Applications of étale comparison

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Section 23 Applications of étale comparison

Reference.

[18], lecture IX; [25], section 9.

In this section, we describe some applications of the étale comparison theorem for prismatic cohomology (Theorem 22.6.1).

Subsection 23.1 Tilting of valuation rings

We prove the converse of Lemma 22.4.4 and recover the tilting correspondence for perfectoid fields (Theorem 8.3.4). This theme will be continued in the treatment of almost purity (Section 25).

Theorem 23.1.1.

Let $V$ be a lens. Then $V$ is an AIC valuation ring if and only if $V^\flat$ is.

Proof.

By Lemma 22.4.2, $V$ is a valuation ring if and only if $V^\flat$ is; it thus remains to show that if $V^\flat$ is AIC, then so is $V\text{.}$ So suppose by way of contradiction that $V$ is not AIC. By Lemma 22.4.3, $V$ and $V^\flat$ have the same (algebraically closed) residue field and the same (divisible) value group. Consequently, any nontrivial finite Galois extension of $\Frac V$ is totally wildly ramified and so has Galois group which is a $p$-group. This in turn implies that $\Frac V$ admits a nontrivial $\ZZ/p\ZZ$-extension, and so $H^1_{\et}(\Spec V[p^{-1}], \underline{\FF_p}) \neq 0\text{.}$ However, this contradicts Theorem 22.6.1: the right-hand side of (22.1) vanishes by Proposition 22.1.1.

Remark 23.1.2.

Theorem 23.1.1 can be used to recover the tilting correspondence for perfectoid fields (Theorem 8.3.4) as follows. Let $K$ be a perfectoid field and let $L$ be a completed algebraic closure of $K\text{.}$ Theorem 23.1.1 implies that $L^\flat$ is an algebraically closed extension of $K^\flat\text{,}$ so it contains a completed algebraic closure $M$ of $K^\flat\text{.}$ Each finite subextension of $M$ over $K^\flat$ untilts to a finite extension of $K$ within $L$ which is perfectoid. The completed union of these extensions is an untilt of $M\text{,}$ so by Lemma 22.4.4 this untilt is algebraically closed. In particular it contains the integral closure of $K$ in $L\text{,}$ and so by completeness it equals $L\text{;}$ in other words, $M = L^\flat\text{.}$

Now let $P \in K[x]$ be an irreducible polynomial with roots $\alpha_1,\dots,\alpha_n \in L\text{.}$ By the previous paragraph, we can find a finite Galois perfectoid extension $K'$ of $K$ within $L$ and an element $\beta \in L$ such that $|\beta-\alpha_1| < |\alpha_i - \alpha_1|$ for $i=2,\dots,n\text{.}$ By Krasner’s lemma, we have $\beta \in K'\text{;}$ it follows that every finite extension of $K$ within $L$ is contained in a finite Galois perfectoid extension of $K$ within $L\text{.}$ Using the Galois correspondence, we deduce that every finite extension of $K$ is the untilt of some finite extension of $K^\flat$ within $L^\flat\text{,}$ and so is perfectoid.

This argument is essentially the proof of Theorem 8.3.4 given in [80], Theorem 1.5.6 except that therein, Theorem 23.1.1 is proved by an explicit computation ([80], Lemma 1.5.4). The novelty here is that arc-descent allows us to deduce this from the much more basic statement that $V$ can be extended to an AIC valuation ring, which is then automatically a lens (Lemma 22.4.1).

Subsection 23.2 Torsion in étale and de Rham cohomology

Lemma 23.2.1.

Let $V$ be a valuation ring with fraction field $F$ and residue field $k\text{.}$ Then for any matrix $A$ over $V\text{,}$ the rank of $A$ as a matrix over $F$ is greater than or equal to the rank of $A$ as a matrix over $k\text{.}$

Proof.

Let $r$ be the rank of $A$ as a matrix over $k\text{.}$ Then there exists an $r \times r$ submatrix of $A$ whose determinant has nonzero image in $k\text{.}$ This determinant also has nonzero image in $A\text{,}$ and so the rank of $A$ as a matrix over $F$ is at least $r\text{.}$

Lemma 23.2.2. Semicontinuity for perfect complexes.

Let $V$ be a valuation ring with fraction field $F$ and residue field $k\text{.}$ Let $K^\bullet$ be a perfect complex in $D(V)\text{.}$ Then for each $i\text{,}$

\begin{equation*} \dim_F H^i(K^\bullet \otimes_V^L F) \leq \dim_k H^i(K^\bullet \otimes_V^L k). \end{equation*}

Remark 23.2.3.

A minimal example of strict inequality in Lemma 23.2.2 is a two-term complex $V \stackrel{\times x}{\to} V$ placed in degrees 0 and 1, where $x \in V$ is a nonzero element of the maximal ideal: over $F$ the cohomology vanishes, but over $k$ we have a nonzero $H^1\text{.}$

Proof.

We may assume that $K^\bullet$ is represented by a bounded complex of finite free $V$-modules. Fix bases of these modules and let $A, B$ be the matrices representing the differentials in and out of degree $i$ in these bases. By Lemma 23.2.1, the rank of $A$ does not increase when passing from $F$ to $k\text{,}$ and the corank of $B$ does not decrease; combining these two points yields the desired inequality. (Compare [117], tag 0BDI.)

Lemma 23.2.4.

Let $k$ be an algebraically closed field of characteristic $p\text{.}$ Let $(M, \phi) \in D(k[F])$ be a pair in which $M$ is perfect as a complex of $k$-modules. Then for each integer $i\text{,}$ the natural map

\begin{equation*} H^i(M^{\phi=1}) \otimes_{\FF_p} k \to H^i(M) \end{equation*}

is injective. Moreover, for each $i\text{,}$ the map is bijective if and only if $\phi\colon H^i(M) \to H^i(M)$ is bijective.

Proof.

Exercise (Lemma 23.2.4).

The following statement recovers Theorem 1.2.2.

Theorem 23.2.5.

Let $\CC$ be a complete algebraically closed extension of the field $\QQ_p\text{.}$ Let $\frako_\CC$ be the valuation ring of $\CC$ and let $k$ be the residue field of $\frako_\CC\text{.}$ Let $X$ be a smooth proper formal scheme over $\frako_\CC$ with generic fiber $X_\eta$ and special fiber $X_k\text{.}$ Then for all $i \geq 0\text{,}$

\begin{equation} \dim_{\FF_p} H^i_{\et}(X_\eta, \underline{\FF_p}) \leq \dim_k H^i_{\dR}(X_k/k).\tag{23.1} \end{equation}

Proof.

By Lemma 22.4.1, $\frako_\CC$ is a lens; let $(A,I)$ be its underlying perfect prism, choose a generator $d$ of $I$ (Theorem 7.2.2), and put $V = A/p = \frako_\CC^\flat\text{.}$ Let $(W, (p))$ be the perfect crystalline prism corresponding to $k\text{.}$ By Theorem 7.3.5, the morphism $\frako_\CC \to k$ lifts to a unique morphism $(A,I) \to (W,(p))\text{.}$

We conflate the underlying spaces of the formal scheme $X$ and the ordinary scheme $X_k\text{;}$ on this space, we may define prismatic cohomology complexes of sheaves $\Prism_{X/A}$ and $\Prism_{X_k/W}\text{.}$ By the Hodge-Tate comparison (Theorem 12.4.1) and its compatibility with base change (Lemma 15.1.3), we have

\begin{equation} \Prism_{X/A} \widehat{\otimes}^L_A W \cong \Prism_{X_k/W}\tag{23.2} \end{equation}

(with the completion being $p$-adic).

Define

\begin{equation*} R\Gamma_A(X) = R\Gamma(X, \Prism_{X/A}) \in D(A). \end{equation*}

Since $R\Gamma(X, \bullet)$ preserves limits, $R \Gamma_A(X)$ is a derived $(p,I)$-complete object of $D(A)\text{.}$ By the Hodge-Tate comparison and the usual finiteness property of coherent cohomology on a proper scheme ([117], tag 02O5), $R\Gamma_A(X) \otimes_A^L \frako_\CC$ is a perfect complex of $\frako_\CC$-modules. By derived Nakayama (Proposition 6.6.2) applied to suitable truncations, we deduce that $R \Gamma_A(X)$ is a perfect complex of $A$-modules.

In particular, $R \Gamma_A(X) \otimes_A^L V$ is a perfect complex of $V$-modules. By Lemma 23.2.2, we obtain

\begin{equation} \dim_{\CC^\flat} H^i(R\Gamma_A(X) \otimes_A^L \CC^\flat) \leq \dim_k H^i(R\Gamma_A(X) \otimes_A^L k).\tag{23.3} \end{equation}

We will deduce (23.1) by comparing its terms with (23.3).

We first check that the right-hand sides of (23.1) and (23.3) coincide. From (23.2) we obtain

\begin{equation*} R\Gamma_A(X) \widehat{\otimes}_A^L W \cong R \Gamma_W(X_k). \end{equation*}

Reduce modulo $p$ (which gets rid of the completion) and applying the crystalline comparison theorem (Corollary 14.4.10) yields

\begin{equation*} R\Gamma_A(X) \otimes_A^L k \cong R\Gamma_W(X_k) \otimes_W^L k \cong \phi_* R\Gamma_{\dR}(X_k/k). \end{equation*}

Since $k$ is perfect, the Frobenius twist $\phi_*$ has no effect on $k$-dimensions; we thus deduce the desired equality.

We next check that the left-hand sides of (23.1) and (23.3) satisfy

\begin{equation*} \dim_{\FF_p} H^i_{\et}(X_\eta, \FF_p) \leq \dim_{\CC^\flat} H^i(R\Gamma_A(X) \otimes_A^L \CC^\flat) \end{equation*}

(in fact equality will hold as perf [18], Lecture IX, Remark 5.3, but this will suffice for now). Apply the étale comparison theorem (Theorem 22.6.1) to the terms of an open affine cover of $X$ to obtain an identification

\begin{equation*} R\Gamma_{\et}(X_\eta, \FF_p) \cong (R\Gamma_A(X) \otimes_A^L \CC^\flat)^{\phi=1}, \end{equation*}

then apply Lemma 23.2.4 to obtain for each $i$ an injective linear map

\begin{equation*} H^i_{\et}(X_\eta, \FF_p) \otimes_{\FF_p} \CC^\flat \to H^i( R\Gamma_A(X) \otimes_A^L \CC^\flat). \end{equation*}

This yields the desired inequality. (Compare [18], Lecture IX, Theorem 5.1.)

Subsection 23.3 Tate twists

Definition 23.3.1.

For any scheme $X$ and any positive integer $n\text{,}$ let $\mu_n$ be the sheaf on $X$ for the flat topology which is the kernel of the multiplication-by-$n$ map on $\GG_{m,X}\text{.}$ (If $n$ is invertible on $X\text{,}$ we may use instead the étale topology.) Define the pro-sheaf

\begin{equation*} \ZZ_p(1) = \lim_m \mu_{p^m}; \end{equation*}

for $n \in \ZZ\text{,}$ set $\ZZ_p(n) = \ZZ_p(1)^{\otimes n}$ (taking the tensor product over the constant pro-sheaf $\ZZ_p$).

For $n \geq 0\text{,}$ define the presheaf $\ZZ_p(n)_{\lens}$ on the category of lenses, valued in $D(\ZZ_p)\text{,}$ by the following formula: for $(A,I)$ a perfect prism with slice $R\text{,}$ let $d$ be a generator of $I$ and set

\begin{equation*} \ZZ_p(n)_{\lens}(R) = \left( \phi^{-1}(d)^n A \stackrel{\phi/d^n - 1}{\to} A \right) \end{equation*}

with the first term placed in degree $0\text{.}$ (Note that the resulting object does not depend on the choice of $d\text{.}$) By Theorem 22.5.2, this construction defines an arc$_p$-sheaf.

Lemma 23.3.2.

Let $R$ be a lens. Then for $n \gt 0\text{,}$ there are natural isomorphisms

\begin{equation*} \ZZ_p(n) \cong \ZZ_p(n)_{\lens} \cong \ZZ_p(1)_{\lens}^{\otimes n} \end{equation*}

of arc$_p$-sheaves on the opposite category of lenses over $R\text{.}$

Proof.

By arc$_p$-descent and Example 20.3.8, we may reduce to the case where $R$ is a product of $p$-complete AIC valuation rings, and then to the case of a single such ring. In this case, the map $\phi^{-1}(d) A \stackrel{\phi/d - 1}{\to} A$ is surjective modulo $p$ (by the AIC property) and hence surjective by derived Nakayama (Proposition 6.3.1).

Suppose that $R$ is of characteristic $p\text{.}$ In this case, we may take $d = p\text{,}$ and then $\ZZ_p(n)_{\lens} \cong \left( A \stackrel{\phi-p^n}{\to} A \right)\text{.}$ The map $\phi-p^n$ on $A$ is visibly injective, so both sides of the desired isomorphism are zero.

Suppose next that $R$ is of characteristic $0\text{.}$ Choose a morphism $\ZZ_p[\mu_{p^\infty}] \to R\text{,}$ let $\epsilon \in R^\flat$ be the element $(1,\zeta_p,\zeta_{p^2},\dots)\text{,}$ and put $q = \epsilon^\sharp\text{.}$ We can then take the generator of $d$ to be $[p]_q = (q^p-1)/(q-1)\text{;}$ we may then identify $\ZZ_p(n)_{\lens}$ with $(q-1)^n \ZZ_p \subset \phi^{-1}(d^n) A\text{.}$ This gives the desired natural isomorphism $\ZZ_p(n)_{\lens} \cong \ZZ_p(1)_{\lens}^{\otimes n}\text{.}$

To specify a natural isomorphism $\ZZ_p(n) \cong \ZZ_p(n)_{\lens}\text{,}$ it now suffices to do so for $n=1\text{.}$ In this case, we must check that the action of $\Gal(\QQ_p(\mu_{p^\infty}))$ on $(q-1)\ZZ_p$ matches the action on $\lim_n \mu_{p^n}\text{;}$ this follows from the fact that

\begin{equation*} q^m - 1 \equiv m(q-1) \pmod{d(q-1)} \qquad (m \in \ZZ). \end{equation*}

Remark 23.3.3.

One can promote Lemma 23.3.2 to the assertion that the two definitions of Tate twists correspond to a single construction on the quasisyntomic site, as per [23], section 7.4. We will not spell this out further here; instead, see [25], section 14.

Theorem 23.3.4.

Let $R$ be a lens.

We have a canonical identification

\begin{equation*} \ZZ_p(0)_{\lens}(R) \cong R\Gamma_{\et}(\Spec R, \ZZ_p(0)). \end{equation*}
For $n \gt 0\text{,}$ we have a canonical identification

\begin{equation*} \ZZ_p(n)_{\lens}(R) \cong R\Gamma_{\et}(\Spec R[1/p], \ZZ_p(n)). \end{equation*}

Proof.

Point (1) follows from Remark 22.6.2. Point (2) follows from Theorem 22.6.1 and Lemma 23.3.2.

Exercises 23.4 Exercises

1.

Prove Lemma 23.2.4.

2.

Let $R$ be a lens. Using Theorem 23.3.4, show that $\Pic(R)$ and $\Pic(R[p^{-1}])$ are both uniquely $p$-divisible.

Hint.

Use Theorem 23.3.4 to compare $H^i_{\et}(\Spec R, \mu_p)$ with $H^i_{\et}(\Spec R[p^{-1}], \mu_p)$ for $i=1,2\text{.}$ For more details, see [25], Corollary 9.5.

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