The Stacks project

Lemma 62.9.3. Let $f : X \to Y$, $g : Y \to Z$, $h : Z \to T$ be separated morphisms of finite type of quasi-compact and quasi-separated schemes. Then the diagram

\[ \xymatrix{ Rh_! \circ Rg_! \circ Rf_! \ar[r]_{\gamma _ C} \ar[d]^{\gamma _ A} & R(h \circ g)_! \circ Rf_! \ar[d]_{\gamma _{A + B}} \\ Rh_! \circ R(g \circ f)_! \ar[r]^{\gamma _{B + C}} & R(h \circ g \circ f)_! } \]

of isomorphisms of Lemma 62.9.2 commutes (for the meaning of the $\gamma $'s see proof).

Proof. To do this we choose a compactification $\overline{Z}$ of $Z$ over $T$, then a compactification $\overline{Y}$ of $Y$ over $\overline{Z}$, and then a compactification $\overline{X}$ of $X$ over $\overline{Y}$. This uses More on Flatness, Theorem 38.33.8 and Lemma 38.32.2. Let $W \subset \overline{Y}$ be the inverse image of $Z$ under $\overline{Y} \to \overline{Z}$ and let $U \subset V \subset \overline{X}$ be the inverse images of $Y \subset W$ under $\overline{X} \to \overline{Y}$. This produces the following diagram

\[ \xymatrix{ X \ar[d]_ f \ar[r] & U \ar[r] \ar[d] \ar@{}[dr]|A & V \ar[d] \ar[r] \ar@{}[rd]|B & \overline{X} \ar[d] \\ Y \ar[d]_ g \ar[r] & Y \ar[r] \ar[d] & W \ar[r] \ar[d] \ar@{}[rd]|C & \overline{Y} \ar[d] \\ Z \ar[d]_ h \ar[r] & Z \ar[d] \ar[r] & Z \ar[d] \ar[r] & \overline{Z} \ar[d] \\ T \ar[r] & T \ar[r] & T \ar[r] & T } \]

Without introducing tons of notation but arguing exactly as in the proof of Lemma 62.9.2 we see that the maps in the first displayed diagram use the maps of Lemma 62.8.1 for the rectangles $A + B$, $B + C$, $A$, and $C$ as indicated in the diagram in the statement of the lemma. Since by Lemmas 62.8.2 and 62.8.3 we have $\gamma _{A + B} = \gamma _ B \circ \gamma _ A$ and $\gamma _{B + C} = \gamma _ B \circ \gamma _ C$ we conclude that the desired equality holds provided $\gamma _ A \circ \gamma _ C = \gamma _ C \circ \gamma _ A$. This is true because the two squares $A$ and $C$ only intersect in one point (similar to the last argument in Remark 62.8.4). $\square$


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