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.
Functoriality Base Change And Generator Change For Koszul Complexes
Statement
Koszul complexes commute with localization and flat base change, and an invertible change of finite generators gives a signed chain isomorphism; the corresponding homology conclusions hold.
Facts & Assumptions
Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Koszul Complex Localises Termwise, Koszul Complex Flat Base Change, Koszul Homology Flat Base Change, Koszul Generator Matrix Chain Map, Koszul Complex Invariant Under Invertible Generator Change.
Proof
Termwise localization and flat-base-change maps are chain isomorphisms and exactness supplies their homology conclusions.
An invertible generator matrix has an inverse exterior chain map; all maps commute with coefficient-module maps by construction.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
10 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)