The étale comparison theorem

\(\frenchspacing \def\AA{\mathbb{A}} \def\CC{\mathbb{C}} \def\FF{\mathbb{F}} \def\GG{\mathbb{G}} \def\NN{\mathbb{N}} \def\PP{\mathbb{P}} \def\QQ{\mathbb{Q}} \def\RR{\mathbb{R}} \def\WW{\mathbb{W}} \def\ZZ{\mathbb{Z}} \def\calA{\mathcal{A}} \def\calC{\mathcal{C}} \def\calE{\mathcal{E}} \def\calF{\mathcal{F}} \def\calI{\mathcal{I}} \def\calO{\mathcal{O}} \def\frakm{\mathfrak{m}} \def\frako{\mathfrak{o}} \def\frakp{\mathfrak{p}} \def\frakX{\mathfrak{X}} \def\dual{\vee} \def\Ainf{\mathbf{A}_{\mathrm{inf}}} \DeclareMathOperator{\ab}{ab} \DeclareMathOperator{\Ab}{\mathbf{Ab}} \DeclareMathOperator{\an}{an} \DeclareMathOperator{\Ann}{Ann} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\Cl}{Cl} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\colim}{colim} \DeclareMathOperator{\comp}{comp} \DeclareMathOperator{\Comp}{\mathbf{Comp}} \DeclareMathOperator{\Cone}{Cone} \DeclareMathOperator{\Cor}{Cor} \DeclareMathOperator{\crys}{crys} \DeclareMathOperator{\CW}{CW} \DeclareMathOperator{\cyc}{cyc} \DeclareMathOperator{\dcrys}{dcrys} \DeclareMathOperator{\dR}{dR} \DeclareMathOperator{\et}{et} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\fin}{fin} \DeclareMathOperator{\Fil}{Fil} \DeclareMathOperator{\Fix}{Fix} \DeclareMathOperator{\fppf}{fppf} \DeclareMathOperator{\Frac}{Frac} \DeclareMathOperator{\Frob}{Frob} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\gr}{gr} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\HT}{HT} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\Ind}{Ind} \DeclareMathOperator{\Inf}{Inf} \DeclareMathOperator{\inv}{inv} \DeclareMathOperator{\Kos}{Kos} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\lens}{lens} \DeclareMathOperator{\Mod}{\mathbf{Mod}} \DeclareMathOperator{\Norm}{Norm} \DeclareMathOperator{\op}{op} \DeclareMathOperator{\perf}{perf} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Poly}{\mathbf{Poly}} \DeclareMathOperator{\prim}{prim} \DeclareMathOperator{\Prm}{\mathbf{Prism}} \DeclareMathOperator{\Pshv}{\mathbf{Pshv}} \DeclareMathOperator{\Rad}{Rad} \DeclareMathOperator{\Real}{Re} \DeclareMathOperator{\red}{red} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\Ring}{\mathbf{Ring}} \DeclareMathOperator{\sep}{sep} \DeclareMathOperator{\Sch}{\mathbf{Sch}} \DeclareMathOperator{\Set}{\mathbf{Set}} \DeclareMathOperator{\sheafHom}{\mathcal{H}\!\mathit{om}} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Spf}{Spf} \DeclareMathOperator{\Sym}{Sym} \DeclareMathOperator{\Tor}{Tor} \DeclareMathOperator{\Tot}{Tot} \DeclareMathOperator{\Trace}{Trace} \DeclareMathOperator{\unr}{unr} \DeclareMathOperator{\Vect}{\mathbf{Vect}} \DeclareMathOperator{\Ver}{Ver} \def\Prism{\mathbf{\Delta}} \def\llbracket{[\![} \def\rrbracket{]\!]} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)

Section 22 The étale comparison theorem

Reference.

[18], lecture IX; [25], section 9. We follow the latter more closely.

In this section, we establish the étale comparison theorem for prismatic cohomology (Theorem 22.6.1). The strategy is to use arc-descent to reduce to a case where everything can be calculated explicitly; this avoids any use of analytic geometry.

Subsection 22.1 The Artin-Schreier-Witt exact sequence

Our entire discussion up to now has involved cohomology theores of “coherent nature”, involving some sort of algebro-geometric structure sheaf. It is reasonable to wonder how then we can hope to say anything meaningful about étale cohomology. The fundamental bridge between these two words is the Artin-Schreier exact sequence, or in a more general form the Artin-Schreier-Witt exact sequence. (See [117], tag 0A3J for more discussion.)

Proposition 22.1.1. Artin-Schreier-Witt exact sequence.

For any scheme $X$ over $\FF_p\text{,}$ the sequence

\begin{equation*} 0 \to \underline{\FF}_p \to \GG_a \stackrel{\phi-1}{\to} \GG_a \to 0 \end{equation*}

of étale sheaves on $X$ is exact. More generally, for any positive integer $n\text{,}$ the sequence

\begin{equation*} 0 \to \underline{\ZZ/p^n \ZZ} \to W_n \stackrel{\phi-1}{\to} W_n \to 0 \end{equation*}

of étale sheaves on $X$ is exact (where $W_n$ denotes $p$-typical Witt vectors of rank $n\text{,}$ viewed as a group scheme with respect to addition).

Proof.

The key point is the surjectivity of $\phi - 1\text{.}$ To check this, we may assume $X = \Spec A$ is affine. It will suffice to check that for $x = (x_0,\dots,x_{n-1}) \in W_n(A)\text{,}$ the morphism

\begin{equation*} A \to B = A[y_0,\dots,y_{n-1}]/((\varphi-1)y - x) \end{equation*}

is étale (and even finite étale). The defining ideal is generated by some elements of the form

\begin{equation*} y_i^p - y_i - P(x_0,\dots,x_i,y_0,\dots,y_{i-1}) \qquad (i=0,\dots,n-1); \end{equation*}

we may thus deduce that $A \to B$ is finite étale using the Jacobian criterion (the Jacobian matrix is triangular with units on the diagonal).

Remark 22.1.2.

The Artin-Schreier-Witt sequence also has a nonabelian analogue; this appears in Fontaine's theory of $(\varphi, \Gamma)$-modules [49].

Subsection 22.2 Frobenius fixed points and coperfections

The Artin-Schreier exact sequence leads us to the following considerations.

Definition 22.2.1.

Fix an $\FF_p$-algebra $B$ and an element $t \in B\text{.}$ Write $D_{\comp}(B)$ for the $t$-complete derived category of $B\text{.}$

Let $D(B[F])$ be the derived category of Frobenius $B$-modules; objects in this category are pairs $(M, \phi)$ where $M \in D(B)$ and $\phi\colon M \to \phi_* M$ is a morphism. Let $D_{\comp}(B[F])$ be the full subcategory of $D(B[F])$ spanned by pairs $(M, \phi)$ with $M \in D_{\comp}(B)\text{.}$

Given $(M, \phi) \in D_{\comp}(B[F])\text{,}$ define $M^{\phi-1}$ as the cocone of $\phi$ (see Definition 10.2.2); that is,

\begin{equation*} M^{\phi=1} = \Cone(M \stackrel{\phi-1}{\to} M)[-1] = R\Hom_{D(B[F])}((B, \phi), (M, \phi))[-1]. \end{equation*}

We call this object the Frobenius fixed points of $M\text{.}$

Lemma 22.2.2.

With notation as in Definition 22.2.1, for any $(N, \phi) \in D_{\comp}(B[F])\text{,}$ the map $N^{\phi=1} \to (N/t)^{\phi=1}$ is an isomorphism. (Here the $\phi$-structure on $N/t$ is the one induced from $N$ using the fact that $\phi(t) = t^p \in tB\text{.}$)

Proof.

If $N$ is derived $t$-complete, then the cone $F$ of $N \to N/t$ is complete for the $t$-adic filtration, and the $\phi$-action is topologically nilpotent because $\phi(t) = t^p \in t^2 B\text{.}$ From this we see that $F^{\phi=1} = 0\text{,}$ proving the claim.

Proposition 22.2.3.

With notation as in Definition 22.2.1, the following statements hold.

The functors $D_{\comp}(B[F]) \to D(\FF_p)$ given by $M \mapsto M^{\phi=1}$ and $M \mapsto (M[t^{-1}])^{\phi=1}$ commute with sequential colimits. (In the realm of $\infty$-categories, this can be upgraded to arbitrary colimits.)
For any $(M, \phi) \in D_{\comp}(B[F])\text{,}$ for

\begin{equation*} (N, \phi) = \colim_\phi (M, \phi) \in D_{\comp}(B[F]), \end{equation*}

the natural maps

\begin{equation*} M^{\phi=1} \to N^{\phi=1}, \qquad (M[t^{-1}])^{\phi=1} \to (N[t^{-1}])^{\phi=1} \end{equation*}

are isomorphisms.

Proof.

For convenience, we take colimits in the ambient category $D(B)$ (in contrast to the original statement).

For (1), we first verify that the two assertions are equivalent. Consider a diagram $\{(M_i, \phi_i)\}$ in $D_{\comp}(B[F])\text{.}$ Let $F$ be the cone of the map from $\colim_i M_i$ to its derived $t$-completion; note that $F$ is uniquely $t$-divisible, so $F$ is also the cone of the map obtained after inverting $t$ on both sides. Now both statements of the lemma are equivalent to the vanishing of $F^{\phi=1}\text{,}$ and hence to each other.

We now prove that $M \mapsto M^{\phi=1}$ commutes with colimits. By Lemma 22.2.2, this functor refactors as

\begin{equation*} D_{\comp}(B[F]) \stackrel{N \mapsto N/t}{\to} D(B[F]) \stackrel{\bullet^{\phi=1}}{\to} D(\FF_p) \end{equation*}

and both factors commute with colimits.

For (2), using (1) it suffices to check that each map $(M, \phi) \stackrel{\phi}{\to} (M, \phi)$ induces an isomorphism upon applying either $\bullet^{\phi=1}$ or $(\bullet[t^{-1}])^{\phi=1}\text{,}$ both of which are clear. (Compare [18], Lecture IX, Proposition 1.2.)

Corollary 22.2.4.

Let $(A,I)$ be a perfect prism and choose a generator $d$ of $I$ (Theorem 7.2.2). Let $R$ be a derived $p$-complete $\overline{A}$-algebra. Then the natural maps

\begin{gather*} (\Prism_{R/A}/p^n)^{\phi=1} \to (\Prism_{R/A,\perf}/p^n)^{\phi=1}\\ (\Prism_{R/A}[d^{-1}]/p^n)^{\phi=1} \to (\Prism_{R/A,\perf}[d^{-1}]/p^n)^{\phi=1} \end{gather*}

are isomorphisms.

Proof.

This formally reduces to the case $n=1\text{.}$ In this case, apply Proposition 22.2.3 with $B = A/p, t = d$ where $d$ is a generator of $I$ (Theorem 7.2.2).

Subsection 22.3 The arc$_p$-topology

We need a variant of the arc-topology that accounts for $p$-completion.

Definition 22.3.1.

Let $f\colon R \to S$ be a morphism of derived $p$-complete rings. We say that $f$ is an arc$_p$-covering if the completion property from Figure 20.3.2 (with $X = \Spec R\text{,}$ $Y = \Spec S$) holds whenever the valuation ring $V$ is $p$-complete (and eudoxian). Note that we can then take $W$ to also be $p$-complete (and eudoxian).

Remark 22.3.2.

A sufficient (but not necessary) condition for a morphism $f\colon R \to S$ to be an arc$_p$-covering is the following: for every $p$-complete AIC eudoxian valuation ring $V\text{,}$ the map

\begin{equation*} \Hom_{\Ring}(R, V) \to \Hom_{\Ring}(S, V) \end{equation*}

is a bijection.

Lemma 22.3.3.

Let $R \to S$ be an arc$_p$-covering of derived $p$-complete rings. Then

\begin{equation*} R \to S \oplus R/p \oplus R[1/p] \end{equation*}

is an arc-covering.

Proof.

Let $R \to V$ be a morphism in $\Ring$ with $V$ a eudoxian valuation ring. The image of $p$ in $V$ is then one of the following.

A nonzero element of the maximal ideal. In this case, we can replace $V$ with its $p$-completion, in which case the map factors through $S$ because $R \to S$ is an arc$_p$-covering.
The zero element. In this case, the map factors through $R/p\text{.}$
A unit. In this case, the map factors through $R[1/p]\text{.}$

Theorem 22.3.4. Arc$_p$-descent for étale cohomology.

For $R \in \Ring\text{,}$ let $\calF$ be a torsion sheaf on $(\Spec R)_{\et}\text{.}$ Then the functor

\begin{equation*} (f\colon \Spec S \to \Spec R) \mapsto R\Gamma(\Spec S^\wedge_{(p)}[p^{-1}], f^* \calF) \end{equation*}

from $\Ring_R^{\op}$ to $D(\ZZ)^{\geq 0}$ satisfies descent for the arc$_p$-topology.

Proof.

It suffices to check descent for an arc$_p$-covering $R \to S$ (so in particular both rings are derived $p$-complete). By Lemma 22.3.3, $R \to S \oplus R/p \oplus R[1/p]$ is an arc-covering. Since derived $p$-completion followed by inverting $p$ kills both $(R/p)$-modules and $R[1/p]$-modules, we may deduce the claim from arc-descent for étale cohomology (Theorem 22.3.4). (Compare [21], Corollary 6.17.)

Subsection 22.4 Tilting valuation rings

In order to further relate the arc-topology with the arc$_p$-topology, we study the effect of tilting on valuation rings. This can be thought of as a continuation of our discussion of perfectoid fields (Subsection 8.3).

Lemma 22.4.1.

Let $V$ be a $p$-complete AIC valuation ring. Then $V$ is a lens.

Proof.

If $V$ is a $p$-complete AIC valuation ring, then $\Frac V$ is a perfectoid field (Definition 8.3.1). We may thus deduce the claim from Lemma 8.3.3.

Lemma 22.4.2.

Let $V$ be a lens. Then $V$ is a valuation ring if and only if $V^\flat$ is. In this case, the value groups of $V$ and $V^\flat$ are isomorphic; in particular, $V$ is eudoxian if and only if $V^\flat$ is.

Proof.

Let $\sharp\colon V^\flat \to V$ be the multiplicative map obtained by composing the constant lift $[\bullet]: V^\flat \to W(V^\flat)$ with the quotient map $W(V^\flat) \to V\text{.}$ It is customary to write the image of $x$ under $\sharp$ as $x^\sharp$ rather than $\sharp(x)\text{.}$

Suppose that $V$ is a valuation ring. If $\Frac V$ has characteristic $p\text{,}$ then $V = V^\flat$ and there is nothing more to check; we may thus assume that $\Frac V$ has characteristic 0. Since $V$ is an integral domain, the $p$-power map on $V$ is injective; hence for $x \in V^\flat\text{,}$ $x^\sharp = 0$ if and only if $x = 0\text{.}$ This in turn implies that $V^\flat$ is an integral domain (if $xy = 0$ then $x^\sharp y^\sharp = 0$) and that the principal ideals of $V^\flat$ are totally ordered with respect to inclusion (if $x^\sharp$ is divisible by $y^\sharp\text{,}$ then the ratio admits a coherent sequence of $p$-power roots and so is itself in the image of $\sharp$). Hence $V^\flat$ is a valuation ring.

Conversely, suppose that $V^\flat$ is a valuation ring. Again, we may assume that $p \neq 0$ in $V\text{;}$ since $V^\flat$ is an integral domain, we may apply Exercise 8.5.3 to deduce that $V$ is $p$-torsion-free. Since $V^\flat$ is a local ring, so are $W(V^\flat)$ and its quotient $V\text{.}$ Choose $\varpi \in V$ as per Lemma 8.2.3.

We need to show that given any two nonzero elements $x,y \in V\text{,}$ one is a multiple of the other. By dividing by powers of $\varpi$ as needed, we may reduce to the case where $x$ and $y$ have nonzero images in $V/\varpi$ and hence in $V/p = V^\flat/d\text{.}$ Since $V^\flat$ is a valuation ring, after possibly swapping terms we can write $x = yz + pu$ for some $z,u \in V\text{.}$ Similarly, we can write $\varpi \equiv yw + pv$ for some $w,v \in V\text{.}$ Since $V$ is classically $\varpi$-complete, $1 - (p/\varpi)v$ is a unit in $V\text{;}$ hence $\varpi$ is divisible by $y\text{,}$ as then is $p\text{.}$ Consequently, $x = yz + pu$ is also divisible by $y\text{,}$ as desired.

Lemma 22.4.3.

Let $V$ be a lens which is a valuation ring. Then $V$ and $V^\flat$ have residue fields isomorphic to each other (and to that of $W(V^\flat)$) and $\sharp: V^\flat \to V$ induces an isomorphism between the value groups of $V$ and $V^\flat\text{.}$ In particular, $V$ is eudoxian if and only if $V'$ is.

Proof.

From the proof of Lemma 22.4.2, we see that $\sharp$ induces an injective map from the value group of $V^\flat$ to that of $V\text{.}$ To prove surjectivity, we must check that every element $x$ of $V$ has an associate in the image of $V^\flat\text{.}$ As in the proof of Lemma 22.4.2, we may prove this by first dividing by a suitable power of $\varpi$ to ensure that $x \not\equiv 0 \pmod{\varpi}\text{,}$ then showing that in this case $x$ is an associate of $[\overline{x}]$ where $\overline{x} \in V/p$ is the image of $x\text{.}$

Lemma 22.4.4.

Let $V$ be a lens. If $V$ is an AIC valuation ring, then so is $V^\flat\text{.}$

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 not AIC, then neither is $V\text{.}$ We may assume that $V$ has characteristic 0, as otherwise there is nothing to check.

Let $R$ be the integral closure of $V^\flat$ in a nontrivial finite Galois extension of its fraction field with Galois group $G\text{.}$ By Theorem 7.3.5, $V' = V \otimes_{W(V^\flat)} W(R)$ is a lens with $V^{\prime \flat} \cong R$ and, by Lemma 22.4.2, again a valuation ring.

By construction, $G$ acts on $V^{\prime \flat}$ with fixed subring $V\text{.}$ By functoriality, $G$ also acts on $V'\text{;}$ since $V'$ is of characteristic $0\text{,}$ we can we can see by averaging over the group action that the fixed subring $V'^{G}$ is equal to $V\text{.}$

By the Artin and Dedekind lemmas in Galois theory, we see that $\Frac V'$ is a finite Galois extension of $\Frac V$ of degree $\#G \gt 1\text{.}$ This proves that $V$ is not AIC.

Remark 22.4.5.

The tilting correspondence for perfectoid fields (Theorem 8.3.4) implies the converse of Lemma 22.4.4: if $V$ is a lens and $V^\flat$ is an AIC valuation ring, then so is $V\text{.}$ We will recover this later as a corollary of the étale comparison theorem (Theorem 23.1.1).

Subsection 22.5 Arc$_p$-descent for lenses

Lemma 22.5.1.

Let $R \to S$ be an arc$_p$-covering of lenses. Write $R$ as the slice of a perfect prism $(A,I)$ with $I = (d)$ (Theorem 7.2.2). Then

\begin{equation*} R^\flat \to R^\flat[d^{-1}] \oplus S^\flat \end{equation*}

is an arc-covering.

Proof.

Let $R^\flat \to V$ be a map to a eudoxian valuation ring, which we may assume is perfect. If $d$ maps to a unit in $V\text{,}$ then the map extends to $R^\flat[d^{-1}]\text{.}$ Otherwise, we may replace $V$ with its $d$-adic completion; by Theorem 7.3.5, the map $R^\flat \to V$ corresponds to a map $R \to V^\sharp = A \otimes_R W(V)$ whose target is a lens and (by Lemma 22.4.2) a $p$-complete eudoxian valuation ring. Since $R \to S$ is an arc-covering, we get an extension to a map $S \to V'$ for some eudoxian valuation ring $V'$ containing $V^\sharp\text{,}$ which we may take to be $p$-complete and AIC and hence a lens (Lemma 22.4.1). Then $S^\flat \to V^{\prime \flat}$ gives the desired extension. (Compare [25], Proposition 8.9.)

Theorem 22.5.2.

Let $R \to S$ be an arc$_p$-covering of lenses. Then the augmented Čech-Alexander complex

\begin{equation*} 0 \to R \to S \to S \widehat{\otimes}_R S \to \cdots \end{equation*}

is acyclic.

Proof.

Write $R$ as the slice of a perfect prism $(A,I)\text{.}$ By applying Corollary 21.1.8 to the arc-covering from Lemma 22.5.1, then taking derived $d$-completions (which kills all terms involving $R^\flat[d^{-1}]$), we deduce the stated result. (Compare [25], Proposition 8.9.)

Subsection 22.6 The comparison theorem

We finally obtain the étale comparison theorem.

Theorem 22.6.1. Étale comparison theorem.

Let $(A,I)$ be a perfect prism and choose a generator $d$ of $I$ (Theorem 7.2.2). Let $R$ be a derived $p$-complete $\overline{A}$-algebra. Then for each positive integer $n\text{,}$ there is a canonical identification

\begin{equation} R\Gamma_{\et}(\Spec R[p^{-1}], \underline{\ZZ/p^n}) \cong (\Prism_{R/A}[d^{-1}]/p^n)^{\phi=1}\tag{22.1} \end{equation}

(in the sense of Definition 22.2.1). In particular, the right-hand side of (22.1) depends only on $R\text{.}$

Proof.

We first observe that both sides of (22.1) admits descent for the arc$p$-topology: for the left-hand side this is Theorem 22.3.4, and for the right-hand side it follows from Corollary 22.2.4 and Theorem 22.5.2. (Compare [25], Corollary 8.10.)

Using arc$_p$-descent and the Artin-Schreier-Witt exact sequence (Proposition 22.1.1), we obtain a map (from left to right in (22.1)) of the desired form. To check that it is an isomorphism, we may apply arc$_p$-descent again: using a v-covering as in Example 20.3.8, we may reduce to a case where $R = \prod_i R_i$ is a product of $p$-complete AIC valuation rings (and in particular a lens, by Lemma 22.4.1). Note that by Theorem 23.1.1, each ring $R_i^\flat$ is an AIC valuation ring.

In this case, the left-hand side of (22.1) equals $(\ZZ/p^n)^I$ concentrated in degree $0\text{.}$ As for the right-hand side, we have $\Prism_{R/A,\perf} \cong W(R^\flat)$ concentrated in degree $0$ (Example 19.2.5). By Proposition 22.1.1, for each $i \in I\text{,}$ $\phi-1$ is surjective on $W(R_i^\flat[d^{-1}])/p^n$ with kernel $\ZZ/p^n\text{.}$ We thus have a canonical exact sequence

\begin{equation*} 0 \to (\ZZ/p^n)^I \to W(R^\flat[d^{-1}])/p^n \stackrel{\phi-1}{\to} W(R^\flat[d^{-1}])/p^n \to 0 \end{equation*}

which is exactly what we needed.

Remark 22.6.2.

As per [25], Remark 9.3, we point out that a similar method can be used to obtain a variant of Theorem 22.6.1 without inverting $p$ or $d\text{:}$ there is a canonical identification of étale sheaves on $\Spec \overline{A}\text{:}$

\begin{equation*} \underline{\ZZ/p^n} \cong (\Prism_{R/A}/p^n)^{\phi=1} \end{equation*}

(so in particular the right-hand side is concentrated in degree $0$). The corresponding exact sequence in the proof would be

\begin{equation*} 0 \to (\ZZ/p^n)^I \to W(R^\flat)/p^n \stackrel{\phi-1}{\to} W(R^\flat)/p^n \to 0. \end{equation*}

Exercises 22.7 Exercises

1.

Let $R$ be a lens. Prove that $R$ is a seminormal ring in the sense of Swan [118]: that is, the map

\begin{equation*} R \to \{(y,z) \in R^2: y^3 = z^2\}, \qquad x \mapsto (x^2, x^3) \end{equation*}

is a bijection.

Hint.

Use Example 20.3.8 and Theorem 22.5.2 to reduce to the case where $R$ is an AIC valuation ring.

Section 22 The étale comparison theorem

Reference.

Subsection 22.1 The Artin-Schreier-Witt exact sequence

Proposition 22.1.1. Artin-Schreier-Witt exact sequence.

Proof.

Remark 22.1.2.

Subsection 22.2 Frobenius fixed points and coperfections

Definition 22.2.1.

Lemma 22.2.2.

Proof.

Proposition 22.2.3.

Proof.

Corollary 22.2.4.

Proof.

Subsection 22.3 The arc\(_p\)-topology

Definition 22.3.1.

Remark 22.3.2.

Lemma 22.3.3.

Proof.

Theorem 22.3.4. Arc\(_p\)-descent for étale cohomology.

Proof.

Subsection 22.4 Tilting valuation rings

Lemma 22.4.1.

Proof.

Lemma 22.4.2.

Proof.

Lemma 22.4.3.

Proof.

Lemma 22.4.4.

Proof.

Remark 22.4.5.

Subsection 22.5 Arc\(_p\)-descent for lenses

Lemma 22.5.1.

Proof.

Theorem 22.5.2.

Proof.

Subsection 22.6 The comparison theorem

Theorem 22.6.1. Étale comparison theorem.

Proof.

Remark 22.6.2.

Exercises 22.7 Exercises

1.