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

$K$ is derived complete, and

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

$\Lambda /I^ n$ is a perfect $\Lambda $-module^{1}.

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 21.50.1). 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 21.23.3). 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.
$\square$

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