Abelian category

Depends on

• Additive category
• The canonical morphism from the coimage to the image exists and is unique

Used by

• An abelian category that is a preorder is trivial Corollary
• Composition of Deligne kernels is balanced tensor product Corollary
• Finite one-sided exactness is equivalent to the existence of the corresponding adjoint Corollary
• In an abelian category, monic means zero kernel and epic means zero cokernel Corollary
• The center is Morita invariant, via natural endomorphisms of the identity Corollary
• The kernel-cokernel sequence of a composite is a snake Corollary
• An exact functor need not be faithful Counterexample
• Filtered vector spaces can be additive with kernels and cokernels without being abelian Counterexample
• Topological abelian groups are additive but not abelian Counterexample
• Torsion-free abelian groups do not form an abelian category Counterexample
• Abelian subcategory and exact embedding Definition
• Cartan-Eilenberg injective resolution of a bounded-below complex Definition
• Chain complex in an abelian category Definition
• Cochain complex in an abelian category Definition
• Cohomological functor on a triangulated category Definition
• Exact couple Definition
• Graded Grothendieck groups, shift action, and Cartan map Definition
• Grothendieck group of an essentially small abelian category Definition
• Homological functor on a triangulated category Definition
• Homological spectral sequence Definition
• Increasing and decreasing filtration of an object Definition
• Injective object Definition
• Left and right Nakayama functors by finite kernel calculus Definition
• Locally finite k-linear abelian categories Definition
• Projective object Definition
• Semisimple objects and semisimple abelian categories Definition
• Small projective generators and progenerators Definition
• Superfluous subobjects and projective covers in an abelian category Definition
• Supplied injective resolution data Definition
• Supplied projective resolution data Definition
• The arrow category of an abelian category Definition
• The Deligne product of finite linear categories Definition
• A nonsplit simple has Hom-pairing diagonal two Example
• The Deligne product of finite vector spaces is finite vector spaces Example
• FALSE: every abelian category has a generator False statement
• Abelian-group model for spectral-sequence computations Lemma
• Bilinear right exact functors are determined by their value on the regular modules Lemma
• Colimits of a graded additive functor equal right exactness plus coproduct preservation Lemma
• Degreewise direct sums and homogeneous free covers in graded modules Lemma
• Finite biproducts of injective objects are injective Lemma

…and 57 more results.

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

• Alexandre Grothendieck, Some aspects of homological algebra, §1.4 (standard reference, not scraped)
• The Stacks Project, Section 12.5, Definition 12.5.1 (standard reference, not scraped)
• Gautam Tamme, Algebra II Lecture 9, §9.4 (standard reference, not scraped)