Some further developments: a whirlwind tour

Section 28 Some further developments: a whirlwind tour

Reference.

See the various sections below.

In this section, we survey some further developments. This is primarily meant to serve as a point of departure for further reading; as we have come nearly to the end of the course, we will not be able to provide much detail on any individual aspect.

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Subsection 28.1 Topological Hochschild homology

Definition 28.1.1.

Let

A \to B

be a morphism in

Ring .

The Hochschild homology of

B

over

A

is the complex of

A

-modules associated to the simplicial object

K_{∙}

{Ring}_{A}

in which

K_{n}

is the

(n + 1)

-fold tensor product of

B

over

A

and the various maps

K_{n} \to K_{n - 1}

act by taking the product of some pair of consecutive factors:

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b_{0} \otimes \dots \otimes b_{i} \otimes b_{i + 1} \otimes \dots \otimes b_{n} \mapsto b_{0} \otimes \dots \otimes b_{i} b_{i + 1} \otimes \dots \otimes b_{n} .

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Remark 28.1.2.

It has been anticipated for some time that there should be deep links between structures arising in

p

-adic Hodge theory and parallel structures arising in algebraic topology, particularly with regard to topological Hochschild homology (THH), the analogue of Hochschild homology with rings replaced by ring spectra; working with THH means that one takes tensor products over the sphere spectrum, which lies “below” the ordinary ring of integers and thus provides a base more closely resembling the “field of one element”. Much of the early work in this direction is due to Hesselholt; see for example [65].

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A systematic link between THH and

p

-adic Hodge theory was developed more systematically in [23], in which the

A_{\inf}

-cohomology of [22] is reconstructed using THH. This link is revisited in [25] using prismatic techniques.

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Subsection 28.2 The absolute prismatic site

This material comes from announcements by Bhatt and Scholze. There is not yet a primary reference; in the interim, the recorded lecture [111] of Scholze will have to suffice.

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Definition 28.2.1.

For

R

a derived

p

-complete ring, the absolute prismatic oppo-site of

R,

denoted

(Spec R)_{Δ}^{op},

is the category in which an object is a prism

(B, J)

equipped with a ring homomorphism

R \to B / J,

and a morphism is a morphism of underlying prisms

(B, J) \to (B^{'}, J^{'})

for which the induced morphism

B / J \to B^{'} / J^{'}

R

-linear. Taking the opposite category yields the absolute prismatic site of

R,

denoted

(Spec R)_{Δ},

which we equip with the indiscrete topology. Note that there is no base prism in the definition.

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Definition 28.2.2.

Let

C

be a site equipped with a sheaf of rings

O

(or more generally a ringed topos). A crystal on

(C, O)

is a sheaf of

O

-modules locally obtained by tensoring

O

with a finite projective module over the ring of global sections.

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We will typically apply this definition in a situation where descent of finite projective modules is effective. In this case, a crystal can be specified by assigning to each

X \in C

to a finite projective

O (X)

-module

M (X)

and to each morphism

Y \to X

C

an isomorphism

M (X) \otimes_{O (X)} O (Y) ≅ M (Y)

in a manner compatible with composition.

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Definition 28.2.3.

For

R

a derived

p

-complete ring, a prismatic

F

-crystal on

R

is a crystal

M

on the absolute prismatic site of

R

equipped with an isomorphism

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F : ϕ^{*} M [I^{- 1}] \to M [I^{- 1}] .

That is, for each object

(B, J) \in (Spec R)_{Δ},

we specify an isomorphism

ϕ^{*} M (B) [J^{- 1}] \to M (B) [J^{- 1}]

compatible with the morphisms in the site (where

M (B) [J^{- 1}] = {colim}_{n} M (B) \otimes_{B} J^{- n};

this makes sense because

J

is an invertible ideal).

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Theorem 28.2.4.

Let

K

be a completely valued field of mixed characteristics

(0, p)

with perfect residue field

k .

Then the category of prismatic

F

-crystals on

O_{K}

is equivalent to the category of crystalline lattices, i.e., finite free

Z_{p}

-modules equipped with continuous

G_{K}

-action whose base extensions from

Z_{p}

Q_{p}

are crystalline Galois representations (see Remark 28.2.5).

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Proof.

See [26]. The key ingredients are the étale comparison theorem (Theorem 22.6.1), Kisin’s description of crystalline Galois representations via Breuil-Kisin modules ([85]), and Beauville-Laszlo glueing (Remark 21.2.7).

Remark 28.2.5.

It would take us well beyond the scope of these notes to explain enough of Fontaine’s theory of

p

-adic representations and

p

-adic periods to define the notion of a crystalline Galois representation. The motivating example is the étale cohomology

H_{et}^{i} (X_{\overset{―}{K}}, Q_{p})

where

X

is a smooth proper

O_{K}

-scheme. For

i > 0

such an extension cannot be unramified, as it would be if

Q_{p}

were replaced by

Q_{ℓ}

for some prime

ℓ \neq p,

because the kernel field of the Galois representation contains the

p

-cyclotomic tower (the determinant of cohomology is a nonzero power of the cyclotomic character); the crystalline condition is a replacement. For approachable treatments of this subject, see [33] or [51].

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Subsection 28.3 Prismatization

The primary reference for this topic is to be a preprint of Bhatt and Lurie which is not yet available; however, in the meantime Drinfeld has produced an independent writeup [42].

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Definition 28.3.1.

Let

W

be the ring scheme of

p

-typical Witt vectors. That is, for any ring

A,

there is a natural (in

A

) bijection between the underlying set of the ring

W (A)

and the set of morphisms

Spec A \to W .

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Write

W

Spec Z [x_{0}, x_{1}, \dots]

in terms of the Witt coordinates, and let

W_{n} = Spec Z [x_{0}, \dots, x_{n - 1}]

be the

n

-th truncation of

W .

For

n \geq 1,

let

W_{prim, n}

be the completion of

W_{n}

along the locally closed subscheme defined by the conditions

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p = x_{0} = 0, x_{1} \neq 0.

Since

W_{n}^{\times}

acts on

W_{prim, n}

by multiplication, we can form the quotient

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{CW}_{n} = W_{prim, n} / W_{n}^{\times}

in the category of sheaves on the category of

p

-adic formal schemes. Similarly, we may form the sheaf

CW = lim_{n} {CW}_{n},

called the Cartier-Witt stack.

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By definition, for any oriented prism

(A, (d))

we get a morphism

Spec A \to W_{prim}

under which the distinguished element

(x_{0}, x_{1}, \dots)

O (W_{prim})

pulls back to

d .

Consequently, for any prism

(A, I),

we get a morphism

Spec A \to CW .

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Remark 28.3.2.

Some caution is in order because the objects

CW

and

{CW}_{n}

are not algebraic stacks but rather formal stacks. We will not elaborate on what this means; see [42].

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Subsection 28.4 Prismatic Dieudonné theory

The reference for this topic is [6].

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Definition 28.4.1.

We say that

R \in Ring

is quasi-syntomic if

R

p

-complete with bounded

p

-power torsion and the cotangent complex

L_{R / Z_{p}}

has

p

-complete Tor amplitude in

[- 1, 0] .

For example, a noetherian lci ring is quasi-syntomic, as in a regular semilens (Definition 18.2.1) with bounded

p

-power torsion.

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Theorem 28.4.2.

For any quasi-syntomic ring

R,

there is an anti-equivalence betewen the category of

p

-divisible groups over

R

and the category of prismatic Dieudonné crystals over

R .

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Proof.

See [6].

Remark 28.4.3.

Theorem 28.4.2 builds upon a long history of describing

p

-divisible groups in terms of objects of semilinear algebra (e.g., see [90]), as well as more recent work classifying

p

-divisible groups over perfectoid spaces ([112], [89]; see also [113], Appendix to Lecture 17).

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Subsection 28.5 Logarithmic prismatic cohomology

The reference for this topic is [88].

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Definition 28.5.1.

A prelog structure on a ring

A

consists of a monoid

M

and a morphism

α : M \to A

of monoids. In general, one prefers to “sheafify” this definition to define a log structure, as in [77].

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Example 28.5.2.

Suppose that

Z

is an effective Cartier divisor on

Spec A .

If the components of

Z

are cut out by some elements

x_{1}, \dots, x_{r}

A,

we can use the monoid generated by these and its inclusion into

A

as a prelog structure. The resulting log structure will then depend only on

Z

and not on the components, and also makes sense even when the components of

Z

are not globally cut out by regular functions (as this is always true locally).

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Note that there is a difference between sheafifying with respect to the Zariski topology versus the étale topology, and we generally prefer the latter. For example, if

Z

is a nodal cubic curve in the plane, we would like the monoid to have two independent generators corresponding to the two branches at the node, and this is true étale-locally but not Zariski-locally.

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Definition 28.5.3.

As per [88], Definition 2.2, a

δ_{\log}

-ring is a tuple

(A, δ, α, δ_{\log})

in which

(A, δ)

is a

δ

-ring,

α : M \to A

defines a prelog structure on

A,

and

δ_{\log} : M \to A

is a function satisfying the following conditions.

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For $e \in M$ the identity element, $δ_{\log} (e) = 0 .$
For $m \in M,$

$δ (α (m)) = α (m)^{p} δ_{\log} (m) .$
For $m, m^{'} \in M,$

$δ_{\log} (m m^{'}) = δ_{\log} (m) + δ_{\log} (m^{'}) + p δ_{\log} (m) δ_{\log} (m^{'}) .$

An important special case is when

δ_{\log} = 0

identically. In this case, we say that the

δ_{\log}

-ring in question is constant (or of rank 1 in Koshikawa’s terminology).

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We report some examples from [88], Example 2.4.

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Example 28.5.4.

For

(A, δ) \in {Ring}_{δ},

consider the canonical log structure where

M = A^{\times}

and

α : A^{\times} \to A

is the canonical inclusion. There is then a unique

δ_{\log}

-ring structure given by

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δ_{\log} (x) = \frac{δ (x)}{x^{p}} (x \in A^{\times}) .

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Example 28.5.5.

Let

R \in {Ring}_{F_{p}}

be perfect and view

W (R)

as a

δ

-ring via the Witt vector Frobenius. The prelog structure given by the constant lift

[∙] : R \to W (R)

then admits a constant

δ_{\log}

-structure.

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Example 28.5.6.

For any monoid

M,

we may view the monoid ring

Z_{(p)} [M]

as a

δ

-ring in such a way that the elements of

M

are all constant. The prelog structure given by the natural map

M \to Z_{(p)} [M]

then admits a constant

δ_{\log}

-structure.

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Example 28.5.7.

Given a

δ_{\log}

-ring

A

and a morphism

A \to B

δ

-rings, we may upgrade to a morphism of

δ_{\log}

-rings by equipping

B

with the prelog structure

M \to A \to B

and the

δ_{\log}

-structure

M \overset{δ_{\log}}{\to} A \to B .

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Remark 28.5.8.

One can continue in this manner to extend much of the formalism of

δ

-rings; define logarithmic prisms and logarthmic prismatic sites; establish crystalline and Hodge-Tate comparison theorems; and obtain

q

-analogues. The purpose of this (not yet fully realized) is to develop a prismatic theory that provides a geometric construction of semistable Breuil-Kisin modules associated to the cohomology of smooth proper schemes over

p

-adic fields that do not have good reduction, building on the adaptation of [22] carried out in [38].

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However, it may be possible to give an alternate development using the formalism of prismatization (Subsection 28.3) and the fact that logarithmic structures on a given scheme

X

can be described locally in terms of morphisms from

X

to the quotient stack

A^{1} / G_{m} .

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