Koszul-Tate resolutions and decorated trees

Aliaksandr Hancharuk; Camille Laurent-Gengoux; Thomas Strobl

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arXiv · 2406.03955

Koszul-Tate resolutions and decorated trees

Aliaksandr Hancharuk·Camille Laurent-Gengoux·Thomas Strobl

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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.

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BibTeX RIS

Aliaksandr Hancharuk, Camille Laurent-Gengoux, Thomas Strobl. 2024-06-06. Koszul-Tate resolutions and decorated trees. https://arxiv.org/abs/2406.03955

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