In this section, we prove the Hodge-Tate comparison theorem (Theorem 12.4.1). Our strategy will be to build up from the special case treated in Section 14, in which we used the crystalline prism \((\ZZ_p, (p))\) as the base and the ring \(R = \FF_p[x_1,\dots,x_r]\text{.}\)
We also assert the crystalline and de Rham comparison theorems. These are technically a bit more involved, so we do not include all of the details here.
Throughout, we fix a bounded prism \((A,I)\) and denote its slice \(A/I\) by \(\overline{A}\text{.}\)
Subsection15.1Étale localization and base change
Remark15.1.1.
Recall that a morphism \(R \to S\) of rings is smooth if and only if locally on \(\Spec(S)\text{,}\) it can be written in the form \(R \to R[x_1,\dots,x_r] \to S\) where the second map is étale (see [117], tag 054L). Similarly, if \(R \to S\) is a \(p\)-completely smooth map, then locally on \(\Spec(S/p)\) it can be written in the form \(R \to R \langle x_1,\dots,x_r\rangle \to S\) where the second map is \(p\)-completely étale (use Proposition 6.5.3 to reduce to the previous statement).
If \(\overline{A} \to R\) is \(p\)-completely smooth, then \(\widehat{\Omega}^1_{R/\overline{A}}\) is a finite projective \(R\)-module (again by Proposition 6.5.3 to reduce to the corresponding statement about differentials for a smooth morphism). Consequently, if \(R \to S\) is \(p\)-completely étale, then \(\widehat{\Omega}^i_{R/\overline{A}} \widehat{\otimes}_R S \cong \widehat{\Omega}^i_{S/\overline{A}}\) for all \(i\text{.}\)
This suggests the strategy of proving the Hodge-Tate comparison for a general \(p\)-completely smooth \(\overline{A}\)-algebra \(R\) by proving the corresponding compatibility with étale maps, and then using this to reduce to the case \(R = \overline{A} \langle X_1,\dots,X_r \rangle\text{.}\) The first step in this program is executed by Lemma 15.1.2.
Lemma15.1.2.Étale localization for Hodge-Tate cohomology.
Let \(R \to S\) be a \(p\)-completely étale map of \(p\)-completely smooth \(\overline{A}\)-algebras. Then the natural map \(\overline{\Prism}_{R/A} \widehat{\otimes}^L_R S \to \overline{\Prism}_{S/A}\) is an isomorphism.
The restriction functor \((S/A)_\Prism \to (R/A)_\Prism\) admits a right adjoint \(F\) taking \((B \to B/IB \leftarrow R)\) to \((B_S \to B_S/IB_S \leftarrow S)\) where \(B_S/IB_S = B/IB \widehat{\otimes}^L_R S\) and \(B \to B_S\) is the unique lift of the étale morphism \(B/IB \to B/IB \widehat{\otimes}^L_R S\) given by the the henselian property of derived completions (see Corollary 6.3.2), promoted from \(\Ring \) to \(\Ring_\delta\) using Exercise 6.7.11. Applying \(F\) to a weakly final object of \((R/A)_\Prism\) (Proposition 11.6.5), we obtain a weakly final object of \((S/A)_\Prism\text{;}\) since \(F\) also preserves finite products, we can take a complex computing \(\overline{\Prism}_{R/A}\) and apply \(F\) term by term to obtain a complex computing \(\overline{\Prism}_{S/A}\text{.}\) It thus remains to compare this with \(\overline{\Prism}_{R/A} \widehat{\otimes}^L_R S\text{;}\) we have a natural isomorphism at the level of simplicial rings, and (since \(R \to S\) is \(I\)-completely flat and thus has finite \(I\)-complete Tor amplitude) we may now deduce the claim from Exercise 10.6.3. (Compare [25], Lemma 4.19.)
Another tool we will use is a base-change assertion. This will allow us to simplify the base ring \(\overline{A}\) in some cases.
Lemma15.1.3.Base change for prismatic and Hodge-Tate cohomology.
Let \(R\) be a \(p\)-completely smooth \(\overline{A}\)-algebra. Let \((A, I) \to (A', I')\) be a map of bounded prisms such that \(A \to A'\) has finite \((p, I)\)-complete Tor amplitude (e.g., a faithfully flat morphism). For \(R' = R \widehat{\otimes}_A A'\text{,}\) we have natural isomorphisms
\begin{equation*}
\Prism_{R/A} \widehat{\otimes}^L_A A' \cong \Prism_{R'/A'}, \qquad
\overline{\Prism}_{R/A} \widehat{\otimes}^L_A A' \cong \overline{\Prism}_{R'/A'}.
\end{equation*}
Let \((P, IP)\) be a weakly final object of \((R/A)_{\Prism}\text{,}\) then apply Remark 11.6.7 to construct a Čech-Alexander complex computing \(\Prism_{R/A}\text{.}\) Then \(\Prism_{R'/A'}\) is computed by the complex obtained by applying \(\bullet \widehat{\otimes}_A A'\) termwise. Under the hypothesis on the Tor amplitude, we may apply Exercise 10.6.3 to conclude. (Compare [25], Lemma 4.18.)
As an immediate application of étale localization and base change, we upgrade our previous statement about the Hodge-Tate comparison for crystalline prisms (Proposition 14.4.12).
Lemma15.1.4.
Suppose that \((A,I)\) is a crystalline prism and \(R\) is a smooth \(\overline{A}\)-algebra. Then the Hodge-Tate comparison map (12.1) is an isomorphism.
By Proposition 14.4.12 and Lemma 15.1.3, the claim holds when \(R = \overline{A}[x_1,\dots,x_r]\text{.}\) We may then deduce the general case using Lemma 15.1.2. (Compare [25], Corollary 5.5.)
Subsection15.2Comparing a universal prism to a crystalline prism
Remark15.2.1.
We reproduce [18], Lecture VI, Remark 2.2, in order to justify why we can't directly transpose the proof of the Hodge-Tate comparison from crystalline prisms to more general prisms. Suppose that \(I = (d)\text{.}\) Consider the object \(P = A[x] \in \Ring_\delta\) with \(\delta(x) = 0\text{.}\) Then the derived \((p,d)\)-completion of \(P\{\phi(x)/d\}\) does not equal the derived \((p,d)\)-completion of the divided power envelope of \((P, (x))\text{.}\) A typical example of this is the case \((A, (d)) = (\ZZ_p\llbracket q-1 \rrbracket, ([p]_q))\text{;}\) in this case, the derived \((p,d)\)-completion of \(P\{\phi(x)/d\}\) will end up coinciding with the derived \((p,d)\)-completion of the \(q\)-divided power envelope of \((P, (x))\text{.}\)
Lemma15.2.2.
Let \((A, (d))\) be the universal oriented prism (Example 5.3.5) and put \(B = A \{ \phi(d)/p\}^\wedge\) (the completion being the derived \(p\)-completion).
The ring \(B\) is classically \((p,d)\)-complete.
The ring \(B\) equals the classical \(p\)-completion of the divided power envelope of \((A, (d))\) (and is \(p\)-torsion-free).
Since \(A\) is \(p\)-torsion-free, \(B\) is not just derived \(p\)-complete but also classically \(p\)-complete. Since \(d^p = p(\phi(d)/p - \delta(p))\) is divisible by \(p\text{,}\)\(B\) is also classically \((p,d)\)-complete. By Corollary 14.3.4, \(B\) equals the classical \(p\)-completion of the divided power envelope of \((A, (d))\text{.}\) (Compare [25], Construction 6.1.)
Remark15.2.3.
In the notation of Lemma 15.2.2, \(B\) is again a \(\delta\)-ring and both \(\phi(d)\) and \(p\) are distinguished elements. Since \(\phi(d)\) is divisible by \(p\text{,}\) we may apply Lemma 5.2.1 to deduce that \(\phi(d)\) and \(p\) generate the same ideal in \(B\text{.}\) In other words, the composition of maps of \(\delta\)-rings
\begin{equation*}
A \to B \stackrel{\phi_B}{\to} B
\end{equation*}
in which the target is crystalline! This will ultimately allow us to transfer information from the crystalline case of the Hodge-Tate comparison to the universal case, and then from there to the general case.
Lemma15.2.4.
With notation as in Lemma 15.2.2, let \(\alpha\colon A \to B\) be the composition of \(\phi\colon A \to A\) with the natural map \(A \to B\text{.}\)
The map \(A/p \to B/p\) induced by \(\alpha\) factors as a composition
in which the first map has finite \((p,d)\)-complete Tor amplitude and the second and third maps are faithfully flat.
The functor \(\widehat{\alpha^*}\colon D_{\comp}(A) \to D_{\comp}(B)\) is conservative (i.e., reflects isomorphisms). Here \(D_{\comp}(*)\) denotes the subcategory of derived \((p,d)\)-complete objects of \(D(*)\text{.}\)
For any \(p\)-completely smooth \(\overline{A}\)-algebra \(R\text{,}\) writing \(R_B = R \widehat{\otimes}_A B\text{,}\) the map \(\widehat{\alpha^*} \Prism_{R/A} \to \Prism_{R_B/B}\) is an isomorphism.
In (1), the first map has finite \((p,d)\)-complete Tor amplitude because \(d\) is not a zero-divisor in \(A/p\text{;}\) the second and third maps are faithfully flat by construction. (Compare [25], Construction 6.1.)
Proposition15.2.5.
Let \((A, (d))\) be the universal oriented prism (Example 5.3.5) and put \(R = \overline{A}\langle x_1,\dots,x_r \rangle\text{.}\) Then the Hodge-Tate comparison map (12.1) is an isomorphism.
Let \((A,(d)) \to (B, (d))\) be the morphism from Lemma 15.2.2. By Lemma 15.2.4, we may apply Lemma 15.1.3 to reduce the claim from the prism \((A, (d))\) to \((B, (d))\text{.}\) The latter is a crystalline prism by Remark 15.2.3, so Lemma 15.1.4 applies.
Subsection15.3Hodge-Tate comparisons
We finally treat the Hodge-Tate comparison theorem (Theorem 12.4.1) in general. Before treating the general case, we give an easier argument that covers many cases of interest.
Proposition15.3.1.
Suppose that \((A,I) = (A,(d))\) is an oriented (bounded) prism and the map from the universal oriented prism has finite \((p,d)\)-complete Tor-amplitude. (For example, this last condition holds if \(d\) is a non-zerodivisor in \(A/p\text{.}\)) Let \(R\) be any \(p\)-completely smooth \(\overline{A}\)-algebra. Then the Hodge-Tate comparison map (12.1) is an isomorphism.
By Lemma 15.1.2, we may reduce to the case \(R = \overline{A} \langle x_1,\dots,x_r \rangle\text{.}\) In this case, since we assumed the map \((A_0, (d)) \to (A, (d))\) from the universal oriented prism has finite \((p,d)\)-complete Tor amplitude, we can use Lemma 15.1.3 to transfer the desired result from \((A_0,(d))\) (to which Proposition 15.2.5 applies) to \((A, (d))\text{.}\) (Compare [25], Proposition 6.2.)
Proposition15.3.2.
In the full generality of Theorem 12.4.1, the Hodge-Tate comparison map (12.1) is an isomorphism.
Again by Lemma 15.1.2, we may reduce to the case \(R = \overline{A} \langle x_1,\dots,x_r \rangle\text{.}\) Using Lemma 5.2.5, we can further reduce to the case where \((A,I) = (A, (d))\) is an oriented prism. Let \((A_0, (d)) \to (A, (d))\) be the morphism from the universal oriented prism. Form a diagram as in Proposition 15.2.5 in which \(\alpha\) is the map from Lemma 15.2.2 and the square is a pushout of \((p,d)\)-complete simplicial commutative rings. (The key technical complication here is that \(E\) is not necessarily an ordinary ring.)
Figure15.3.3.
The arrow \(\ZZ_p \to D_0\) promotes to a map \((\ZZ_p, (p)) \to (D_0, (p))\) of prisms, so we also have a map \(\gamma\colon (\ZZ_p,(p)) \to (E, (p))\) of oriented prisms. Using the explicit description of prismatic cohomology given in Corollary 14.4.4, we may produce a natural isomorphism \(\widehat{\beta^*} \overline{\Prism}_{R/A} \cong \widehat{\gamma^*} \overline{\Prism}_{\FF_p[x_1,\dots,x_r]/\ZZ_p}\text{.}\) By Proposition 14.4.12, we know that the Hodge-Tate comparison map is an isomorphism for the prism \((\ZZ_p,(p))\) and the ring \(\FF_p[x_1,\dots,x_r]\text{;}\) combining this with the previous isomorphism, we deduce that the original Hodge-Tate comparison map becomes an isomorphism after applying \(\widehat{\beta^*}\text{.}\) Since this last functor is conservative (because \(\widehat{\alpha^*}\) is conservative by Lemma 15.2.4), it is itself an isomorphism. (Compare [25], Proposition 6.2.)
Subsection15.4The crystalline and de Rham comparisons
On a related note, we describe the comparison between prismatic, crystalline, and de Rham cohomology under some mild restrictions (in addition to our running condition that \((A,I)\) is bounded). One complication is that we do not have an analogue of Lemma 15.1.2 for prismatic cohomology: for \(R \to S\) an \(I\)-completely etale morphism of \(\overline{A}\)-algebras, there is no obvious base change functor to relate \(\Prism_{R/A}\) to \(\Prism_{S/A}\text{.}\)
Theorem15.4.1.
Assume that \((A,(p))\) is a crystalline prism and let \(J\) be an ideal of \(A\) containing \(p\) on which \(A\) admits divided powers valued in \(I\) (that is, \(J\) is a pd-ideal of \(A\)). Let \(\psi\colon A/J \to A/p\) be the morphism induced by the Frobenius on \(A/p\text{.}\) Let \(R\) be a smooth \(A/J\)-algebra and put \(R^{(1)} = R \otimes_{A/I, \psi} A/p\text{.}\) Then there is a canonical isomorphism of \(\Prism_{R^{(1)}/A}\) with the crystalline cohomology of \(\Spf(R)\) relative to the pd-thickening \(\Spec(A/J) \subset \Spf(A)\text{;}\) more precisely, this is an isomorphism of \(E_\infty-A\)-algebras compatible with Frobenius.
The key point is to construct the map, as thereafter one can compute in local coordinates as in the proof of Lemma 14.4.8. See [25], Theorem 5.2.
Remark15.4.2.
Before continuing, we make an observation that will explain a somewhat odd condition in Theorem 15.4.3. Recall that by construction, the \(p\)-typical Witt vector functor \(W\) is a right adjoint to the forgetful functor \(\Ring_\delta \to \Ring\) (Definition 3.1.1). We may thus apply adjunction to the canonical map \(A \to \overline{A}\) to obtain a morphism \(A \to W(\overline{A})\) in \(\Ring_\delta\text{.}\)
Now let \(\psi\colon A \to W(\overline{A})\) be the composition of the resulting map with the Frobenius \(\phi\) on \(W(\overline{A})\text{.}\) This map carries \(I\) into \((p)\text{:}\) the original map \(A \to W(\overline{A})\) carries \(I\) into the image of the Verschiebung \(V\) on \(W(\overline{A})\text{,}\) and the composition \(\phi \circ V\) is multiplication by \(p\) (Definition 3.2.3). Hence the map \(\psi\) induces a morphism \((A, I) \to (W(\overline{A}), (p))\) of prisms provided that \(W(\overline{A})\) is \(p\)-torsion-free (so that the target is actually a prism).
Theorem15.4.3.
Assume that the prism \((A,I)\) is bounded and that \(W(\overline{A})\) is \(p\)-torsion-free. Let \(R\) be a \(p\)-completely smooth \(\overline{A}\)-algebra. Then there is a natural isomorphism \(\Prism_{R/A} \widehat{\otimes}^L_{A,\phi} \overline{A} \cong \Omega^\bullet_{R/\overline{A}}\) of commutative ring objects in \(D(\overline{A})\) (where the completion is the derived \(p\)-adic completion).
In light of Remark 15.4.2, it is enough to construct a functorial isomorphism of \(\Prism_{R/A} \widehat{\otimes}^L_{A,\psi} W(\overline{A}) \cong \Prism_{R'/W(\overline{A})}\text{,}\) where \(R' = R \widehat{\otimes}_{\overline{A},\psi} W(\overline{A})/p\) (and the stated isomorphism is given by Lemma 15.1.3), with the crystalline cohomology of \(R/p\) with coefficients in \(W(\overline{A})\text{.}\) This amounts to an application of Theorem 15.4.1. (Compare [25], Theorem 6.4.)
Remark15.4.4.
In Theorem 15.4.3, the condition that \(W(\overline{A})\) is \(p\)-torsion-free holds in two natural cases of interest: when \(A/I\) is \(p\)-torsion-free, or when \(I = (p)\) and \(\overline{A}\) is reduced. The result remains true without this condition, but this is more difficult and falls outside the scope of these notes; see [25], Corollary 15.4.
