Lemma 59.89.3. Let $S$ be a scheme. Let $S' = \mathop{\mathrm{lim}}\nolimits S_ i$ be a directed inverse limit of schemes $S_ i$ smooth over $S$ with affine transition morphisms. Let $f : X \to S$ be quasi-compact and quasi-separated and form the fibre square

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

Then

$g^{-1}Rf_*E = R(f')_*(g')^{-1}E$

for any $E \in D^+(X_{\acute{e}tale})$ whose cohomology sheaves $H^ q(E)$ have stalks which are torsion of orders invertible on $S$.

Proof. Consider the spectral sequences

$E_2^{p, q} = R^ pf_*H^ q(E) \quad \text{and}\quad {E'}_2^{p, q} = R^ pf'_*H^ q((g')^{-1}E) = R^ pf'_*(g')^{-1}H^ q(E)$

converging to $R^ nf_*E$ and $R^ nf'_*(g')^{-1}E$. These spectral sequences are constructed in Derived Categories, Lemma 13.21.3. Combining the smooth base change theorem (Theorem 59.89.2) with Lemma 59.86.3 we see that

$g^{-1}R^ pf_*H^ q(E) = R^ p(f')_*(g')^{-1}H^ q(E)$

Combining all of the above we get the lemma. $\square$

Comment #5906 by Rubén Muñoz--Bertrand on

Typo in the statement: quasi-compact.

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