Remark 91.10.9 (0CYC)—The Stacks project

Remark 91.10.9. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$, $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_ i), \mathcal{O}'_ i)$, $\mathcal{I}_ i$, and $h : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_1), \mathcal{O}'_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}_2), \mathcal{O}'_2)$ be as in Remark 91.10.7. Observe that $h^\sharp : \mathcal{O}'_2 \to \mathcal{O}'_1$ in particular induces an $\mathcal{O}$-module map $\mathcal{I}_2 \to \mathcal{I}_1$. Let $\mathcal{F}$ be an $\mathcal{O}$-module. Let $(\mathcal{K}_ i, c_ i)$, $i = 1, 2$ be a pair consisting of an $\mathcal{O}$-module $\mathcal{K}_ i$ and a map $c_ i : \mathcal{I}_ i \otimes _\mathcal {O} \mathcal{F} \to \mathcal{K}_ i$. Assume furthermore given a map of $\mathcal{O}$-modules $\mathcal{K}_2 \to \mathcal{K}_1$ such that

\[ \xymatrix{ \mathcal{I}_2 \otimes _\mathcal {O} \mathcal{F} \ar[r]_-{c_2} \ar[d] & \mathcal{K}_2 \ar[d] \\ \mathcal{I}_1 \otimes _\mathcal {O} \mathcal{F} \ar[r]^-{c_1} & \mathcal{K}_1 } \]

is commutative. Then we claim the map

\[ \mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {O}(\mathcal{F}, \mathcal{K}_2) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^2_\mathcal {O}(\mathcal{F}, \mathcal{K}_1) \]

sends $o(\mathcal{F}, \mathcal{K}_2, c_2)$ to $o(\mathcal{F}, \mathcal{K}_1, c_1)$.

To prove this claim choose an embedding $j_2 : \mathcal{K}_2 \to \mathcal{K}_2'$ where $\mathcal{K}_2'$ is an injective $\mathcal{O}$-module. As in the proof of Lemma 91.10.4 we can choose an extension of $\mathcal{O}_2$-modules

\[ 0 \to \mathcal{K}_2' \to \mathcal{E}_2 \to \mathcal{F} \to 0 \]

such that $c_{\mathcal{E}_2} = j_2 \circ c_2$. The proof of Lemma 91.10.4 constructs $o(\mathcal{F}, \mathcal{K}_2, c_2)$ as the Yoneda extension class (in the sense of Derived Categories, Section 13.27) of the exact sequence of $\mathcal{O}$-modules

\[ 0 \to \mathcal{K}_2 \to \mathcal{K}_2' \to \mathcal{E}_2/\mathcal{K}_2 \to \mathcal{F} \to 0 \]

Let $\mathcal{K}_1'$ be the cokernel of $\mathcal{K}_2 \to \mathcal{K}_1 \oplus \mathcal{K}_2'$. There is an injection $j_1 : \mathcal{K}_1 \to \mathcal{K}_1'$ and a map $\mathcal{K}_2' \to \mathcal{K}_1'$ forming a commutative square. We form the pushout:

\[ \xymatrix{ 0 \ar[r] & \mathcal{K}_2' \ar[r] \ar[d] & \mathcal{E}_2 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{K}_1' \ar[r] & \mathcal{E}_1 \ar[r] & \mathcal{F} \ar[r] & 0 } \]

There is a canonical $\mathcal{O}_1$-module structure on $\mathcal{E}_1$ and for this structure we have $c_{\mathcal{E}_1} = j_1 \circ c_1$ (this uses the commutativity of the diagram involving $c_1$ and $c_2$ above). The procedure of Lemma 91.10.4 tells us that $o(\mathcal{F}, \mathcal{K}_1, c_1)$ is the Yoneda extension class of the exact sequence of $\mathcal{O}$-modules

\[ 0 \to \mathcal{K}_1 \to \mathcal{K}_1' \to \mathcal{E}_1/\mathcal{K}_1 \to \mathcal{F} \to 0 \]

Since we have maps of exact sequences

\[ \xymatrix{ 0 \ar[r] & \mathcal{K}_2 \ar[d] \ar[r] & \mathcal{K}_2' \ar[d] \ar[r] & \mathcal{E}_2/\mathcal{K}_2 \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar@{=}[d] & 0 \\ 0 \ar[r] & \mathcal{K}_2 \ar[r] & \mathcal{K}_2' \ar[r] & \mathcal{E}_2/\mathcal{K}_2 \ar[r] & \mathcal{F} \ar[r] & 0 } \]

we conclude that the claim is true.

The Stacks project

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