Lemma 12.31.4 (070B)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 12: Homological Algebra
Section 12.31: Inverse systems
Lemma 12.31.4 (cite)

numberstags

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

be an exact sequence of inverse systems of abelian groups. If the system $(A_ i)$ is ML, then the sequence

$\mathop{\mathrm{lim}}\nolimits _ i B_ i \to \mathop{\mathrm{lim}}\nolimits _ i C_ i \to \mathop{\mathrm{lim}}\nolimits _ i D_ i$

is exact.

Proof. Let $Z_ i = \mathop{\mathrm{Ker}}(C_ i \to D_ i)$ and $I_ i = \mathop{\mathrm{Im}}(A_ i \to B_ i)$ . Then $\mathop{\mathrm{lim}}\nolimits Z_ i = \mathop{\mathrm{Ker}}(\mathop{\mathrm{lim}}\nolimits C_ i \to \mathop{\mathrm{lim}}\nolimits D_ i)$ and we get a short exact sequence of systems

$0 \to (I_ i) \to (B_ i) \to (Z_ i) \to 0$

Moreover, by Lemma 12.31.3 we see that $(I_ i)$ has (ML), thus another application of Lemma 12.31.3 shows that $\mathop{\mathrm{lim}}\nolimits B_ i \to \mathop{\mathrm{lim}}\nolimits Z_ i$ is surjective which proves the lemma. $\square$

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