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 $\bigoplus_j \mathrm{H}^i(\mathscr{X}, \Omega^j_{\mathscr{X}/\mathscr{O}_K})\{-j\}$.


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