Abelian groups form an abelian category

Depends on

• Abelian groups and $\mathbb Z$-modules have the same objects and morphisms
• Abelian category
• Left modules over a fixed ring and module homomorphisms form the large locally small category $R\text{-}\mathbf{Mod}$
• Module homomorphism and isomorphism, kernel, image and cokernel
• First isomorphism theorem for modules: $M/\ker f\cong\operatorname{im}f$
• For every ring R, the category R-Mod is complete and cocomplete

Used by

• Exact module tensor functors correspond to right-flat bimodules Corollary
• A coproduct-preserving left exact module functor is not tensor Counterexample
• A degreewise split cone sequence with no chain splitting Counterexample
• A general member of an abelian group need not come from an element Counterexample
• A non-split short exact sequence of abelian groups Counterexample
• A quasi-isomorphism that is not an isomorphism of complexes Counterexample
• A snake configuration whose kernel row is not short exact Counterexample
• An acyclic noncontractible complex from a nonsplit extension Counterexample
• Hom is not exact Counterexample
• The kernel row of a morphism of short exact sequences need not be short exact Counterexample
• The members of an object do not form a group Counterexample
• Two distinct chain maps inducing the same homology map Counterexample
• Two morphisms agreeing on every member need not be equal Counterexample
• A chain map computed on cycles, boundaries, and homology Example
• A contracting homotopy for the two-term identity complex Example
• A member chase verifying monicity Example
• A split exact complex and its contraction Example
• An ordinary element as the member from the integers Example
• Euler-Poincare for a finite complex Example
• Kernels, cokernels, images, and coimages in abelian groups are the familiar subgroup and quotient constructions Example
• Members of an abelian group correspond to its subgroups Example
• Relative homology of a map between stalk complexes Example
• Representations of the quiver 1 -> 2 in abelian groups form an abelian category Example
• Shifting a three-term complex with all signs Example
• The cone of multiplication by m on the integers Example
• The cone of zero and of the identity Example
• The connecting morphism computed for a short exact sequence of abelian groups Example
• The covering criterion checked in abelian groups Example
• The Hom complex of two two-term complexes Example
• The kernel row failure for multiplication by two computed Example
• The mapping cylinder of an inclusion of two-term complexes Example
• The multiplication-by-m complex computes a cyclic group Example
• The nine lemma verified on a diagram of cyclic groups Example
• The short five lemma chased with members Example
• The snake lemma applied to multiplication by an integer Example
• The three-cone calculation for a composite Example
• FALSE: a chain map is determined by its maps on homology False statement
• FALSE: any sequence of morphisms is a chain complex False statement
• FALSE: every quasi-isomorphism is an isomorphism of complexes False statement
• FALSE: the degreewise splitting of the cone sequence is a chain splitting False statement

…and 5 more results.

Dependency tree · two levels

29 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

• Gautam Tamme, Algebra II Lecture 9, §9.4 (standard reference, not scraped)
• The Stacks Project, Section 12.5 (standard reference, not scraped)