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Notes on prismatic cohomology

Kiran S. Kedlaya

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.)

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 \(M\text{.}\)

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.

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.)

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.

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.
  1. 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.
  2. The zero element. In this case, the map factors through \(R/p\text{.}\)
  3. A unit. In this case, the map factors through \(R[1/p]\text{.}\)

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).

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.

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.

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 \(x\text{.}\)

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

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.)

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.

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.
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