Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06
How 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.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Exterior Algebra Of A Finite Free Module
Definition
Let be a commutative unital ring and let be a finite free -module with ordered basis . Define , graded by tensor degree; write for .
Depends on
Used by
- Koszul Betti Numbers Over A Local Ring Definition
- Koszul Complex Of A Sequence With Coefficients Definition
- Integral cohomology ring of a torus Example
- The exterior square of the permutation FI-module has character polynomial C(X₁, 2)-X₂ Example
- The zero quadratic form gives the exterior algebra Example
- Exterior Algebra Basis Monomials Lemma
- Exterior Multiplication Koszul Sign Rule Lemma
- K-theory of projective space and of projective bundles Lemma
- Tensor products, exterior powers and Hom spaces of finite-dimensional rational representations are rational Lemma
- The top exterior power detects stabilizers of a subspace Lemma
- Wedge and contraction operators satisfy the Clifford relations Lemma
- PBW basis and exterior associated graded for a Clifford algebra Theorem
Dependency tree · two levels
3 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
- The Stacks Project, Koszul complexes and regular sequences (standard reference, not scraped)