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.
An additive functor is exact exactly when it preserves kernels and cokernels
Statement
Let be an additive functor between abelian categories. Then is exact if and only if it preserves kernels and cokernels.
Facts & Assumptions
Given: An additive functor between abelian categories.
An additive functor between additive categories is left exact exactly when it preserves kernels (An additive functor is left exact exactly when it preserves kernels).
The opposite of an abelian category is abelian (The opposite of an abelian category is abelian).
Exact means additive, left exact, and right exact (Exact functor between abelian categories).
Proof
If is exact, then [L3] says it is left exact and right exact. The left-exact half and [L1] show that preserves kernels. Applying the same argument to and using [L2] shows that preserves cokernels as well.
Conversely, assume preserves kernels and cokernels. By [L1], kernel preservation makes left exact. Passing to opposites and using [L2], cokernel preservation makes right exact. Since is additive by hypothesis, [L3] shows that is exact.
Depends on
Used by
- The degree-zero projection is exact and cocontinuous but not a graded tensor functor Counterexample
- The Hom functor of a small projective generator is exact, coproduct-preserving, and faithful Lemma
- An exact functor commutes with homology Theorem
- An exact functor transports every diagram lemma Theorem
- Module reconstruction from a small projective generator with supplied copowers and cokernels Theorem
- Tensor product in a multitensor category is biexact Theorem
Dependency tree · two levels
12 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)