Notes on prismatic cohomology

We reduce at once to the case where all of the schemes in question are affine. In this case, the claim reduces at once to Exercise 7.4.4. (Compare [24], Lemma 3.18.)

For both assertions, we may assume that $X=\mathrm{Spec}A$ is affine; write $Z=\mathrm{Spec}A/I\text{.}$ Write $nE$ for the subscheme of $Y$ cut out by $I^n\text{.}$

For (1), we may assume $\mathcal{F}=\mathcal{O}\text{.}$ By our hypotheses, we have $\mathcal{O}(X)\cong \mathcal{O}(Y)$ and $\mathcal{O}(Z)\cong \mathcal{O}(E)$ by Stein factorization, and similarly after taking perfections. Since $X$ and $Z$ are both affine, it remains to check that $H^i(Y_{\mathrm{perf}},\mathcal{O})\to H^i(E_{\mathrm{perf}},\mathcal{O})$ is an isomorphism for each $i>0\text{.}$

To begin with, both assertions hold for the flat (fpqc) topology in place of the arc-topology thanks to classical faithfully flat descent ([117], tag 0238).

We next upgrade both assertions from the flat topology to the v-topology. Every v-covering is a cofiltered limit of h-coverings, so we may reduce to considering perfections of h-coverings of finite type ${\mathbb{F}}_p$-schemes. Since the h-topology is generated by faithfully flat coverings and proper surjective morphisms, and we already know descent for the former. we may reduce to considering the perfection of a proper surjective morphism. Moreover, by Raynaud-Gruson flattening [103], we may further reduce to considering the case of a blowup, to which we may apply Lemma 21.1.5.

Finally, we upgrade both assertions from the v-topology to the arc-topology. By passing to affines and then pulling back along a cover as in Example 20.3.8, we may reduce to considering a covering as in Example 20.3.10 (compare [21], Theorem 4.1). For this, apply Corollary 21.1.4 and Lemma 21.2.1.

This follows at once from Theorem 21.1.7. (Compare [21], Proposition 8.9.)

Let $M_1,M_2,M_{12}$ be objects of $\mathbf{Vect}(R_1),\mathbf{Vect}(R_2),\mathbf{Vect}(R_{12})$ equipped with isomorphisms $M_i{\otimes }_{R_i}{R}_{12}\cong M_{12}$ and put $M=\ker (M_1\oplus M_2\to M_{12})\text{;}$ we will show that $M\in \mathbf{Vect}(R)$ and that the induced maps $M{\otimes }_R{R}_i\to M_i$ are isomorphisms.

We next check that $M$ is a finite $R$-module. From the previous discussion, we see that there exist a finite free $R$-module $F$ and a morphism $F\to M$ of $R$-modules such that, for $F_i=F{\otimes }_R{R}_{\ast }\text{,}$ the induced map $F_i\to M_i$ is surjective. Put $N=\ker (F\to M)$ and $N_i=\ker (F_i\to M_i)\text{.}$ We have a diagram as in Figure 21.2.4 in which all of the squares commute and all of the rows and columns are exact, except possibly for the dashed arrows. However, because the modules $M_i$ are projective, the maps $N_i{\otimes }_{R_i}{R}_{12}\to N_{12}$ are isomorphisms, so all of the preceding logic applies to them also; this allows us to add the dashed horizontal arrow to the diagram, and hence also the dashed vertical arrow.

We next check that for each $i\text{,}$ $M{\otimes }_i{R}_i\to M_i$ is an isomorphism. Consider the commutative diagram as in Figure 21.2.5 with exact rows. By the previous logic, we know that both of the outside vertical maps are surjective. By the five lemma, the right vertical arrow is an isomorphism.

We finally check that $M$ is a projective $R$-module. By repeating the logic used to construct Figure 21.2.4, we obtain another commutative diagram as in Figure 21.2.6 with exact rows and columns. The element of ${\mathrm{Hom}}_{R_1}(M_1,M_1)\oplus {\mathrm{Hom}}_2(M_2,M_2)$ corresponding to the identity maps has zero horizontal image, so by the snake lemma it lifts to some ${\mathrm{Hom}}_{R_1}(M_1,F_1)\oplus {\mathrm{Hom}}_{R_2}(M_2,F_2)$ which maps to zero in ${\mathrm{Hom}}_{R_{12}}(M_{12},F_{12})\text{.}$ This gives us maps $M_1\to F_1,M_2\to F_2$ which agree on $M$ and map it into $F\text{;}$ the resulting map $M\to F$ splits the surjection $F\to M\text{,}$ showing that $M$ is projective. (Compare [82], Lemma 1.3.8, Lemma 1.3.9.)

We first verify descent for a v-covering $f:Y\to X\text{,}$ in which we may assume both schemes are qcqs. We can then write $Y$ as a filtered limit of some finitely presented $X$-schemes, each of which is itself a v-covering, with affine transition maps; we may thus reduce to dealing with a finitely presented $v$-covering. By arguing as in [106], Theorem 3.12, we may refine this covering by a composition of a quasicompact open covering with a proper surjective morphism. As descent for the former is immediate, we may further assume that $f$ is proper surjective. In this case, we are in the usual setting of cohomological descent for étale cohomology. For this, we may assume that $X$ is the spectrum of a strictly henselian local ring with closed point $x\text{.}$ By the proper base change theorem, $\mathcal{F}(Y)\cong \mathcal{F}(Y_x)\text{,}$ so we may check the claim after pulling back along $x\to X\text{.}$ But the resulting map $Y_x\to x$ has a section, so it satisfies descent for purely formal reasons. See [21], Proposition 5.2 for more details.

To obtain descent for the arc-topology, as in the proof of Theorem 21.1.7 we may use v-descent to reduce to a covering as in Example 20.3.10 in which $V$ is AIC. In this case, $V/\mathfrak{p}$ is also AIC, so both $V$ and $V/\mathfrak{p}$ are strictly henselian with the same residue field. It follows that the functor in question takes the same values on $V$ and $V/\mathfrak{p}\text{,}$ and takes the same values on $V_{\mathfrak{p}}$ and $\kappa (\mathfrak{p})\text{.}$ (Compare [21], Theorem 5.4.)