TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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Objects of finite length form an abelian subcategory

Statement

In an abelian category, the full subcategory whose objects have finite length is an abelian subcategory.

Facts & Assumptions

Given: An abelian category $A$ .

[L1]

Finite length is the property defined in Object of finite length.

[L2]

Finite length is stable under passing to subobjects and quotients, and is additive across a subobject (Length is additive along a subobject).

[L3]

A full subcategory is abelian precisely when it is closed under kernels, cokernels, and finite biproducts computed in the ambient abelian category (Abelian subcategory and exact embedding).

Proof

technique · direct

1.1L1L2construct

Let $f : X \to Y$ be a morphism between finite-length objects. Since $\ker (f) \leq X$ and $im (f) \leq Y$ , [L2] makes both $\ker (f)$ and $im (f)$ finite length. The quotient $Y / im (f)$ is then finite length by [L2], so the cokernel of $f$ is finite length as well.

1.2L2construct

If $X$ and $Y$ have finite length, then the inclusion $X ↣ X \oplus Y$ has quotient $Y$ . Applying [L2] to that inclusion shows that $X \oplus Y$ has finite length. So the finite-length objects are closed under finite biproducts.

2.1L3step 1.1step 1.2∎

Steps 1.1 and 1.2 are exactly the kernel, cokernel, and finite-biproduct closures required by [L3]. Therefore the full subcategory of finite-length objects is an abelian subcategory.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor Categories, Section 1.5 (standard reference, not scraped)