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.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Exact sequences and short exact sequences of modules
Definition
A sequence of left -modules and homomorphisms is exact at if (Module homomorphism and isomorphism, kernel, image and cokernel). It is exact if it is exact at every displayed module at which two arrows meet.
A short exact sequence is an exact sequence Thus is injective, is surjective, and (Kernels and images of module homomorphisms are submodules, and injectivity is equivalent to trivial kernel).
Depends on
Used by
- For flat M, one has IM∩ JM=(I∩ J)M Corollary
- Finitely presented modules and finitely presented algebras Definition
- Flat and faithfully flat modules and ring homomorphisms Definition
- Left and right flat modules over an arbitrary ring Definition
- Split short exact sequences, sections, and retractions Definition
- Schanuel's lemma for two presentations of a module Example
- Associated primes of a submodule lie in those of the ambient module Lemma
- Associated primes of the middle term lie in those of the ends Lemma
- Finite-group invariants are exact when the group order is invertible Lemma
- The endpoints of a short exact sequence encode injectivity and surjectivity Lemma
- The injective and surjective Four Lemmas Lemma
- A short exact sequence with flat quotient remains short exact after tensoring Theorem
- Assuming the Axiom of Choice, a sequence of modules is exact exactly when all prime localisations are exact Theorem
- Complexification preserves kernels, images, finite rank, nullity, and short exact sequences Theorem
- Covariant and contravariant Hom are left exact Theorem
- Localisation of modules is exact Theorem
- Noetherian and Artinian conditions are each exact in short exact sequences Theorem
- Over a Noetherian ring a module is Noetherian exactly when it is finitely generated, exactly when it is finitely presented Theorem
- Support in a short exact sequence is the union of the outer supports Theorem
- Tensoring is right exact Theorem
- The Snake Lemma for modules Theorem
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
- 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)