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Section 28 Some further developments: a whirlwind tour


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.

Subsection 28.1 Topological Hochschild homology

Definition 28.1.1.

Let \(A \to B\) be a morphism in \(\Ring\text{.}\) The Hochschild homology of \(B\) over \(A\) is the complex of \(A\)-modules associated to the simplicial object \(K_\bullet\) of \(\Ring_A\) 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:
\begin{equation*} b_0 \otimes \cdots \otimes b_i \otimes b_{i+1} \otimes \cdots \otimes b_n \mapsto b_0 \otimes \cdots \otimes b_ib_{i+1} \otimes \cdots \otimes b_n. \end{equation*}

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].
A systematic link between THH and \(p\)-adic Hodge theory was developed more systematically in [23], in which the \(A_{\mathrm{inf}}\)-cohomology of [22] is reconstructed using THH. This link is revisited in [25] using prismatic techniques.

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.

Definition 28.2.1.

For \(R\) a derived \(p\)-complete ring, the absolute prismatic oppo-site of \(R\text{,}\) denoted \((\Spec R)^{\op}_{\Prism}\text{,}\) is the category in which an object is a prism \((B,J)\) equipped with a ring homomorphism \(R \to B/J\text{,}\) and a morphism is a morphism of underlying prisms \((B,J) \to (B', J')\) for which the induced morphism \(B/J \to B'/J'\) is \(R\)-linear. Taking the opposite category yields the absolute prismatic site of \(R\text{,}\) denoted \((\Spec R)_{\Prism}\text{,}\) which we equip with the indiscrete topology. Note that there is no base prism in the definition.

Definition 28.2.2.

Let \(\calC\) be a site equipped with a sheaf of rings \(\calO\) (or more generally a ringed topos). A crystal on \((\calC, \calO)\) is a sheaf of \(\calO\)-modules locally obtained by tensoring \(\calO\) with a finite projective module over the ring of global sections.
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 \(X \in \calC\) to a finite projective \(\calO(X)\)-module \(M(X)\) and to each morphism \(Y \to X\) in \(\calC\) an isomorphism \(M(X) \otimes_{\calO(X)} \calO(Y) \cong M(Y)\) in a manner compatible with composition.

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
\begin{equation*} F\colon \phi^* M[I^{-1}] \to M[I^{-1}]. \end{equation*}
That is, for each object \((B,J) \in (\Spec R)_{\Prism}\text{,}\) we specify an isomorphism \(\phi^* M(B)[J^{-1}] \to M(B)[J^{-1}]\) compatible with the morphisms in the site (where \(M(B)[J^{-1}] = \colim_n M(B) \otimes_B J^{-n}\text{;}\) this makes sense because \(J\) is an invertible ideal).
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 \(H^i_{\et}(X_{\overline{K}}, \QQ_p)\) where \(X\) is a smooth proper \(\calO_K\)-scheme. For \(i \gt 0\) such an extension cannot be unramified, as it would be if \(\QQ_p\) were replaced by \(\QQ_\ell\) for some prime \(\ell \neq p\text{,}\) 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].

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].

Definition 28.3.1.

Let \(W\) be the ring scheme of \(p\)-typical Witt vectors. That is, for any ring \(A\text{,}\) there is a natural (in \(A\)) bijection between the underlying set of the ring \(W(A)\) and the set of morphisms \(\Spec A \to W\text{.}\)
Write \(W\) as \(\Spec \ZZ[x_0, x_1,\dots]\) in terms of the Witt coordinates, and let \(W_n = \Spec \ZZ[x_0,\dots,x_{n-1}]\) be the \(n\)-th truncation of \(W\text{.}\) For \(n \geq 1\text{,}\) let \(W_{\prim,n}\) be the completion of \(W_n\) along the locally closed subscheme defined by the conditions
\begin{equation*} p = x_0 = 0, \qquad x_1 \neq 0. \end{equation*}
Since \(W_n^\times\) acts on \(W_{\prim,n}\) by multiplication, we can form the quotient
\begin{equation*} \CW_n = W_{\prim,n}/W_n^\times \end{equation*}
in the category of sheaves on the category of \(p\)-adic formal schemes. Similarly, we may form the sheaf \(\CW = \lim_n \CW_n\text{,}\) called the Cartier-Witt stack.
By definition, for any oriented prism \((A, (d))\) we get a morphism \(\Spec A \to W_{\prim}\) under which the distinguished element \((x_0, x_1, \dots)\) of \(\calO(W_{\prim})\) pulls back to \(d\text{.}\) Consequently, for any prism \((A,I)\text{,}\) we get a morphism \(\Spec A \to \CW\text{.}\)

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].

Subsection 28.4 Prismatic Dieudonné theory

The reference for this topic is [6].

Definition 28.4.1.

We say that \(R \in \Ring\) is quasi-syntomic if \(R\) is \(p\)-complete with bounded \(p\)-power torsion and the cotangent complex \(L_{R/\ZZ_p}\) has \(p\)-complete Tor amplitude in \([-1,0]\text{.}\) For example, a noetherian lci ring is quasi-syntomic, as in a regular semilens (Definition 18.2.1) with bounded \(p\)-power torsion.

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).

Subsection 28.5 Logarithmic prismatic cohomology

The reference for this topic is [88].

Definition 28.5.1.

A prelog structure on a ring \(A\) consists of a monoid \(M\) and a morphism \(\alpha\colon M \to A\) of monoids. In general, one prefers to “sheafify” this definition to define a log structure, as in [77].

Example 28.5.2.

Suppose that \(Z\) is an effective Cartier divisor on \(\Spec A\text{.}\) If the components of \(Z\) are cut out by some elements \(x_1,\dots,x_r\) of \(A\text{,}\) 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).
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.

Definition 28.5.3.

As per [88], Definition 2.2, a \(\delta_{\log}\)-ring is a tuple \((A, \delta, \alpha, \delta_{\log})\) in which \((A, \delta)\) is a \(\delta\)-ring, \(\alpha\colon M \to A\) defines a prelog structure on \(A\text{,}\) and \(\delta_{\log}\colon M \to A\) is a function satisfying the following conditions.
  1. For \(e \in M\) the identity element, \(\delta_{\log}(e) = 0\text{.}\)
  2. For \(m \in M\text{,}\)
    \begin{equation*} \delta(\alpha(m)) = \alpha(m)^p \delta_{\log}(m). \end{equation*}
  3. For \(m,m' \in M\text{,}\)
    \begin{equation*} \delta_{\log}(mm') = \delta_{\log}(m) + \delta_{\log}(m') + p \delta_{\log}(m) \delta_{\log}(m'). \end{equation*}
An important special case is when \(\delta_{\log} = 0\) identically. In this case, we say that the \(\delta_{\log}\)-ring in question is constant (or of rank 1 in Koshikawa's terminology).
We report some examples from [88], Example 2.4.

Example 28.5.4.

For \((A, \delta) \in \Ring_\delta\text{,}\) consider the canonical log structure where \(M = A^\times\) and \(\alpha\colon A^\times \to A\) is the canonical inclusion. There is then a unique \(\delta_{\log}\)-ring structure given by
\begin{equation*} \delta_{\log}(x) = \frac{\delta(x)}{x^p} \qquad (x \in A^\times). \end{equation*}

Example 28.5.5.

Let \(R \in \Ring_{\FF_p}\) be perfect and view \(W(R)\) as a \(\delta\)-ring via the Witt vector Frobenius. The prelog structure given by the constant lift \([\bullet]\colon R \to W(R)\) then admits a constant \(\delta_{\log}\)-structure.

Example 28.5.6.

For any monoid \(M\text{,}\) we may view the monoid ring \(\ZZ_{(p)}[M]\) as a \(\delta\)-ring in such a way that the elements of \(M\) are all constant. The prelog structure given by the natural map \(M \to \ZZ_{(p)}[M]\) then admits a constant \(\delta_{\log}\)-structure.

Example 28.5.7.

Given a \(\delta_{\log}\)-ring \(A\) and a morphism \(A \to B\) of \(\delta\)-rings, we may upgrade to a morphism of \(\delta_{\log}\)-rings by equipping \(B\) with the prelog structure \(M \to A \to B\) and the \(\delta_{\log}\)-structure \(M \stackrel{\delta_{\log}}{\to} A \to B\text{.}\)

Remark 28.5.8.

One can continue in this manner to extend much of the formalism of \(\delta\)-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].
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 \(\AA^1/\GG_m\text{.}\)