Notes on prismatic cohomology

Definition 28.1.

Let \(A \to B\) be a morphism in \(\Ring\) The Hochschild homology of \(B\) over \(A\) is the simplicial object \(K_\bullet\) 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:

Definition 28.3.

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 28.4.

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 28.5.

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

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).

Definition 28.8.

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

Since \(W_n^\times\) acts on \(W_{\prim,n}\) by multiplication, we can form the quotient

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 \(\Spec A \to \CW\text{.}\)

Definition 28.10.

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 18.6) with bounded \(p\)-power torsion.

Definition 28.13.

A prelog structure on a ring \(A\) consists of a monoid \(M\) and a morphism \(\alpha: M \to A\) of monoids. In general, one prefers to “sheafify” this definition to define a log structure, as in [72].

Definition 28.15.

As per [83], Definition 2.2, a \(\delta_{\log}\)-ring is a tuple \((A, \delta, \alpha, \delta_{\log})\) in which \((A, \delta)\) is a \(\delta\)-ring, \(\alpha: M \to A\) defines a prelog structure on \(A\text{,}\) and \(\delta_{\log}: M \to A\) 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).