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 -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 21.50.1). 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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