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.
- 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 Basis Monomials
Statement
For every , the wedges with form an -basis of ; in particular it is zero for .
Facts & Assumptions
Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Exterior Algebra Of A Finite Free Module.
Proof
Sort wedges using anticommutation and delete repetitions using , so the increasing wedges span.
The alternating determinant map to the free module on -subsets kills the exterior relations and sends these wedges to distinct basis elements, proving independence.
Depends on
Used by
- Complete Intersection Betti Numbers Binomial Corollary
- Diagonal Hochschild homology of a polynomial ring Corollary
- 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 Multiplication Koszul Sign Rule Lemma
- Koszul Differential Coordinate Formula Lemma
- Koszul sheaf Ext of a smooth regular immersion is concentrated in codimension Lemma
- Tensor products, exterior powers and Hom spaces of finite-dimensional rational representations are rational Lemma
- PBW basis and exterior associated graded for a Clifford algebra Theorem
- The diagonal Koszul complex is a finite free resolution of R Theorem
Dependency tree · two levels
2 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)