Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Exact sequence and short exact sequence in an abelian category
Definition
A sequence of morphisms in an abelian category is exact when it is exact at every interior object in the sense of Exactness at a node.
In particular, a composable triple is a short exact sequence when it is exact at , at , and at .
This page uses the sequence language literally: the definition names exactness of displayed chains of morphisms, not chain complexes as objects.
Depends on
Used by
- An extension of an object by an object in an abelian category Definition
- Cohomological delta functor Definition
- Exact sequences of sheaves Definition
- Grothendieck group of an essentially small abelian category Definition
- Homological delta functor Definition
- Short exact sequence of complexes Definition
- Snake data Definition
- Split short exact sequence in an abelian category Definition
- The Grothendieck ring of a tensor category Definition
- Rank bookkeeping for a long exact sequence of finite-dimensional vector spaces Lemma
- The degree-zero horseshoe lift Lemma
- An exact sequence is a complex, and its exactness agrees with the earlier notion Proposition
- A short exact sequence is a kernel-cokernel pair Theorem
- A square is cartesian exactly when a short sequence is exact Theorem
- AB5 is equivalent to exactness of filtered colimits Theorem
- Degenerate exactness criteria Theorem
- Short five lemma in an abelian category Theorem
Dependency tree · two levels
4 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
- The Stacks Project, Section 12.5, Definition 12.5.7 (standard reference, not scraped)
- David Mehrle, Category Theory, Part III, Definition 7.20 (standard reference, not scraped)