Alphabeta MathHow statement and proof provenance work
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Projective modules and the lifting property
Definition
A left -module is projective if it has the lifting property for epimorphisms: whenever is a surjective module homomorphism and is a module homomorphism, there exists a module homomorphism such that (Module homomorphism and isomorphism, kernel, image and cokernel, Injection, surjection, bijection).
The lift need not be unique. Projectivity asks for a lift in every such square.
Depends on
Used by
- Over a finite group algebra in defining characteristic, finite-dimensional projective and injective modules coincide Corollary
- A module is relatively H-projective when it is a direct summand of one induced from H Definition
- An essential epimorphism is a surjection with superfluous kernel, and a projective cover is a projective source with such a map Definition
- Integral cohomological dimension Definition
- The idempotent completion of the matrix category gives the finitely generated projective modules Example
- Flat modules need not have projective dimension zero False statement
- fs-every-complex-of-projectives-is-homotopically-projective.md False statement
- Tor one of R modulo I and M is not always the I-torsion submodule of M False statement
- Collapse with projective associated graded pieces splits the finite filtration noncanonically Proposition
- Positive Tor vanishes when the resolved variable is projective Proposition
- Restriction and induction along a subgroup preserve projective modules Proposition
- Equivalent characterizations of projective modules Theorem
- Equivalent module-theoretic characterizations of semisimple rings Theorem
- Free modules are projective, with the exact choice boundary Theorem
- The integers have global dimension one Theorem
- Tor one of a cyclic abelian group detects n-torsion Theorem
Dependency tree · two levels
6 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)