Koszul-Tate resolutions and decorated trees
Aliaksandr Hancharuk, Camille Laurent-Gengoux, Thomas Strobl

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 , a proper ideal , and a resolution of by projective -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 -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 is not a complete intersection. As a by-product of our construction, the initial projective -module resolution becomes equipped with an explicit -algebra.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology · Commutative Algebra and Its Applications
