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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