The Stacks project

61.30 Proper base change

In this section we explain how to prove the proper base change theorem for derived complete objects on the pro-étale site using the proper base change theorem for étale cohomology following the general theme that we use the pro-étale topology only to deal with “limit issues” and we use results proved for the étale topology to handle everything else.

Theorem 61.30.1. Let $f : X \to Y$ be a proper morphism of schemes. Let $g : Y' \to Y$ be a morphism of schemes giving rise to the base change diagram

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]

Let $\Lambda $ be a Noetherian ring and let $I \subset \Lambda $ be an ideal such that $\Lambda /I$ is torsion. Let $K$ be an object of $D(X_{pro\text{-}\acute{e}tale})$ such that

  1. $K$ is derived complete, and

  2. $K \otimes _\Lambda ^\mathbf {L} \underline{\Lambda /I^ n}$ is bounded below with cohomology sheaves coming from $X_{\acute{e}tale}$,

  3. $\Lambda /I^ n$ is a perfect $\Lambda $-module1.

Then the base change map

\[ Lg_{comp}^*Rf_*K \longrightarrow Rf'_*L(g')^*_{comp}K \]

is an isomorphism.

Proof. We omit the construction of the base change map (this uses only formal properties of derived pushforward and completed derived pullback, compare with Cohomology on Sites, Remark 21.19.3). Write $K_ n = K \otimes ^\mathbf {L}_\Lambda \underline{\Lambda /I^ n}$. By Lemma 61.20.1 we have $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ because $K$ is derived complete. By Lemmas 61.20.2 and 61.20.1 we can unwind the left hand side

\[ Lg_{comp}^* Rf_* K = R\mathop{\mathrm{lim}}\nolimits Lg^*(Rf_*K)\otimes ^\mathbf {L}_\Lambda \underline{\Lambda /I^ n} = R\mathop{\mathrm{lim}}\nolimits Lg^* Rf_* K_ n \]

the last equality because $\Lambda /I^ n$ is a perfect module and the projection formula (Cohomology on Sites, Lemma 21.48.1). Using Lemma 61.20.2 we can unwind the right hand side

\[ Rf'_* L(g')^*_{comp} K = Rf'_* R\mathop{\mathrm{lim}}\nolimits L(g')^* K_ n = R\mathop{\mathrm{lim}}\nolimits Rf'_* L(g')^* K_ n \]

the last equality because $Rf'_*$ commutes with $R\mathop{\mathrm{lim}}\nolimits $ (Cohomology on Sites, Lemma 21.22.3). Thus it suffices to show the maps

\[ Lg^* Rf_* K_ n \longrightarrow Rf'_* L(g')^* K_ n \]

are isomorphisms. By Lemma 61.19.8 and our second condition we can write $K_ n = \epsilon ^{-1}L_ n$ for some $L_ n \in D^+(X_{\acute{e}tale}, \Lambda /I^ n)$. By Lemma 61.23.1 and the fact that $\epsilon ^{-1}$ commutes with pullbacks we obtain

\[ Lg^* Rf_* K_ n = Lg^* Rf_* \epsilon ^*L_ n = Lg^* \epsilon ^{-1} Rf_* L_ n = \epsilon ^{-1} Lg^* Rf_* L_ n \]

and

\[ Rf'_* L(g')^* K_ n = Rf'_* L(g')^* \epsilon ^{-1} L_ n = Rf'_* \epsilon ^{-1} L(g')^* L_ n = \epsilon ^{-1} Rf'_* L(g')^* L_ n \]

(this also uses that $L_ n$ is bounded below). Finally, by the proper base change theorem for étale cohomology (Étale Cohomology, Theorem 59.91.11) we have

\[ Lg^* Rf_* L_ n = Rf'_* L(g')^* L_ n \]

(again using that $L_ n$ is bounded below) and the theorem is proved. $\square$

[1] This assumption can be removed if $K$ is a constructible complex, see [BS].

Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09C8. Beware of the difference between the letter 'O' and the digit '0'.