Lemma 15.95.7. Let $A$ be a ring and let $f \in A$ be a nonzerodivisor. Let $M^\bullet$ be a complex of $f$-torsion free $A$-modules. For $i \in \mathbf{Z}$ the following are equivalent

1. $\mathop{\mathrm{Ker}}(d^ i \bmod f^2)$ surjects onto $\mathop{\mathrm{Ker}}(d^ i \bmod f)$,

2. $\beta : H^ i(M^\bullet \otimes _ A f^ iA/f^{i + 1}A) \to H^{i + 1}(M^\bullet \otimes _ A f^{i + 1}A/f^{i + 2}A)$ is zero.

These equivalent conditions are implied by the condition $H^{i + 1}(M^\bullet )[f] = 0$.

Proof. The equivalence of (1) and (2) follows from the definition of $\beta$ as the boundary map on cohomology of a short exact sequence of complexes isomorphic to the short exact sequence of complexes $0 \to fM^\bullet /f^2M^\bullet \to M^\bullet /f^2M^\bullet \to M^\bullet /fM^\bullet \to 0$. If $\beta \not= 0$, then $H^{i + 1}(M^\bullet )[f] \not= 0$ because of the factorization (15.95.5.1). $\square$

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