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The endpoints of a short exact sequence encode injectivity and surjectivity

Statement

For a module homomorphism $f : A \to B$ , the sequence $0 \to A \overset{f}{\to} B$ is exact at $A$ if and only if $f$ is injective. The sequence $A \overset{f}{\to} B \to 0$ is exact at $B$ if and only if $f$ is surjective. Consequently $0 \to A \overset{i}{\to} B \overset{p}{\to} C \to 0$ is short exact if and only if $i$ is injective, $p$ is surjective, and $im i = \ker p$ .

Facts & Assumptions

Given: Module homomorphisms $f : A \to B$ , $i : A \to B$ , and $p : B \to C$ .

[F1]

Exactness at a term means equality of the incoming image and outgoing kernel (Exact sequences and short exact sequences of modules).

[F2]

Injective means equal images have equal inputs, and surjective means every target element has a preimage (Injection, surjection, bijection).

[L1]

A module homomorphism is injective exactly when its kernel is zero (Kernels and images of module homomorphisms are submodules, and injectivity is equivalent to trivial kernel).

Proof

technique · direct

1.1

The zero map $0 \to A$ has image ${0}$ , so by [F1] the sequence $0 \to A \overset{f}{\to} B$ is exact at $A$ exactly when $\ker f = {0}$ , which is equivalent to injectivity by [L1].

F1L1

1.2

The zero map $B \to 0$ has kernel $B$ , so by [F1] the sequence $A \overset{f}{\to} B \to 0$ is exact at $B$ exactly when $im f = B$ , which is equivalent to surjectivity by [F2].

F1F2

2.1

A four-term sequence $0 \to A \overset{i}{\to} B \overset{p}{\to} C \to 0$ is exact at $A, B, C$ exactly when the endpoint conditions of steps 1.1 and 1.2 hold and $im i = \ker p$ holds at $B$ .

step 1.1step 1.2F1

3.1

This proves both endpoint equivalences and both directions of the short-exact characterization.

step 1.1step 1.2step 2.1∎

Depends on

Used by

The Snake Lemma for modules Theorem
The splitting lemma for short exact sequences of modules Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 38 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

A. Kleshchev, Lectures on Abstract Algebra for Graduate Students, sections 3.6, 3.14, and 3.15 (standard reference, not scraped)
The Stacks Project, Algebra (standard reference, not scraped)
P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)