Lemma 63.9.2. Let $f : X \to Y$ and $g : Y \to Z$ be separated morphisms of finite type of quasi-compact and quasi-separated schemes. Then there is a canonical isomorphism $Rg_! \circ Rf_! \to R(g \circ f)_!$.
Proof. Choose a compactification $i : Y \to \overline{Y}$ of $Y$ over $Z$. Choose a compactification $X \to \overline{X}$ of $X$ over $\overline{Y}$. This uses More on Flatness, Theorem 38.33.8 and Lemma 38.32.2 twice. Let $U$ be the inverse image of $Y$ in $\overline{X}$ so that we get the commutative diagram
\[ \xymatrix{ X \ar[r]_ j \ar[d]_ f & U \ar[dl]^{f'} \ar[r]_{j'} & \overline{X} \ar[dl]^{\overline{f}} \\ Y \ar[r]_ i \ar[d]_ g & \overline{Y} \ar[dl]^{\overline{g}} \\ Z } \]
Then we have
\begin{align*} R(g \circ f)_! & = R(\overline{g} \circ \overline{f})_* \circ (j' \circ j)_! \\ & = R\overline{g}_* \circ R\overline{f}_* \circ j'_! \circ j_! \\ & = R\overline{g}_* \circ i_! \circ Rf'_* \circ j_! \\ & = Rg_! \circ Rf_! \end{align*}
The first equality is the definition of $R(g \circ f)_!$. The second equality uses the identifications $R(\overline{g} \circ \overline{f})_* = R\overline{g}_* \circ R\overline{f}_*$ and $(j' \circ j)_! = j'_! \circ j_!$ of Lemma 63.3.13. The identification $i_! \circ Rf'_* \to R\overline{f}_* \circ j_!$ used in the third equality is Lemma 63.8.1. The final fourth equality is the definition of $Rg_!$ and $Rf_!$. To finish the proof we show that this isomorphism is independent of choices made.
Suppose we have two diagrams
\[ \vcenter { \xymatrix{ X \ar[r]_{j_1} \ar[d] & U_1 \ar[dl]^{f_1} \ar[r]_{j'_1} & \overline{X}_1 \ar[dl]^{\overline{f}_1} \\ Y \ar[r]_{i_1} \ar[d] & \overline{Y}_1 \ar[dl]^{\overline{g}_1} \\ Z } } \quad \text{and}\quad \vcenter { \xymatrix{ X \ar[r]_{j_2} \ar[d] & U_2 \ar[dl]^{f_2} \ar[r]_{j'_2} & \overline{X}_2 \ar[dl]^{\overline{f}_2} \\ Y \ar[r]_{i_2} \ar[d] & \overline{Y}_2 \ar[dl]^{\overline{g}_2} \\ Z } } \]
We can first choose a compactification $i : Y \to \overline{Y}$ of $Y$ over $Z$ which dominates both $\overline{Y}_1$ and $\overline{Y}_2$, see More on Flatness, Lemma 38.32.1. By More on Flatness, Lemma 38.32.3 and Categories, Lemmas 4.27.13 and 4.27.14 we can choose a compactification $X \to \overline{X}$ of $X$ over $\overline{Y}$ with morphisms $\overline{X} \to \overline{X}_1$ and $\overline{X} \to \overline{X}_2$ and such that the composition $\overline{X} \to \overline{Y} \to \overline{Y}_1$ is equal to the composition $\overline{X} \to \overline{X}_1 \to \overline{Y}_1$ and such that the composition $\overline{X} \to \overline{Y} \to \overline{Y}_2$ is equal to the composition $\overline{X} \to \overline{X}_2 \to \overline{Y}_2$. Thus we see that it suffices to compare the maps determined by our diagrams when we have a commutative diagram as follows
\[ \xymatrix{ X \ar[rr]_{j_1} \ar@{=}[d] & & U_1 \ar[d]^{h'} \ar[ddll] \ar[rr]_{j'_1} & & \overline{X}_1 \ar[d]^ h \ar[ddll] \\ X \ar '[r][rr]^-{j_2} \ar[d] & & U_2 \ar '[dl][ddll] \ar '[r][rr]^-{j'_2} & & \overline{X}_2 \ar[ddll] \\ Y \ar[rr]^{i_1} \ar@{=}[d] & & \overline{Y}_1 \ar[d]^ k \\ Y \ar[rr]^{i_2} \ar[d] & & \overline{Y}_2 \ar[dll] \\ Z } \]
Each of the squares
\[ \xymatrix{ X \ar[r]_{j_1} \ar[d]_{\text{id}} \ar@{}[dr]|A & U_1 \ar[d]^{h'} \\ X \ar[r]^{j_2} & U_2 } \quad \xymatrix{ U_2 \ar[r]_{j_2'} \ar[d]_{f_2} \ar@{}[dr]|B & \overline{X}_2 \ar[d]^{\overline{f}_2} \\ Y \ar[r]^{i_2} & \overline{Y}_2 } \quad \xymatrix{ U_1 \ar[r]_{j_1'} \ar[d]_{f_1} \ar@{}[dr]|C & \overline{X}_1 \ar[d]^{\overline{f}_1} \\ Y \ar[r]^{i_1} & \overline{Y}_1 } \quad \xymatrix{ Y \ar[r]_{i_1} \ar[d]_{\text{id}} \ar@{}[dr]|D & \overline{Y}_1 \ar[d]^ k \\ Y \ar[r]^{i_2} & \overline{Y}_2 } \quad \xymatrix{ X \ar[r]_{j_1' \circ j_1} \ar[d]_{\text{id}} \ar@{}[dr]|E & \overline{X}_1 \ar[d]^ h \\ X \ar[r]^{j_2} & \overline{X}_2 } \]
gives rise to an isomorphism as follows
\begin{align*} \gamma _ A & : j_{2, !} \to Rh'_* \circ j_{1, !} \\ \gamma _ B & : i_{2, !} \circ Rf_{2, *} \to R\overline{f}_{2, *} \circ j'_{2, !} \\ \gamma _ C & : i_{1, !} \circ Rf_{1, *} \to R\overline{f}_{1, *} \circ j'_{1, !} \\ \gamma _ D & : i_{2, !} \to Rk_* \circ i_{1, !} \\ \gamma _ E & : j_{2, !} \to Rh_* \circ (j'_1 \circ j_1)_! \end{align*}
by applying the map from Lemma 63.8.1 (which is the same as the map in Lemma 63.8.6 in case the left vertical arrow is the identity). Let us write
\begin{align*} F_1 & = Rf_{1, *} \circ j_{1, !} \\ F_2 & = Rf_{2, *} \circ j_{2, !} \\ G_1 & = R\overline{g}_{1, *} \circ i_{1, !} \\ G_2 & = R\overline{g}_{2, *} \circ i_{2, !} \\ C_1 & = R(\overline{g}_1 \circ \overline{f}_1)_* \circ (j'_1 \circ j_1)_! \\ C_2 & = R(\overline{g}_2 \circ \overline{f}_2)_* \circ (j'_2 \circ j_2)_! \end{align*}
The construction given in the first paragraph of the proof and in Lemma 63.9.1 uses
$\gamma _ C$ for the map $G_1 \circ F_1 \to C_1$,
$\gamma _ B$ for the map $G_2 \circ F_2 \to C_2 $,
$\gamma _ A$ for the map $F_2 \to F_1$,
$\gamma _ D$ for the map $G_2 \to G_1$, and
$\gamma _ E$ for the map $C_2 \to C_1$.
This implies that we have to show that the diagram
\[ \xymatrix{ C_2 \ar[rr]_{\gamma _ E} & & C_1 \\ G_2 \circ F_2 \ar[rr]^{\gamma _ D \circ \gamma _ A} \ar[u]^{\gamma _ B} & & G_1 \circ F_1 \ar[u]_{\gamma _ C} } \]
is commutative. We will use Lemmas 63.8.2 and 63.8.3 and with (abuse of) notation as in Remark 63.8.4 (in particular dropping $\star $ products with identity transformations from the notation). We can write $\gamma _ E = \gamma _ F \circ \gamma _ A$ where
\[ \xymatrix{ U_1 \ar[r]_{j'_1} \ar[d]_{h'} \ar@{}[rd]|F & \overline{X}_1 \ar[d]^ h \\ U_2 \ar[r]^{j'_2} & \overline{X}_2 } \]
Thus we see that
\[ \gamma _ E \circ \gamma _ B = \gamma _ F \circ \gamma _ A \circ \gamma _ B = \gamma _ F \circ \gamma _ B \circ \gamma _ A \]
the last equality because the two squares $A$ and $B$ only intersect in one point (similar to the last argument in Remark 63.8.4). Thus it suffices to prove that $\gamma _ C \circ \gamma _ D = \gamma _ F \circ \gamma _ B$. Since both of these are equal to the map for the square
\[ \xymatrix{ U_1 \ar[r] \ar[d] & \overline{X}_1 \ar[d] \\ Y \ar[r] & \overline{Y}_2 } \]
we conclude. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: