Koszul-Tate resolutions and decorated trees

Given a commutative algebra $\mathcal{O}$, a proper ideal $\mathcal{I}$, and a resolution of $\mathcal{O}\mathrm{/}\mathcal{I}$ 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 $A_{\mathrm{\infty }}$-algebra.