Lemma 15.95.7 (0F7Y)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.95: An operator introduced by Berthelot and Ogus
Lemma 15.95.7 (cite)

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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

$\mathop{\mathrm{Ker}}(d^ i \bmod f^2)$ surjects onto $\mathop{\mathrm{Ker}}(d^ i \bmod f)$ ,
$\beta : H^ i(M^\bullet \otimes _ A f^ iA/f^{i + 1}A) \to H^{i + 1}(M^\bullet \otimes _ A f^{i + 1}A/f^{i + 2}A)$ 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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