Notes on prismatic cohomology

As indicated in Figure 14.1.3, the construction of relative Frobenius extends to an arbitrary morphism of schemes $f:Y\to X$ in characteristic $p\text{.}$ The example of a polynomial ring, and its description in the proof of Corollary 14.1.4, may be misleading: in general $Y$ and $Y^{(1)}$ will not be isomorphic over $X\text{.}$ For example, if $X$ is the spectrum of an algebraically closed field $k$ and $Y$ is an elliptic curve over $k\text{,}$ then the $j$-invariants $j(Y)$ and $j(Y^{(1)})$ will differ in general (the latter being the image of the former under ${\varphi }_k$).

Yet another construction can be given (for $R={\mathbb{F}}_p\text{,}$ then deducing the general case by base change) by lifting from ${\mathbb{F}}_p[x_1,\dots ,x_r]$ to ${\mathbb{Z}}_p[x_1,\dots ,x_r]\text{.}$ Given an element $f\in {\mathbb{Z}}_p[x_1^p,\dots ,x_r^p]$ lifting $\bar{f}\in {\mathbb{F}}_p[x_1^p,\dots ,x_r^p]\cong S^{(1)}\text{,}$ the element $p^{-1}df$ reduces to an element of ${\mathrm{\Omega }}_{S/R}^1$ independent of the choice of $f\text{,}$ and this is the image of $d\bar{f}$ under the Cartier map.

This last construction is quite similar to how the Cartier isomorphism will appear in the proof of the Hodge-Tate comparison for crystalline prisms (Proposition 14.4.12). In fact that result will itself establish the canonicality of the Cartier isomorphism, so we don’t need to worry much about it right now.

In any case, once canonicality is established by some means, we can easily promote Lemma 14.1.6 to a similar statement for any smooth morphism $R\to S$ in ${\mathbf{Ring}}_{{\mathbb{F}}_p}\text{.}$ We omit the details here, as we will see the same argument again soon (Lemma 15.1.2).

Our definition of “admits divided powers” is not quite the usual one: normally one also requires that ${\gamma }_n$ maps $J$ into $J$ for each $n\ge 1\text{.}$ For example, this occurs in Example 14.2.2 because (14.2) includes $p^{n-1}$ rather than $p^n\text{.}$ However, the last statement in Definition 14.2.4 ensures that this discrepancy doesn’t affect anything later.

When studying divided powers, it is common to use the initialism pd for the French phrase puissances divisées. For example, the divided power envelope is also called the pd-envelope.

Proposition 14.2.6 amounts to the computation of the crystalline cohomology of a point. We will see in Subsection 14.4 that the proof of the Hodge-Tate comparison isomorphism for a crystalline prism naturally passes through crystalline cohomology.

One potentially confusing point is that unlike de Rham cohomology in characteristic 0, the crystalline cohomology of a higher-dimensional affine space is not the same as that of a point! In fact, the crystalline cohomology of $\mathrm{Spec}{\mathbb{F}}_p[x_1,\dots ,x_r]$ is computed by the complex ${\hat{\mathrm{\Omega }}}_{P/{\mathbb{Z}}_p}^{\bullet }$ where $P={\mathbb{Z}}_p[x_1,\dots ,x_r{]}_{(p)}^{\wedge }$ (this ring already admits divided powers on $(p)$ by Example 14.2.2); taking the derived base change from ${\mathbb{Z}}_p$ to ${\mathbb{F}}_p$ yields ${\mathrm{\Omega }}_{R/{\mathbb{F}}_p}\text{,}$ whose cohomology we already know is quite large (Lemma 14.1.6). What Proposition 14.2.6 tells us is that the answer does not change if we include some extra “divided power variables”; see Proposition 14.2.8 for a concrete statement.

In any case, none of this has much meaning without an actual definition of crystalline cohomology itself. For that, see [15].

If $R$ is a ${\mathbb{Z}}_{(p)}$-algebra which admits a $\delta$-ring structure, then $R$ admits divided powers on some ideal $J$ if and only if ${\gamma }_p(x)\in R$ for all $x\in J\text{.}$ See Exercise 2.5.10.

Corollary 14.3.4 implies that the subring $A\{\varphi (f_1)/p,\dots ,\varphi (f_r)/p\}$ of $A[p^{-1}]$ is independent of the choice of $\delta$-structure on $A\text{,}$ as the characterization via the divided power envelope of $(A,I)$ makes no reference to $\delta$ or $\varphi \text{.}$ By contrast, $A\{f_1/p,\dots ,f_r/p\}$ is not independent of this choice; see [25], Warning 2.40.

Lemma 14.3.2 asserts that the divided power envelope of $(R,J)$ as a ring is also the divided power envelope as a $\delta$-ring. The corresponding statement for $\lambda$-rings is false: for $A$ the free $\lambda$-ring on $x$ (over $\mathbb{Z}$), the divided power envelope of $(A,(x))$ is not a $\lambda$-subring of $A{\otimes }_{\mathbb{Z}}\mathbb{Q}\text{.}$ The issue here is with the use of Example 14.2.2: the ring $\mathbb{Z}$ does not admit divided powers on $(p)$ for any prime $p\text{.}$

This then leads to the question of describing the smallest $\lambda$-subring of $A{\otimes }_{\mathbb{Z}}\mathbb{Q}$ containing $A$ which admits divided powers on $(x)\text{.}$ We do not know the answer, but as a partial result we note that this ring contains the elements ${\delta }_p(x{)}^n/m$ where $p$ is a prime, $n$ is a positive integer, and $m$ is the prime-to-$p$ factor of $n!\text{.}$

A related question is whether Remark 14.3.5 admits a $\lambda$-ring analogue. That is, if $A\in {\mathbf{Ring}}_{\lambda }$ is $\mathbb{Z}$-flat and $f_1,\dots ,f_r\in A$ form a regular sequence in $A/p$ for every prime $p\text{,}$ does the minimal $\lambda$-subring of $A{\otimes }_{\mathbb{Z}}\mathbb{Q}$ containing $A$ which admits divided powers on $(f_1,\dots ,f_r)$ depend only on the underlying ring structure of $A$ and not its $\lambda$-structure?

Note that $P^{\bullet }$ is itself a Čech-Alexander complex, namely the one associated to the covering $\mathrm{Spf}P\to \mathrm{Spf}A$ in the category of $p$-adic formal schemes. In particular, $A\to P^{\bullet }$ is an isomorphism in $K(A)$ by the Čech-Alexander construction in the category of $p$-formal schemes (and even a homotopy equivalence of cosimplicial $A$-algebras, as per Definition 16.2.5).

Note that while $J^0$ is generated by ${\delta }^m(x_i)$ for all $m>0\text{,}$ $J^n$ is not generated by ${\delta }^m(x_{ij})$ for all $m>0\text{;}$ we must also add the generators $x_{ij}-x_{i{j}^{\prime }}$ for $j\ne j^{\prime }\text{.}$

To summarize, the rows of Figure 14.4.7 are quasi-isomorphic to each other and to ${\mathbf{\Delta }}_{R/A}\text{.}$

In Corollary 14.4.10, we write ${\varphi }^{\ast }{\mathbf{\Delta }}_{R/A}$ instead of ${\mathbf{\Delta }}_{R/A}$ to keep track of the fact that prismatic cohomology computes not crystalline cohomology per se, but rather a canonical Frobenius descent of it.