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Notes on prismatic cohomology
Kiran S. Kedlaya
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Front Matter
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Abstract
1
Introduction and overview
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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
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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
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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
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4.1
The big Witt vector functor
4.2
λ
-rings
4.3
Exercises
5
Distinguished elements and prisms
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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
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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
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7.1
Distinguished elements in perfect
δ
-rings
7.2
Perfect prisms
7.3
Tilting and slicing
7.4
Exercises
8
Lenses
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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18.1
Derived prismatic cohomology
18.2
Regular semilenses
18.3
Exercises
19
Coperfections in mixed characteristic
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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
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20.1
Grothendieck topologies
20.2
Valuation rings
20.3
The arc-topology
20.4
Exercises
21
Descent for the arc-topology
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Bibliography
Bibliography
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