Lemma 47.2.2 (08XK)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 47: Dualizing Complexes
Section 47.2: Essential surjections and injections
Lemma 47.2.2 (cite)

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Lemma 47.2.2. Let $\mathcal{A}$ be an abelian category.

If $A \subset B$ and $B \subset C$ are essential extensions, then $A \subset C$ is an essential extension.
If $A \subset B$ is an essential extension and $C \subset B$ is a subobject, then $A \cap C \subset C$ is an essential extension.
If $A \to B$ and $B \to C$ are essential surjections, then $A \to C$ is an essential surjection.
Given an essential surjection $f : A \to B$ and a surjection $A \to C$ with kernel $K$ , the morphism $C \to B/f(K)$ is an essential surjection.

Proof. Omitted. $\square$

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