Modules over a ring form an abelian category

Depends on

• 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 finite tensor functors have projective right-module kernels Corollary
• Exact module tensor functors correspond to right-flat bimodules Corollary
• Finite one-sided exactness is equivalent to the existence of the corresponding adjoint Corollary
• The center is Morita invariant, via natural endomorphisms of the identity Corollary
• A category with enough injectives but not enough projectives Counterexample
• A right exact module functor without coproduct preservation is not tensor Counterexample
• A module homomorphism factors as quotient by its kernel followed by inclusion of its image Example
• A nonsplit simple has Hom-pairing diagonal two Example
• A split exact complex contracts degree by degree Example
• A subcomplex and its quotient complex Example
• A two-term complex has kernel and cokernel homology Example
• An exact short sequence as an acyclic three-term complex Example
• Localization of modules gives an exact functor between module categories Example
• The dual numbers have Cartan map multiplication by two Example
• The kernel-cokernel sequence of a composite of module maps Example
• Vector spaces over a field form an abelian category Example
• FALSE: a short exact sequence splits whenever its middle object is isomorphic to the biproduct of the outer two False statement
• FALSE: every abelian category is equivalent to a module category False statement
• An additive module functor is cocontinuous exactly when it is right exact and preserves coproducts Lemma
• Bilinear right exact functors are determined by their value on the regular modules Lemma
• Finite-support families of finite-dimensional vector spaces are locally finite but not finite Lemma
• Natural transformations of additive cocontinuous module functors are determined at the regular module Lemma
• Small projective modules are exactly finitely generated projective modules; the progenerator identification Lemma
• The copower presentation construction is left adjoint to the generator Hom functor Lemma
• The opposite Deligne product is the category of finite bimodules Lemma
• Degree-zero Hochschild homology is bimodule coinvariants Proposition
• Elementwise formula for the connecting map in module categories Proposition
• Finite-dimensional module categories satisfy the intrinsic finiteness conditions Proposition
• Category O is abelian and extension closed among weight modules Theorem
• Exact adjoints induce adjoint operators on Grothendieck groups Theorem
• Finite Deligne products exist via tensor-product algebras Theorem
• Finite left exact functors are Hom functors with dual bimodule kernels Theorem
• Functoriality and coefficient long exact sequences for Hochschild homology Theorem
• Projective and simple classes are dual bases under splitting Theorem
• Projective Hom pairing descends and is graded sesquilinear Theorem
• Sheaves of abelian groups, and likewise sheaves of modules on a ringed space, form abelian categories Theorem
• The diagonal Koszul complex is a finite free resolution of R Theorem
• The two-sided bar complex is a projective Aᵉ-resolution Theorem

Dependency tree · two levels

19 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)