Lemma 15.86.2 (0H31)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.86: Rlim of abelian groups
Lemma 15.86.2 (cite)

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$0 \to (A_ i) \to (B_ i) \to (C_ i) \to 0$

be a short exact sequence of inverse systems of abelian groups. Then there is an associated $6$ term exact sequence $0 \to \mathop{\mathrm{lim}}\nolimits A_ i \to \mathop{\mathrm{lim}}\nolimits B_ i \to \mathop{\mathrm{lim}}\nolimits C_ i \to R^1\mathop{\mathrm{lim}}\nolimits A_ i \to R^1\mathop{\mathrm{lim}}\nolimits B_ i \to R^1\mathop{\mathrm{lim}}\nolimits C_ i \to 0$ .

Proof. Follows from the vanishing in Lemma 15.86.1. $\square$

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