Notes on prismatic cohomology

Take $A={\mathbb{Z}}_p$ with $d=p\text{.}$ Then $\delta (d)=1-p^{p-1}\equiv 1(\mathrm{mod}p)\text{,}$ so $p$ is distinguished. By the same token, by Remark 2.2.3, $p$ is distinguished in any $\delta$-ring.

This example is closely related to Fontaine’s theory of $(\phi ,\mathrm{\Gamma })$-modules. The original construction of Fontaine [49] described an equivalence of categories between the continuous representations of the absolute Galois group of ${\mathbb{Q}}_p$ on finite ${\mathbb{Z}}_p$-modules and a certain category of finite modules over the $p$-adic completion of $A[(q-1{)}^{-1}]\text{,}$ in which the continuous action of the monoid ${\mathbb{Z}}_p\setminus \{0\}$ on $A$ characterized by $\gamma (q)=q^{\gamma }$ is extended to the module. The elements of the $p$-adic completion can be viewed as formal Laurent series in $q-1$ with coefficients in ${\mathbb{Z}}_p\text{;}$ it was later shown by Cherbonnier and Colmez [39] that the base ring can be shrunk down to the subring consisting of Laurent series whose negative tails converge on some region (see also [80]).

The ring $A$ itself is the base ring of the theory of Wach modules [124], [13]; the $(\phi ,\mathrm{\Gamma })$-module associated to a Galois representation descends to a Wach module if and only if the representation is crystalline in Fontaine’s sense. Similar considerations apply if we enlarge $A$ by replacing ${\mathbb{Z}}_p$ with an unramified extension ${\mathfrak{o}}_K\text{,}$ where now the Galois group in question is that of $K\text{.}$

Let $K/{\mathbb{Q}}_p$ be a finite extension. Let $\pi$ be a uniformizer of $K\text{.}$ Let $W\subseteq {\mathfrak{o}}_K$ be the maximal unramified subring (i.e., the ring $W(k)$ where $k$ is the residue field of ${\mathfrak{o}}_K$). Take $A=W[[u]]$ with the $\delta$-structure extending the canonical one on $W$ for which $\varphi (u)=u^p\text{.}$ Take $d$ to be a generator of the kernel of the map $A\to {\mathfrak{o}}_K$ taking $u$ to $\pi \text{;}$ by projecting along the map $u\mapsto 0$ as in Example 5.1.4, we see that $d$ is distinguished.

The ring $A$ is the base ring of the theory of Breuil-Kisin modules ([85]), which provides an alternative to Wach modules that can be used to classify crystalline representations of the Galois group of a ramified extension of ${\mathbb{Q}}_p\text{.}$ See [37] for more on the parallel between the two constructions.

Let $A$ be the $(p,q-1)$-adic completion of ${\mathbb{Z}}_p[q^{p^{-\mathrm{\infty }}}]\text{.}$ By Proposition 3.3.6, we have an isomorphism $A\cong W(R)$ where $R$ is the $(q-1)$-adic completion of the coperfection of ${\mathbb{F}}_p[q-1]\text{.}$ In particular, $A$ has a unique $\delta$-ring structure, for which $\varphi (q)=q^p\text{;}$ note that in this case $\varphi$ is an automorphism. By Example 5.1.4, $d=[p{]}_q$ is a distinguished element, as is ${\varphi }^n(d)$ for any $n\in \mathbb{Z}\text{.}$

Let $K$ be the $p$-adic completion of the $p$-cyclotomic extension ${\mathbb{Q}}_p({\mu }_{p^{\mathrm{\infty }}})\text{.}$ The ring $R$ can then be identified with the perfection of ${\mathfrak{o}}_K/(p)$ by fixing a choice of a coherent sequence $({\zeta }_{p^n})$ of $p$-power roots of unity and identifying $q$ with this sequence; this identifies $R$ with the tilt of ${\mathfrak{o}}_K$ (see Section 7 for further discussion). In this context, the ring $A$ arises in Fontaine’s notation as the value of the functor ${\mathbf{A}}_{\inf }$ evaluated at the valuation ring ${\mathfrak{o}}_K\text{.}$

A $\delta$-pair $(A,I)$ with $I=(p)$ is a prism if and only if $A$ is $p$-torsion-free and classically $p$-complete. We say that such a prism is crystalline.

Let $A_0={\mathbb{Z}}_{(p)}\{d\}$ be the free $\delta$-ring in a single variable $d$ over ${\mathbb{Z}}_{(p)}\text{.}$ Let $S$ be the multiplicative subset of $A_0$ generated by ${\varphi }^n(\delta (d))$ for all $n\ge 0\text{.}$ By Lemma 2.4.8, the localization $A_1=S^{-1}{A}_0$ is also a $\delta$-ring. Let $A$ be the derived $(p,d)$-completion of $A_1\text{;}$ since $A_1$ is $p$-torsion-free, $A$ is classically $p$-complete. By construction, $d$ is a distinguished element of $A$ and $(A,dA)$ is a bounded prism. Moreover, $p,d$ is a regular sequence in $A$ and $\varphi :\bar{A}\to \bar{A}$ is $p$-completely flat (see Definition 6.5.1).