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.
A left exact functor preserves monomorphisms and a right exact functor preserves epimorphisms
Statement
A left exact functor between abelian categories preserves monomorphisms, and a right exact functor preserves epimorphisms.
Facts & Assumptions
Given: A functor between abelian categories.
One-sided exactness is characterized by the corresponding short exact sequence test (Left exactness, right exactness, and exactness are characterized by short exact sequences).
In an abelian category, monomorphisms are exactly the zero-kernel maps and epimorphisms are exactly the zero-cokernel maps (In an abelian category, monic means zero kernel and epic means zero cokernel).
Proof
If is monic, then [L2] says is left exact. A left exact functor carries this to another left exact sequence by [L1], so again has zero kernel. By [L2], is monic.
The epimorphism claim is dual: if is epic, then [L2] says is right exact, and a right exact functor carries it to a right exact sequence. Hence the cokernel of is zero, so [L2] makes epic.
Depends on
Used by
- A coproduct-preserving left exact module functor is not tensor Counterexample
- Coherently shift-compatible functors and transformations form k-linear hom categories Lemma
- Equivalences preserve small projective generators Lemma
- Homogeneous free presentations prove the graded comparison is an isomorphism Lemma
- Natural transformations of additive cocontinuous module functors are determined at the regular module Lemma
- An exact functor transports every diagram lemma Theorem
Dependency tree · two levels
11 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 12.7, Lemma 12.7.2 (standard reference, not scraped)