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A module is relatively H-projective when it is a direct summand of one induced from H
Definition
Let be a finite group, let , let be a field, and let be a -module. The module is relatively -projective if there exists a -module such that is a direct summand of .
Assuming the Axiom of Choice (The Axiom of Choice), a basis of exists by Every vector space has a basis. When it gives , which may have infinite rank. Thus a relatively -projective module is projective. Conversely, the free-summand characterization of Equivalent characterizations of projective modules realizes every projective -module as a summand of . Hence, under Choice, relative -projectivity recovers ordinary projectivity in the sense of Projective modules and the lifting property.
Depends on
Used by
- A vertex is a minimal p-subgroup for relative projectivity, and a source is an indecomposable inducing summand there Definition
- Green exceptional intersection families Definition
- Green correspondence for a trivial intersection subgroup Example
- Green correspondence identity boundaries Example
- Relative projectivity mackey intersections for finite modules Lemma
- Tensoring preserves relative projectivity for finite-group modules Lemma
- Transitive p-group permutation modules have point-stabilizer vertices Lemma
- Higman's criterion characterizes relative projectivity through the relative trace idempotent test Theorem
Dependency tree · two levels
19 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
- Peter Webb, A Course in Finite Group Representation Theory (23 Feb 2016 draft) (standard reference, not scraped)
- J. MacQuarrie, Modular Representations of Profinite Groups (standard reference, not scraped)