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Koszul Complex Invariant Under Invertible Generator Change
Statement
If and the matrix is invertible, the induced generator-matrix chain map is an isomorphism .
Facts & Assumptions
Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Koszul Generator Matrix Chain Map.
Proof
The matrix chain map is defined by exterior powers of the generator map.
The inverse matrix induces its inverse in every exterior degree, so the complexes are isomorphic.
Depends on
Used by
- Generator Change Koszul Isomorphism Example
- A closed MOY resolution's Koszul complex computes Hochschild homology of its Soergel bimodule Lemma
- Koszul sheaf Ext of a smooth regular immersion is concentrated in codimension Lemma
- Regular Sequence Permutation Adjacent Swap Lemma
- Functoriality Base Change And Generator Change For Koszul Complexes Theorem
Dependency tree · two levels
4 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)