Statements (51)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:category |
Abelian category
Category of modules |
| gptkbp:characterizedBy |
Direct summand of a free module
Every short exact sequence 0 → A → B → P → 0 splits |
| gptkbp:citation |
gptkb:Algebra_by_Serge_Lang
gptkb:Homological_Algebra_by_Cartan_and_Eilenberg gptkb:Introduction_to_Commutative_Algebra_by_Atiyah_and_Macdonald |
| gptkbp:class |
Over a local ring, projective modules are free
Over a semisimple ring, all modules are projective Over a Dedekind domain, finitely generated projective modules are classified by their rank and class group Over a principal ideal domain, projective modules are free |
| gptkbp:contrastsWith |
Flat modules
Injective modules |
| gptkbp:definedIn |
gptkb:King
|
| gptkbp:defines |
A module P is projective if for every surjective module homomorphism f: M → N and every module homomorphism g: P → N, there exists a module homomorphism h: P → M such that f ∘ h = g.
|
| gptkbp:example |
gptkb:Finitely_generated_projective_module
gptkb:Free_module gptkb:Stably_free_module Module over a semisimple ring |
| gptkbp:field |
gptkb:algebra
|
| gptkbp:generalizes |
gptkb:Vector_bundles_(in_algebraic_geometry)
Free modules |
| gptkbp:hasSubfield |
Module theory
|
| gptkbp:homological_property |
gptkb:Tor_functor
Ext functor vanishes for projective modules Projective dimension |
| https://www.w3.org/2000/01/rdf-schema#label |
Projective modules
|
| gptkbp:important_theorem |
gptkb:Serre's_problem
gptkb:Kaplansky's_theorem gptkb:Bass's_cancellation_theorem gptkb:Quillen–Suslin_theorem |
| gptkbp:notation |
P (often used for projective modules)
|
| gptkbp:property |
Direct summand of a projective module is projective
Direct sum of projective modules is projective Every free module is projective Every projective module is flat |
| gptkbp:relatedConcept |
Homological algebra
Projective resolution Projective cover |
| gptkbp:studiedBy |
gptkb:David_Hilbert
gptkb:Emmy_Noether gptkb:Irving_Kaplansky gptkb:Jean-Pierre_Serre gptkb:Hyman_Bass |
| gptkbp:used_in |
gptkb:Algebraic_topology
gptkb:Algebraic_geometry gptkb:Representation_theory Homological algebra |
| gptkbp:bfsParent |
gptkb:K-theory
|
| gptkbp:bfsLayer |
5
|