Notes on prismatic cohomology

Definition 28.1.

Let $A\to B$ be a morphism in $\mathbf{Ring}$ 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 $(\mathrm{Spec}R{)}_{\mathbf{\Delta }}^{\mathrm{op}}\text{,}$ 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 $(B,J)\to (B^{\prime },J^{\prime })$ for which the induced morphism $B/J\to B^{\prime }/J^{\prime }$ is $R$-linear. Taking the opposite category yields the absolute prismatic site of $R\text{,}$ denoted $(\mathrm{Spec}R{)}_{\mathbf{\Delta }}\text{,}$ which we equip with the indiscrete topology. Note that there is no base prism in the definition.

Definition 28.4.

Let $\mathcal{C}$ be a site equipped with a sheaf of rings $\mathcal{O}$ (or more generally a ringed topos). A crystal on $(\mathcal{C},\mathcal{O})$ is a sheaf of $\mathcal{O}$-modules locally obtained by tensoring $\mathcal{O}$ 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 $X\in \mathcal{C}$ to a finite projective $\mathcal{O}(X)$-module $M(X)$ and to each morphism $Y\to X$ in $\mathcal{C}$ an isomorphism $M(X){\otimes }_{\mathcal{O}(X)}\mathcal{O}(Y)\cong M(Y)$ 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 $(B,J)\in (\mathrm{Spec}R{)}_{\mathbf{\Delta }}\text{,}$ we specify an isomorphism ${\varphi }^{\ast }M(B)[J^{-1}]\to M(B)[J^{-1}]$ compatible with the morphisms in the site (where $M(B)[J^{-1}]={\mathrm{colim}}_{n}M(B){\otimes }_B{J}^{-n}\text{;}$ 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 $\mathrm{Spec}A\to W\text{.}$

Write $W$ as $\mathrm{Spec}\mathbb{Z}[x_0,x_1,\dots ]$ in terms of the Witt coordinates, and let $W_n=\mathrm{Spec}\mathbb{Z}[x_0,\dots ,x_{n-1}]$ be the $n$-th truncation of $W\text{.}$ For $n\ge 1\text{,}$ let $W_{\mathrm{prim},n}$ be the completion of $W_n$ along the locally closed subscheme defined by the conditions

Since $W_n^{\times }$ acts on $W_{\mathrm{prim},n}$ 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 $\mathrm{CW}=\underset{n}{lim}{\mathrm{CW}}_n\text{,}$ called the Cartier-Witt stack.

By definition, for any oriented prism $(A,(d))$ we get a morphism $\mathrm{Spec}A\to W_{\mathrm{prim}}$ under which the distinguished element $(x_0,x_1,\dots )$ of $\mathcal{O}(W_{\mathrm{prim}})$ pulls back to $d\text{.}$ Consequently, for any prism $(A,I)\text{,}$ we get a morphism $\mathrm{Spec}A\to \mathrm{CW}\text{.}$

Definition 28.10.

We say that $R\in \mathbf{Ring}$ is quasi-syntomic if $R$ is $p$-complete with bounded $p$-power torsion and the cotangent complex $L_{R/{\mathbb{Z}}_p}$ 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 $m\in M\text{,}$

   $$\delta (\alpha (m))=\alpha (m{)}^p{\delta }_{\log }(m).$$

3. For $m,m^{\prime }\in M\text{,}$

   $${\delta }_{\log }(m{m}^{\prime })={\delta }_{\log }(m)+{\delta }_{\log }(m^{\prime })+p{\delta }_{\log }(m){\delta }_{\log }(m^{\prime }).$$

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).