Lemma 51.22.2. Let $0 \to K \to L \to M \to 0$ be a short exact sequence of $A$-modules such that $K$ and $L$ are annihilated by $I^ n$ and $M$ is an $(A, n, c)$-module. Then the kernel of $p^*K \to p^*L$ is scheme theoretically supported on $Y_ c$.

Proof. Let $\mathop{\mathrm{Spec}}(B) \subset X$ be an affine open. The restriction of the exact sequence over $\mathop{\mathrm{Spec}}(B)$ corresponds to the sequence of $B$-modules

$K \otimes _ A B \to L \otimes _ A B \to M \otimes _ A B \to 0$

which is isomorphismic to the sequence

$K \otimes _{A/I^ n} B/I^ nB \to L \otimes _{A/I^ n} B/I^ nB \to M \otimes _{A/I^ n} B/I^ nB \to 0$

Hence the kernel of the first map is the image of the module $\text{Tor}_1^{A/I^ n}(M, B/I^ nB)$. Recall that the exceptional divisor $Y$ is cut out by $I\mathcal{O}_ X$. Hence it suffices to show that $\text{Tor}_1^{A/I^ n}(M, B/I^ nB)$ is annihilated by $I^ c$. Since multiplication by $a \in I^ c$ on $M$ factors through a finite free $A/I^ n$-module, this is clear. $\square$

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