The Stacks project

Lemma 62.8.3. Consider a commutative diagram of schemes

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

with $f$, $f'$, and $f''$ proper and $g$, $g'$, $h$, and $h'$ separated and locally quasi-finite. Then the isomorphisms of Lemma 62.8.1 for the two squares compose to give the isomorphism for the outer rectangle (see proof for a precise statement).

Proof. The statement means that if we write $(h \circ h')_! = h_! \circ h'_!$ and $(g \circ g')_! = g_! \circ g'_!$ using the equalities of Lemma 62.3.13, then the isomorphism $h_! \circ h'_! \circ Rf''_* \to Rf_* \circ g_! \circ g'_!$ of the outer rectangle is equal to the composition

\[ h_! \circ h'_! \circ Rf''_* \to h_! \circ Rf'_* \circ g'_! \to Rf_* \circ g_! \circ g'_! \]

of the two maps of the squares in the diagram. To prove this choose a K-injective complex $\mathcal{I}^\bullet $ of $\Lambda $-modules on $X''_{\acute{e}tale}$ and a quasi-isomorphism $g'_!\mathcal{I}^\bullet \to \mathcal{J}^\bullet $ to a K-injective complex $\mathcal{J}^\bullet $ of $\Lambda $-modules on $X'_{\acute{e}tale}$. Next, choose a quasi-isomorphism $g_!\mathcal{J}^\bullet \to \mathcal{K}^\bullet $ to a K-injective complex $\mathcal{K}^\bullet $ of $\Lambda $-modules on $X_{\acute{e}tale}$. The proof of Lemma 62.8.1 shows that the canonical maps

\[ h'_!f''_*\mathcal{I}^\bullet \to f'_*\mathcal{J}^\bullet \quad \text{and}\quad h_!f'_*\mathcal{J}^\bullet \to f_*\mathcal{K}^\bullet \]

are quasi-isomorphisms and these quasi-isomorphisms define the first and second arrow above. Since $g_!$ is an exact functor (Lemma 62.3.17) we find that $g_!g'_!\mathcal{I}^\bullet \to \mathcal{K}^\bullet $ is a quasi-ismorphism and hence the canonical map

\[ h_!h'_!f''_*\mathcal{I}^\bullet \to f_*\mathcal{K}^\bullet \]

is a quasi-isomorphism and represents the map for the outer rectangle in the derived category. Clearly this map is the composition of the other two and the proof is complete. $\square$


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