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.
Koszul Generator Matrix Chain Map
Statement
If , the transpose matrix defines an exterior map inducing a chain map .
Facts & Assumptions
Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Koszul Complex Of A Sequence With Coefficients, Koszul Differential Coordinate Formula, Chain map.
Proof
Let send the th basis vector to . The assumed identity makes the degree-one squares commute.
Extending by exterior powers respects wedge products, and the derivation rule then makes the resulting map commute with the differentials in every degree.
Depends on
Used by
- Koszul Complex Invariant Under Invertible Generator Change Corollary
- Generator Change Koszul Isomorphism Example
- Koszul sheaf Ext of a smooth regular immersion is concentrated in codimension Lemma
- Rational-point Koszul residue normalization for a smooth projective embedding Lemma
- Functoriality Base Change And Generator Change For Koszul Complexes Theorem
Dependency tree · two levels
7 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)