Projective modules and the lifting property

Depends on

• Module homomorphism and isomorphism, kernel, image and cokernel
• Injection, surjection, bijection

Used by

• Euler characteristic in a proper flat family is locally constant Corollary
• Exact finite tensor functors have projective right-module kernels Corollary
• Over a finite group algebra in defining characteristic, finite-dimensional projective and injective modules coincide Corollary
• Upper semicontinuity of fibre cohomology dimensions Corollary
• A projective generator need not be small Counterexample
• The left-to-right exact equivalence need not preserve the identity Counterexample
• 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
• Perfect complexes over a ring and its graded version Definition
• Small projective generators and progenerators Definition
• A contractible two-term bimodule complex induces the zero tensor functor Example
• A nonsplit simple has Hom-pairing diagonal two Example
• The diagonal bimodule k[x] is projective on both sides but not over its enveloping algebra Example
• The dual numbers have Cartan map multiplication by two Example
• The dual-numbers tensor functor is right exact but not left exact Example
• 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
• Choice-free Wall chain finiteness from an explicit finite domination Lemma
• Euler class of a bounded projective complex is derived invariant and triangle additive Lemma
• Perfect complexes form an essentially small triangulated subcategory Lemma
• PTFA finite projective base change and Euler characteristic Lemma
• Small projective modules are exactly finitely generated projective modules; the progenerator identification Lemma
• The dual-basis isomorphism for a finitely generated projective bimodule Lemma
• Three projective degrees suffice for twisted first-homology comparison Lemma
• Universal finite projective cohomology complex over any base Lemma
• Collapse with projective associated graded pieces splits the finite filtration noncanonically Proposition
• Finite-dimensional module categories satisfy the intrinsic finiteness conditions Proposition
• Positive Tor vanishes when the resolved variable is projective Proposition
• Restriction and induction along a subgroup preserve projective modules Proposition
• The projective Nakayama pairing and the symmetric-algebra specialization Proposition
• Cohomology and base change for proper flat coherent families Theorem
• Equivalent characterizations of projective modules Theorem
• Equivalent module-theoretic characterizations of semisimple rings Theorem
• Every finite-dimensional module has a projective cover, unique up to isomorphism over the target Theorem
• Exact adjoints induce adjoint operators on Grothendieck groups Theorem
• Free modules are projective, with the exact choice boundary Theorem
• Indecomposable projective classes form a basis of split K0 Theorem

…and 5 more results.

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)