Alphabeta MathHow statement and proof provenance work
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The abelian group and maps induced by pre- and postcomposition
Definition
For left -modules , the set of module homomorphisms is an abelian group under pointwise addition, with zero the zero homomorphism and inverse (Module homomorphism and isomorphism, kernel, image and cokernel, Group and abelian group).
For a homomorphism and any module , postcomposition and precomposition give homomorphisms Composition is associative, so and ; identity maps induce identity maps.
Depends on
Used by
- Over a Noetherian ring the homomorphism module between two finitely generated modules is finitely generated Corollary
- A covariant hom functor on an additive category need not preserve cokernels Counterexample
- Hom_ℤ(-,ℤ) need not preserve surjections on the right Counterexample
- The endomorphism ring End_R(M) under addition and composition Definition
- The R-module Hom_R(M,N) over a commutative ring Definition
- Hom_ℤ(ℤ/m,ℤ/n)≅ℤ/gcd(m,n) for n≥1 Example
- Coinduction sends injective abelian groups to injective modules Lemma
- Over a commutative ring the homomorphism group Hom_R(M,N) is an R-module Lemma
- Module endomorphisms form a ring under pointwise addition and composition Proposition
- Covariant and contravariant Hom are left exact Theorem
- Endomorphisms of a finite direct sum are matrices of Hom-groups Theorem
- Induction is left adjoint to restriction for finite-group modules over a commutative ring Theorem
- The character dual of a flat module is injective Theorem
Dependency tree · two levels
10 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
- A. Kleshchev, Lectures on Abstract Algebra for Graduate Students, sections 3.6, 3.14, and 3.15 (standard reference, not scraped)
- The Stacks Project, Algebra (standard reference, not scraped)
- P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)