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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Projective and injective modules coincide over every ring
Statement
Projective and injective modules coincide over every ring.
Facts & Assumptions
Given: The finite group algebra setting of the preceding corollary.
Projective and injective coincide here only for finite-dimensional modules over the symmetric group algebra (Over a finite group algebra in defining characteristic, finite-dimensional projective and injective modules coincide).
Refutation
The corollary [L1] is explicitly a special finite-dimensional group-algebra statement.
Removing those hypotheses widens the claim far beyond what was proved. Standard counterexamples over rings such as show that injective and projective need not agree. Hence the universal statement is false.
Therefore projective and injective modules do not coincide over every ring.
Depends on
Used by
Nothing in the library uses this result yet.
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
- Peter Webb, A Course in Finite Group Representation Theory (23 Feb 2016 draft) (standard reference, not scraped)