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Notes on prismatic cohomology
Kiran S. Kedlaya
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Front Matter
Abstract
1
Introduction and overview
1.1
(Co)homology of complex varieties
1.2
The trouble with torsion
1.3
The
p
-adic situation
1.4
The role of prisms
2
δ
-rings
2.1
p
-derivations and Frobenius lifts
2.2
Examples of
δ
-rings
2.3
Truncated Witt vectors
2.4
The category of
δ
-rings
2.5
Exercises
3
Witt vectors
3.1
p
-typical Witt vectors via adjunction
3.2
Ghost coordinates
3.3
Witt vectors and perfect
δ
-rings
3.4
Beyond the perfect case in characteristic
p
3.5
Additional remarks
3.6
Exercises
4
Big Witt vectors and
λ
-rings
4.1
The big Witt vector functor
4.2
λ
-rings
4.3
Exercises
5
Distinguished elements and prisms
5.1
Distinguished elements and examples
5.2
Properties of distinguished elements
5.3
Prisms
5.4
The category of prisms
5.5
Exercises
6
Derived completeness
6.1
The trouble with classical completion
6.2
Derived completeness
6.3
The category of derived-complete modules
6.4
Derived
f
-completion
6.5
Flatness and smoothness
6.6
Derived completeness in the derived category
6.7
Exercises
7
Perfect prisms
7.1
Distinguished elements in perfect
δ
-rings
7.2
Perfect prisms
7.3
Tilting and slicing
7.4
Exercises
8
Lenses
8.1
The category of lenses
8.2
On the structure of lenses
8.3
Perfectoid fields
8.4
Glueing of lenses
8.5
Exercises
9
Homotopy categories
9.1
A bit of motivation
9.2
Categories of chain complexes
9.3
Split exact sequences
9.4
Chain complexes and the homotopy category
9.5
Derived functors revisited
10
Derived categories
10.1
Localization in a category
10.2
Distinguished triangles
10.3
Localization at quasi-isomorphisms
10.4
Truncation
10.5
Pseudocoherent and perfect complexes
10.6
Exercises
11
The prismatic site
11.1
Indiscrete Grothendieck topologies
11.2
A word on (co)simplicial objects
11.3
The prismatic site and “oppo-site”
11.4
The case of a perfect prism
11.5
Prismatic and Hodge-Tate cohomology
11.6
More on the prismatic site
11.7
Additional remarks
11.8
Exercises
12
The Hodge-Tate comparison map
12.1
Graded commutativity for graded rings
12.2
The de Rham complex
12.3
Construction of the Hodge-Tate comparison map
12.4
The Hodge-Tate comparison theorem
12.5
Exercises
13
Double complexes
13.1
Double complexes and totalization
13.2
Interchanging the rows and columns
13.3
The spectral sequence(s) of a double complex
13.4
Totalization in the derived category
14
Hodge-Tate comparison for crystalline prisms
14.1
de Rham cohomology in characteristic
p
14.2
Divided powers
14.3
Divided powers in
δ
-rings
14.4
Prismatic cohomology for a crystalline prism
14.5
Exercises
15
Proof of the Hodge-Tate comparison
15.1
Étale localization and base change
15.2
Comparing a universal prism to a crystalline prism
15.3
Hodge-Tate comparisons
15.4
The crystalline and de Rham comparisons
16
Nonabelian derived functors
16.1
More on simplicial objects
16.2
Simplicial resolutions
16.3
Standard resolution
16.4
Nonabelian derived functors
16.5
Under the hood:
∞
-categories
16.6
Exercises
17
Derived de Rham cohomology
17.1
The cotangent complex
17.2
Derived de Rham cohomology
17.3
Regular semiperfect rings
17.4
Derived crystalline cohomology
17.5
Exercises
18
Derived prismatic cohomology
18.1
Derived prismatic cohomology
18.2
Regular semilenses
18.3
Exercises
19
Coperfections in mixed characteristic
19.1
Coperfections in characteristic
p
revisited
19.2
The mixed characteristic case
19.3
More properties of coperfection
19.4
André flatness
19.5
Examples of lens coperfection
19.6
Exercises
20
The arc-topology and friends
20.1
Grothendieck topologies
20.2
Valuation rings
20.3
The arc-topology
20.4
Exercises
21
Descent for the arc-topology
21.1
Descent for perfect schemes
21.2
Additional descent arguments
21.3
Arc-descent for étale cohomology
21.4
Exercises
22
The étale comparison theorem
22.1
The Artin-Schreier-Witt exact sequence
22.2
Frobenius fixed points and coperfections
22.3
The arc
p
-topology
22.4
Tilting valuation rings
22.5
Arc
p
-descent for lenses
22.6
The comparison theorem
22.7
Exercises
23
Applications of étale comparison
23.1
Tilting of valuation rings
23.2
Torsion in étale and de Rham cohomology
23.3
Tate twists
23.4
Exercises
24
Almost commutative algebra
24.1
A bit of motivation
24.2
A context for almost commutative algebra
24.3
Almost commutative algebra for lenses
24.4
Almost Galois extensions of rings
24.5
Exercises
25
Almost purity
25.1
Some initial remarks
25.2
Almost purity (first version)
25.3
Almost purity (second version)
25.4
An application to cohomological dimension
25.5
The direct summand conjecture
25.6
Exercises
26
q
-de Rham cohomology
26.1
A brief history of
q
26.2
Jackson's
q
-calculus
26.3
The
q
-de Rham complex of Aomoto
27
q
-crystalline cohomology
27.1
q
-divided powers
27.2
q
-divided power pairs and envelopes
27.3
Comparison with prismatic cohomology
27.4
Frobenius is an isogeny
27.5
Étale localization
28
Some further developments: a whirlwind tour
28.1
Topological Hochschild homology
28.2
The absolute prismatic site
28.3
Prismatization
28.4
Prismatic Dieudonné theory
28.5
Logarithmic prismatic cohomology
29
Some global speculation
29.1
Divided power envelopes of
λ
-rings
29.2
q
-divided powers for
λ
-rings
29.3
A global site
29.4
Okay, now what?
29.5
Exercises
Back Matter
Bibliography
🔗
Bibliography
