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.
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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