Skip to main content

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)\text{;} \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]\text{.} \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.

🔗

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.

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\text{.} \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.
🔗
🔗
🔗
🔗
🔗
PrevTopNext