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Section 21 Descent for the arc-topology


[21]; [25], section 8.2.
We establish some descent properties for the arc-topology (Section 20) which will be used to establish the étale comparison theorem (Section 22).

Subsection 21.1 Descent for perfect schemes

Definition 21.1.1.

The functor from perfect \(\FF_p\)-schemes (i.e., those on which Frobenius is an isomorphism) to arbitrary \(\FF_p\)-schemes admits a right adjoint, called perfection; for affine schemes, this corresponds to coperfection of rings. Let \(X_{\perf}\) denote the perfection of \(X\text{.}\)
Let \(\Vect(X)\) denote the category of vector bundles on the scheme \(X\text{.}\)
We reduce at once to the case where all of the schemes in question are affine. In this case, the claim reduces at once to Exercise 7.4.4. (Compare [24], Lemma 3.18.)
By Lemma 21.1.2, this reduces to the exactness of the sequence
\begin{equation*} 0 \to V \to V_\frakp \oplus V/\frakp \to \kappa(\frakp) \to 0, \end{equation*}
which we leave as an exercise (Exercise 21.4.1). (Compare [24], Lemma 6.3.)
For both assertions, we may assume that \(X = \Spec A\) is affine; write \(Z = \Spec A/I\text{.}\) Write \(nE\) for the subscheme of \(Y\) cut out by \(I^n\text{.}\)
For (1), we may assume \(\calF = \calO\text{.}\) By our hypotheses, we have \(\calO(X) \cong \calO(Y)\) and \(\calO(Z) \cong \calO(E)\) by Stein factorization, and similarly after taking perfections. Since \(X\) and \(Z\) are both affine, it remains to check that \(H^i(Y_{\perf}, \calO) \to H^i(E_{\perf}, \calO)\) is an isomorphism for each \(i \gt 0\text{.}\)
At this point, we follow [27], Lemma 3.9 (which is written using the Nisnevich topology, but the Zariski topology works equally well). By [117], tag 02OB, point (1), there exists a constant \(c\) such that for \(n \geq c\text{,}\)
\begin{equation*} \ker(H^i(Y, \calO) \to H^i(E_n, \calO)) \subseteq I^{n-c} H^i (Y,\calO). \end{equation*}
Note that \(H^i(Y, \calO)\) is a finitely generated \(A\)-module which, since \(f\) is a blowup and \(i \gt 0\text{,}\) is supported entirely on \(Z\text{.}\) Hence for \(n \gg 0\text{,}\) \(I^{n-c}\) annihilates \(H^i(Y, \calO)\) and so
\begin{equation} H^i(Y, \calO) \hookrightarrow H^i(E_n, \calO) \qquad (n \gg 0).\tag{21.1} \end{equation}
On the other hand, by [117], tag 020B, point (3), for \(m \gg n \gg 0\) we have
\begin{equation} \im(H^i(E_m, \calO) \to H^i(E_n, \calO)) = \im(H^i(Y, \calO) \to H^i(E_n, \calO)).\tag{21.2} \end{equation}
Fix a value \(n \gg 0\) that is large enough for both (21.1) and (21.2) to hold. Then for \(e \gg 0\text{,}\) the image of \(\phi^e\colon H^i(E_n, \calO) \to H^i(E_n, \calO)\) is contained in the image of \(H^i(Y, \calO) \to H^i(E_n, \calO)\text{:}\) to see this, refactor the former map as
\begin{equation*} H^i(E_n, \calO) \stackrel{\phi^e}{\to} H^i(E_{p^e n}, \calO) \to H^i(E_n, \calO) \end{equation*}
and then apply (21.2). By this plus (21.1), we see that
\begin{equation*} \colim_\phi H^i(Y, \calO) = \colim_\phi H^i(E_n, \calO) \end{equation*}
and hence
\begin{align*} H^i(Y_{\perf}, \calO) &= \colim_\phi H^i(E_n, \calO)\\ &= \colim_\phi H^i(E, \calO) = H^i(E_{\perf}, \calO) \end{align*}
as claimed.
For (2), we follow [24], Lemma 4.6. By the Beauville-Laszlo theorem (see Remark 21.2.7), we may assume that \(A\) is (classically) \(I\)-complete. We may also assume that we start with an object in \(\Vect(Y) \times_{\Vect(E)} \Vect(Z)\text{.}\) Let \(\calI\) be the inverse image ideal sheaf of \(I\text{;}\) by the construction of the blowup, \(\calI\) is an ample invertible sheaf on \(Y\text{.}\) Consequently, by Serre vanishing, we may choose some \(n\) such that
\begin{equation} H^i(Y, \calI^k/\calI^{k+1}) = 0 \qquad (k \geq n).\tag{21.3} \end{equation}
Since \(X\) is affine and complete along \(Z\text{,}\) \(\Vect(X) \to \Vect(Z)\) is an equivalence of categories (Exercise 6.7.9). We thus have objects \(\calE \in \Vect(X)\text{,}\) \(\calF \in \Vect(Y)\) and an isomorphism \(\psi\colon f^* \calE|_E \cong \calF|_E\text{.}\) By pulling back by a suitable power of \(\phi\text{,}\) we may construct another isomorphism \(\psi_n\colon f^* \calE|_{nE} \cong \calF|_{nE}\text{.}\)
We now observe that for \(m \geq n\text{,}\) an isomorphism \(\psi_m\colon f^* \calE|_{mE} \cong \calF|_{mE}\) can be promoted to an isomorphism \(\psi_{m+1}\colon f^* \calE|_{(m+1)E} \cong \calF|_{(m+1)E}\text{:}\) namely, the obstruction to lifting belongs to
\begin{equation*} H^1(Y, \calI^m/\calI^{m+1} \otimes \sheafHom(f^* \calE, \calF)) \end{equation*}
which vanishes by (21.3). Since
\begin{equation*} \Vect(Y) \to \lim_m \Vect(mE) \end{equation*}
is an equivalence by the formal existence theorem ([117], tag 0885), we deduce the desired result.

Remark 21.1.6.

Point (1) of Lemma 21.1.5 can also be formulated as follows: for \(j\colon Z \to X\) the inclusion and \(g\colon E \to X\) the induced map (and reusing the names \(f,g,j\) for the images of these maps under the perfection functor), we have a distinguished triangle
\begin{equation*} \calF \to Rf_* f^* \calF \oplus Rj_* j^* \calF \to Rg_* g^* \calF \to \end{equation*}
in the derived category of coherent sheaves on \(X_{\perf}\text{.}\)
To begin with, both assertions hold for the flat (fpqc) topology in place of the arc-topology thanks to classical faithfully flat descent ([117], tag 0238).
We next upgrade both assertions from the flat topology to the v-topology. Every v-covering is a cofiltered limit of h-coverings, so we may reduce to considering perfections of h-coverings of finite type \(\FF_p\)-schemes. Since the h-topology is generated by faithfully flat coverings and proper surjective morphisms, and we already know descent for the former. we may reduce to considering the perfection of a proper surjective morphism. Moreover, by Raynaud-Gruson flattening [103], we may further reduce to considering the case of a blowup, to which we may apply Lemma 21.1.5.
Finally, we upgrade both assertions from the v-topology to the arc-topology. By passing to affines and then pulling back along a cover as in Example 20.3.8, we may reduce to considering a covering as in Example 20.3.10 (compare [21], Theorem 4.1). For this, apply Corollary 21.1.4 and Lemma 21.2.1.

Subsection 21.2 Additional descent arguments

We record here an argument that was used in the proof of Theorem 21.1.7 to promote a statement about acyclicity of the structure sheaf to a statement about descent for vector bundles.
Let \(M_1, M_2, M_{12}\) be objects of \(\Vect(R_1), \Vect(R_2), \Vect(R_{12})\) equipped with isomorphisms \(M_i \otimes_{R_i} R_{12} \cong M_{12}\) and put \(M = \ker(M_1 \oplus M_2 \to M_{12})\text{;}\) we will show that \(M \in \Vect(R)\) and that the induced maps \(M \otimes_R R_i \to M_i\) are isomorphisms.
We first check that the maps \(M \otimes_R R_i \to M_i\) are all surjective.
  • Given \(x \in M_1\text{,}\) we can write the image of \(x\) in \(M_{12}\) as \(y/f\) for some \(y \in M_2\) and some \(f \in R\) which becomes a unit in \(R_1\text{.}\) Then \((fx, y)\) is an element of \(M\) mapping to \(fx \in M_1\text{,}\) so \(M \otimes_R R_1 \to M_1\) is surjective.
  • Since \(R_1 \oplus R_2 \to R_{12}\) is surjective, \(M \otimes_R (R_1 \oplus R_2) \to M_{12}\) is surjective.
  • Given \(x \in M_2\text{,}\) we may map \(x\) to \(M_{12}\) and then lift it to \((x_1, x_2) \in M_1 \oplus M_2\) in the image of \(M \otimes_R (R_1 \oplus R_2)\text{.}\) By construction, \((x_1, x_2-x) \in M\text{,}\) so the image of \(M \otimes_{R_1} \to R_1\) contains both \(x_2\) and \(x_2-x\text{.}\) Hence \(M \otimes_R R_2 \to M_2\) is also surjective.
We next check that \(M\) is a finite \(R\)-module. From the previous discussion, we see that there exist a finite free \(R\)-module \(F\) and a morphism \(F \to M\) of \(R\)-modules such that, for \(F_i = F \otimes_R R_*\text{,}\) the induced map \(F_i \to M_i\) is surjective. Put \(N = \ker(F \to M)\) and \(N_i = \ker(F_i \to M_i)\text{.}\) We have a diagram as in Figure 21.2.4 in which all of the squares commute and all of the rows and columns are exact, except possibly for the dashed arrows. However, because the modules \(M_i\) are projective, the maps \(N_i \otimes_{R_i} R_{12} \to N_{12}\) are isomorphisms, so all of the preceding logic applies to them also; this allows us to add the dashed horizontal arrow to the diagram, and hence also the dashed vertical arrow.
Figure 21.2.4.
We next check that for each \(i\text{,}\) \(M \otimes_i R_i \to M_i\) is an isomorphism. Consider the commutative diagram as in Figure 21.2.5 with exact rows. By the previous logic, we know that both of the outside vertical maps are surjective. By the five lemma, the right vertical arrow is an isomorphism.
Figure 21.2.5.
We finally check that \(M\) is a projective \(R\)-module. By repeating the logic used to construct Figure 21.2.4, we obtain another commutative diagram as in Figure 21.2.6 with exact rows and columns. The element of \(\Hom_{R_1}(M_1,M_1) \oplus \Hom_{2}(M_2,M_2)\) corresponding to the identity maps has zero horizontal image, so by the snake lemma it lifts to some \(\Hom_{R_1}(M_1,F_1) \oplus \Hom_{R_2}(M_2,F_2)\) which maps to zero in \(\Hom_{R_{12}}(M_{12},F_{12})\text{.}\) This gives us maps \(M_1 \to F_1, M_2 \to F_2\) which agree on \(M\) and map it into \(F\text{;}\) the resulting map \(M \to F\) splits the surjection \(F \to M\text{,}\) showing that \(M\) is projective. (Compare [82], Lemma 1.3.8, Lemma 1.3.9.)
Figure 21.2.6.

Remark 21.2.7.

A well-known instance of Lemma 21.2.1 is the Beauville-Laszlo theorem: this is the case where
\begin{equation*} R_1 = R_t, \quad R_2 = \lim_n R/t^n, \quad R_{12} = R_{2,t} \end{equation*}
for some non-zerodivisor \(t \in R\text{.}\) Compare [117], tag 05E5.

Remark 21.2.8.

In Lemma 21.2.1, the hypothesis that \(R \to R_1\) and \(R_2 \to R_{12}\) are localizations at the same multiplicative subset is only needed to ensure that \(M \otimes_R R_1 \to M_1\) is surjective. In some cases one can run the same argument with a different condition; see for example [82], Theorem 2.7.7 for an application to vector bundles on adic spaces.

Subsection 21.3 Arc-descent for étale cohomology

We record another form of descent for the arc-topology, this time in the realm of étale cohomology.
We first verify descent for a v-covering \(f\colon Y \to X\text{,}\) in which we may assume both schemes are qcqs. We can then write \(Y\) as a filtered limit of some finitely presented \(X\)-schemes, each of which is itself a v-covering, with affine transition maps; we may thus reduce to dealing with a finitely presented \(v\)-covering. By arguing as in [106], Theorem 3.12, we may refine this covering by a composition of a quasicompact open covering with a proper surjective morphism. As descent for the former is immediate, we may further assume that \(f\) is proper surjective. In this case, we are in the usual setting of cohomological descent for étale cohomology. For this, we may assume that \(X\) is the spectrum of a strictly henselian local ring with closed point \(x\text{.}\) By the proper base change theorem, \(\calF(Y) \cong \calF(Y_x)\text{,}\) so we may check the claim after pulling back along \(x \to X\text{.}\) But the resulting map \(Y_x \to x\) has a section, so it satisfies descent for purely formal reasons. See [21], Proposition 5.2 for more details.
To obtain descent for the arc-topology, as in the proof of Theorem 21.1.7 we may use v-descent to reduce to a covering as in Example 20.3.10 in which \(V\) is AIC. In this case, \(V/\frakp\) is also AIC, so both \(V\) and \(V/\frakp\) are strictly henselian with the same residue field. It follows that the functor in question takes the same values on \(V\) and \(V/\frakp\text{,}\) and takes the same values on \(V_\frakp\) and \(\kappa(\frakp)\text{.}\) (Compare [21], Theorem 5.4.)

Exercises 21.4 Exercises


Let \(V\) be a perfect valuation ring over \(\FF_p\text{.}\) Let \(\frakp\) be a prime ideal of \(V\text{.}\) Prove directly that the sequence
\begin{equation*} 0 \to V \to V_{\frakp} \oplus V/\frakp \to \kappa(\frakp) \to 0 \end{equation*}
is exact.
See [24], Lemma 6.3.