The Stacks project

Lemma 62.9.4. Consider a cartesian square

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

of quasi-compact and quasi-separated schemes with $f$ separated and of finite type. Then there is a canonical isomorphism

\[ g^{-1} \circ Rf_! \to Rf'_! \circ (g')^{-1} \]

Moreover, these isomorphisms are compatible with the isomorphisms of Lemma 62.9.2.

Proof. Choose a compactification $j : X \to \overline{X}$ over $Y$ and denote $\overline{f} : \overline{X} \to Y$ the structure morphism. Let $j' : X' \to \overline{X}'$ and $\overline{f}' : \overline{X}' \to Y'$ denote the base changes of $j$ and $\overline{f}$. Since $Rf_! = R\overline{f}_* \circ j_!$ and $Rf'_! = R\overline{f}'_* \circ j'_!$ the isomorphism can be constructed via

\[ g^{-1} \circ R\overline{f}_* \circ j_! \to R\overline{f}'_* \circ (\overline{g}')^{-1} \circ j_! \to R\overline{f}'_* \circ j'_! \circ (g')^{-1} \]

where the first arrow is the isomorphism given to us by the proper base change theorem (Étale Cohomology, Lemma 59.91.12 in the bounded below torsion case and Étale Cohomology, Lemma 59.92.3 in the case that $\Lambda $ is torsion) and the second arrow is the isomorphism of Lemma 62.3.12.

To finish the proof we have to show two things: first we have to show that the isomorphism of functors so obtained does not depend on the choice of the compactification and second we have to show that if we vertically stack two base change diagrams as in the lemma, then these base change isomorphisms are compatible with the isomorphisms of Lemma 62.9.2. A straightforward argument which we omit shows that both follow if we can show that the isomorphisms

  1. $Rg_* \circ Rf_* = R(g \circ f)_*$ for $f : X \to Y$ and $g : Y \to Z$ proper,

  2. $g_! \circ f_! = (g \circ f)_!$ for $f : X \to Y$ and $g : Y \to Z$ separated and quasi-finite, and

  3. $g_! \circ Rf'_* = Rf_* \circ g'_!$ for $f : X \to Y$ and $f' : X' \to Y'$ proper and $g : Y' \to Y$ and $g' : X' \to X$ separated and quasi-finite with $f \circ g' = g \circ f'$

are compatible with base change. This holds for (1) by Cohomology on Sites, Remark 21.19.4, for (2) by Remark 62.3.14, and (3) by Lemma 62.8.5. $\square$


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