Koszul-Tate resolutions and decorated trees

TL;DR

This paper introduces an explicit construction of Koszul-Tate resolutions for commutative algebras using decorated trees, providing finite procedures for certain cases and linking to $A_$-algebra structures.

Contribution

It presents a novel arborescent approach to Koszul-Tate resolutions, simplifying computations and extending classical methods with explicit $A_$-algebra structures.

Findings

Finite-length resolutions require finitely many operations.

Constructs explicit $A_$-algebra structures.

Comparison with classical Tate algorithm highlights efficiency.

Abstract

Given a commutative algebra $\mathcal O$, a proper ideal $\mathcal I$, and a resolution of $\mathcal O/ \mathcal I$ by projective $\mathcal O $-modules, we construct an explicit Koszul-Tate resolution. We call it the arborescent Koszul-Tate resolution since it is indexed by decorated trees. When the $ \mathcal O$-module resolution has finite length, only finitely many operations are needed in our constructions -- this is to be compared with the classical Tate algorithm, which requires infinitely many such computations if $ \mathcal I$ is not a complete intersection. As a by-product of our construction, the initial projective $\mathcal O $-module resolution becomes equipped with an explicit $A_\infty$-algebra.