Prisms and Hodge-Tate comparisons

A few weeks ago, Bhatt and Scholze uploaded a preprint of their paper 'Prisms and Prismatic Cohomology' to arxiv.

In Theorem 6.3 they state their Hodge-Tate comparison. Recently, I started reading on Hodge decomposition and Hodge-Tate composition. However, all Hodge-Tate comparison theorems I encountered so far (see for example Theorem 1.3 here) are of a different form and neither could I relate the two to each other nor do I see a connection to Hodge-Tate composition. I would be happy if someone could give a (short) explanation for the name or point out references which I could consult.

tl;dr The Hodge-Tate comparison isomorphism relates the reduction mod $$I$$ of prismatic cohomology to something similar to the "Hodge-Tate cohomology" $$\bigoplus_{i+j = k} H^i(X, \Omega^j_{X/K})$$. Together with the étale comparison theorem relating prismatic cohomology away from $$V(I)$$ to étale cohomology, this gives the integral $$p$$-adic Hodge theory version of the usual Hodge-Tate comparison theorem.

Let $$X$$ be a smooth and proper variety over a $$p$$-adic field $$K$$ (i.e. a finite extension of $$\mathbf{Q}_p$$). Let $$C = \mathbf{C}_p$$ be the $$p$$-adic completion of an algebraic closure of $$\overline{K}$$.

The usual Hodge-Tate comparison theorem relates the $$p$$-adic étale cohomology of $$X$$ to its "Hodge-Tate cohomology".

On one side, we consider the étale cohomology $$\mathrm{H}^k_{\mathrm{et}}(X_{\overline{K}}, \underline{\mathbf{Z}_p})$$: this is a $$\mathbf{Z}_p$$-module with an action of $$G_K$$, the absolute Galois group of $$K$$.

On the other side, we consider the "Hodge-Tate cohomology" of $$X$$. This is: $$\mathrm{H}^k_{\mathrm{HT}}(X/K) = \bigoplus_{i + j = k} \mathrm{H}^i(X, \Omega^j_{X/K})$$ It is a graded $$K$$-vector space with $$\mathrm{gr}^j \ \mathrm{H}^k_{\mathrm{HT}}(X/K) = \mathrm{H}^i(X, \Omega^j_{X/K})$$.

Now, there is a Hodge-Tate spectral sequence, compatible with the Galois actions on both sides. $$\mathrm{E}_2^{i,j} = \mathrm{H}^i(X, \Omega^j_{X/K}) \otimes_K C(-j) \Rightarrow \mathrm{H}^k_{\mathrm{et}}(X_{\overline{K}}, \underline{\mathbf{Z}_p}) \otimes_{\mathbf{Z}_p} C$$

This spectral sequence is defined more generally when $$X$$ is any rigid-analytic space over $$C$$, and is known to always degenerate at $$\mathrm{E}_2$$ (Thm. 1.7 in "Integral $$p$$-adic Hodge theory" by Bhatt, Morrow, and Scholze). In the case that $$X$$ is actually defined over $$K$$, there is even a canonical splitting, giving a canonical $$G_K$$-equivariant isomorphism $$\mathrm{H}^k_{\mathrm{et}}(X_{\overline{K}}, \underline{\mathbf{Z}_p}) \otimes_{\mathbf{Z}_p} C \simeq \bigoplus_j \mathrm{H}^i(X, \Omega^j_{X/K}) \otimes_K C(-j)$$

Now, prismatic cohomology is about integral $$p$$-adic Hodge theory, meaning that we want to get comparison theorems between cohomology theories without having to invert $$p$$. More precisely, assume that $$X$$ is the generic fiber of a smooth proper (formal) scheme $$\mathscr{X}$$ over $$\mathscr{O}_K$$. Then we can ask for a relationship between the $$\mathbf{Z}_p$$-module $$\mathrm{H}^k_{\mathrm{et}}(X_{\overline{K}}, \underline{\mathbf{Z}_p})$$ and the $$\mathscr{O}_K$$-module $$\mathrm{H}^k_\mathrm{HT}(\mathscr{X}/\mathscr{O}_K) := \bigoplus_{i+j =k} \mathrm{H}^i(\mathscr{X}, \Omega^j_{\mathscr{X}/\mathscr{O}_K})$$. Unfortunately, a comparison theorem of the above form does not hold in this context.

Instead, prismatic cohomology constructs a "universal" cohomology object which "interpolates" between étale cohomology and Hodge-Tate cohomology (as well as de Rham cohomology, crystalline cohomology, etc).

Assume for ease of notation that $$K$$ is unramified over $$\mathbf{Q}_p$$. Let $$A$$ be the ring $$\mathscr{O}_K[[u]]$$, equipped with a natural Frobenius endomorphism sending $$u$$ to $$u^p$$. There is an $$\mathscr{O}_K$$-linear surjection $$A \rightarrow \mathscr{O}_K$$ defined by $$u \mapsto p$$, with kernel $$I = (u - p)$$. Then $$(A, I)$$ is an example of a prism. Cohomology of the structure sheaf on the prismatic site of $$\mathscr{X}$$ relative to $$A$$ gives a complex $$\mathrm{R} \Gamma_{\mathrm{prism}}(\mathscr{X}/A)$$ in the derived category of $$A$$-modules.

Loosely, $$\mathrm{R} \Gamma_{\mathrm{prism}}(\mathscr{X}/A)$$ recovers Hodge-Tate cohomology on the closed set $$\mathrm{Spec }\ \mathscr{O}_K = V(u - p) \subseteq \mathrm{Spec }\ A$$, and recovers $$p$$-adic étale cohomology of $$\mathscr{X}$$ on the open set $$\mathrm{Spec }\ A[\frac{1}{u-p}] \subseteq \mathrm{Spec }\ A$$. Bhatt and Scholze refer to the former as the Hodge-Tate comparison theorem, and the latter as the étale comparison theorem. Putting these two statements together describes the integral version of the relationship between Hodge-Tate and étale cohomologies.

More precisely, there is an object $$\Delta_{\mathscr{X}/A}$$ in the derived category of Zariski (or étale) sheaves of $$A$$-modules on $$\mathscr{X}$$ such that $$\mathrm{R} \Gamma(\Delta_{\mathscr{X}/A}) = \mathrm{R} \Gamma_{\mathrm{prism}}(\mathscr{X}/A)$$. We consider its (derived) restriction to $$\mathrm{Spec} \ A/I$$, given by $$\overline{\Delta}_{\mathscr{X}/A} := \Delta_{\mathscr{X}/A} \otimes^L_A (A/I)$$, so we have $$\mathrm{R} \Gamma_{\mathrm{prism}}(\mathscr{X}/A) \otimes^L_A (A/I) \simeq \mathrm{R} \Gamma(\overline{\Delta}_{\mathscr{X}/A})$$.

Now, the Hodge-Tate comparison theorem as stated in the Bhatt-Scholze paper gives a canonical isomorphism of $$\mathscr{O}_K$$-modules. Here, if $$M$$ is an $$\mathscr{O}_K$$-module, $$M\{j\} = M \otimes_{\mathscr{O}_K} (I/I^2)^{\otimes j}$$. $$\mathrm{H}^j(\overline{\Delta}_{\mathscr{X}/A}) \simeq \Omega^j_{\mathscr{X}/\mathscr{O}_K}\{-j\}$$ This gives us a hypercohomology spectral sequence $$\mathrm{E}_2^{i,j} = \mathrm{H}^i(\mathscr{X}, \Omega^j_{\mathscr{X}/\mathscr{O}_K})\{-j\} \Rightarrow \mathbf{H}^{i+j}(\mathscr{X}, \overline{\Delta}_{\mathscr{X}/A})$$ The right side should be thought of as a "derived correction" of $$\mathrm{H}^{i+j}_{\mathrm{prism}}(\mathscr{X}/A) \otimes_A (A/I)$$.

To relate this to étale cohomology, we must pass to a bigger prism. Let $$A_{\mathrm{inf}} = \mathrm{W}(\mathscr{O}_{C^\flat})$$. This has a natural Frobenius automorphism $$\varphi$$ lifting the one on $$\mathscr{O}_{C^\flat}$$ and a surjection $$\widetilde{\theta} \colon A_{\mathrm{inf}} \rightarrow \mathscr{O}_C$$ with kernel $$J = (d)$$ for a certain element $$d \in A_{\mathrm{inf}}$$. ($$\widetilde{\theta} = \varphi^{-1} \circ \theta$$, where $$\theta$$ is the usual map as defined by Fontaine). There is a map of prisms $$(A, I) \rightarrow (A_{\mathrm{inf}}, J)$$ which lifts the inclusion $$\mathscr{O}_K \hookrightarrow \mathscr{O}_C$$.

Then we have $$\mathrm{R} \Gamma_{\mathrm{prism}}(\mathscr{X}_{\mathscr{O}_C}/A_{\mathrm{inf}}) \simeq \mathrm{R} \Gamma_{\mathrm{prism}}(\mathscr{X}/A) \otimes_A^L A_{\mathrm{inf}}$$. In particular, the above gives us a spectral sequence $$\mathrm{E}_2^{i,j} = \mathrm{H}^i(\mathscr{X}, \Omega^j_{\mathscr{X}/\mathscr{O}_K}) \otimes_{\mathscr{O}_K} \mathscr{O}_C\{-j\} \Rightarrow \mathbf{H}^{i+j}(\mathscr{X}_{\mathscr{O}_C}, \overline{\Delta}_{\mathscr{X}_{\mathscr{O}_C}/A_\mathrm{inf}})$$ where the right side should be thought of as a "derived correction" of $$\mathrm{H}^{i+j}_{\mathrm{prism}}(\mathscr{X}_{\mathscr{O}_C}/A_{\mathrm{inf}}) \otimes_{A_{\mathrm{inf}}, \widetilde{\theta}} \mathscr{O}_C$$

On the other hand, the étale comparison theorem gives an isomorphism, equivariant with respect to the natural $$G_K$$-actions on both sides: $$\mathrm{R}\Gamma_{\mathrm{et}}(X_C, \underline{\mathbf{Z}_p}) \otimes_{\mathbf{Z}_p} A_{\mathrm{inf}}[1/d] \simeq \mathrm{R}\Gamma_{\mathrm{prism}}(\mathscr{X}_{\mathscr{O}_C}/A_{\mathrm{inf}})\otimes_{A_{\mathrm{inf}}}^L A_{\mathrm{inf}}[1/d]$$ Note that the right side is isomorphic to $$\mathrm{R} \Gamma_{\mathrm{prism}}(\mathscr{X}/A) \otimes_A^L A_{\mathrm{inf}}[1/d]$$.

Thus, the étale and Hodge-Tate comparison theorems together show that $$\mathrm{R} \Gamma_{\mathrm{prism}}(\mathscr{X}/A)$$ determines a deformation from $$\mathrm{H}^{i+j}_{\mathrm{et}}(X, \underline{\mathbf{Z}_p})$$ to an object related to the Hodge-Tate cohomology $$\bigoplus_j \mathrm{H}^i(\mathscr{X}, \Omega^j_{\mathscr{X}/\mathscr{O}_K})\{-j\}$$.