduction, a famous example being the Tate curve. So replace with semistable reduction.

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Periods

attachments/Pasted%20image%2020220318201405.png One can recover real de Rham cohomology by taking fixed points on the right hand side.

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Algebraic de Rham to etale

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See B_dr

Etale to crystalline

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See Dieudonne module, abelian variety, Tate module, Galois representations, mysterious functor, generic fiber, Hodge filtration.

Etale to log crystalline

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See monodromy operator, attachments/Pasted%20image%2020220318195735.png

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Notes

Relation to admissible representation: attachments/Pasted%20image%2020220318210100.png

Use of Hilbert 90 and Faltings theorem: attachments/Pasted%20image%2020220318210138.png

🗓️ Timeline

Prismatic cohomology

\(A \coloneqq{\mathbf{Z}}_p\) and \(\phi = \operatorname{id}\) with \(I = \left\langle{ p }\right\rangle\) yields crystalline cohomology.

Links to this page

unresolved links output

Tate curve in Projects/2022 AWS/2022 Arizona Winter School, -Unsorted/crystalline cohomology

de Rham representation in Unsorted/crystalline cohomology

divided power in Unsorted/crystalline cohomology

generic fiber in Projects/2021 Fargues Fontaine/Fargues Fontaine Notes, -Unsorted/crystalline cohomology, -Unsorted/semistable reduction

monodromy operator in Unsorted/crystalline cohomology

mysterious functor in Unsorted/crystalline cohomology

potentially semistable representation in Unsorted/crystalline cohomology

proper schemes in Unsorted/crystalline cohomology, -Unsorted/semistable reduction

semistable represenation in Unsorted/crystalline cohomology
p-adic Hodge theory

crystalline cohomology
motive

crystalline cohomology,
isocrystal

crystalline cohomology
etale comparison theorem

See crystalline cohomology.
de Rham-Witt

Motivation: take $X\in {\mathsf{sm}} \mathsf{Alg} {\mathsf{Var}}{/ {k}} $, and define \({ {H}^{\scriptscriptstyle \bullet}} _\mathrm{dR}(X_{/ {k}} ) \coloneqq { {{\mathbb{H}}}^{\scriptscriptstyle \bullet}} ( { {\Omega}^{\scriptscriptstyle \bullet}} _{X/k})\) using hypercohomology. Over \({\mathbf{C}}\) this will be isomorphic to singular cohomology and take values in ${}{k}{\mathsf{Mod}} $, but if \(\operatorname{ch}k = p\) then the cohomology is entirely \(p{\hbox{-}}\)torsion. So Grothendieck/Berthelot define crystalline cohomology which takes values in ${}_{W(k)}{\mathsf{Mod}} $, the \(p{\hbox{-}}\)typical Witt vectors. Originally this was defined in terms of the structure sheaf of the crystalline topos of \(X\), but a more modern definition realizes it as \({ {H}^{\scriptscriptstyle \bullet}} _{\mathrm{crys}}(X_{/ {k}} ) = { {{\mathbb{H}}}^{\scriptscriptstyle \bullet}} ( { {\Omega}^{\scriptscriptstyle \bullet}} _{X, \mathrm{drW}})\), the hypercohomology of the de Rham-Witt complex. This is a lift of algebraic de Rham in the following sense: \({\operatorname{cofib}}(p { {\Omega}^{\scriptscriptstyle \bullet}} _{X, \mathrm{drW}} \to { {\Omega}^{\scriptscriptstyle \bullet}} _{X, \mathrm{drW}}) \simeq { {\Omega}^{\scriptscriptstyle \bullet}} _{X/k}\) is a quasi-isomorphism of cochain complexes of sheaves.
Weil cohomology

crystalline cohomology
Topological cyclic homology

Recovering crystalline cohomology:
K3 Surfaces

Using crystalline cohomology:
Hodge-Tate representation

crystalline
Hodge polygon

crystalline cohomology
Galois representations

See semistable and admissible representation,cyclotomic character, crystalline
2021-11-03-quick_note

Prismatic cohomology: a \(p{\hbox{-}}\)adic analog of crystalline cohomology

#arithmetic-geometry #arithmetic-geometry/p-adic-hodge-theory #resources/notes