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Koszul Complex Resolves A Regular Quotient
Statement
If is finite free and is -regular, then is a finite free resolution of .
Facts & Assumptions
Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Regular Sequences Give Acyclic Koszul Complexes, Basic Koszul Homology, with the product basis, and .
Proof
Regularity gives zero positive homology and the degree-zero calculation gives .
Each term is finite free because both factors are finite free, so this is a finite free resolution.
Depends on
Used by
- Koszul Complex Polynomial Variables Example
- Koszul Resolution Complete Intersection Example
- Finite-length duality over a regular local base Lemma
- Koszul Resolution Minimality Maximal Ideal Sequence Lemma
- Koszul sheaf Ext of a smooth regular immersion is concentrated in codimension Lemma
- regular local residue field koszul resolution Lemma
- The coinvariant algebra is a complete intersection with one-dimensional socle Lemma
- The first-layer relations of a closed MOY resolution form a regular sequence Lemma
- The local affine-Hecke socle argument for residue translations Lemma
- The nilpotent central fiber is reduced and integral Lemma
- Chevalley–Eilenberg cohomology computes Ext of the trivial module Proposition
Dependency tree · two levels
13 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)