Lemma 15.84.9 (0G9J)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.84: Two term complexes
Lemma 15.84.9 (cite)

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Lemma 15.84.9. Let $I$ be an ideal of a ring $R$ . Let $\alpha : K \to K'$ be a morphism of $D(R)$ . Assume

$H^ i(K) = H^ i(K') = 0$ for $i \not\in \{ -1, 0\}$
$H^0(\alpha )$ is an isomorphism and $H^{-1}(\alpha )$ is surjective.

If $K$ satisfies the equivalent conditions (1), (2), and (3) of Lemma 15.84.5, then $K'$ does too.

Proof. Set $M = \mathop{\mathrm{Ker}}(H^{-1}(\alpha ))$ . Then $\alpha$ fits into a distinguished triangle

$M[1] \to K \to K' \to M[2]$

For any $R$ -module $N$ this determines an exact sequence

$\mathop{\mathrm{Ext}}\nolimits ^0_ R(M[1], N) \to \mathop{\mathrm{Ext}}\nolimits ^1_ R(K', N) \to \mathop{\mathrm{Ext}}\nolimits ^1_ R(K, N)$

Since $\mathop{\mathrm{Ext}}\nolimits ^0_ R(M[1], N) = \mathop{\mathrm{Ext}}\nolimits ^{-1}_ R(M, N) = 0$ we see that $\mathop{\mathrm{Ext}}\nolimits ^1_ R(K', N)$ is a submodule of $\mathop{\mathrm{Ext}}\nolimits ^1_ R(K, N)$ . Hence if $\mathop{\mathrm{Ext}}\nolimits ^1_ R(K, N)$ is annihilated by $I$ so is $\mathop{\mathrm{Ext}}\nolimits ^1_ R(K', N)$ . $\square$

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