The Koszul–Tate type resolution for Gerstenhaber–Batalin–Vilkovisky algebras

Abstract

Tate provided an explicit way to kill a nontrivial homology class of a commutative differential graded algebra over a commutative noetherian ring R in Tate (Ill J Math 1:14–27, 1957). The goal of this article is to generalize his result to the case of GBV (Gerstenhaber–Batalin–Vilkovisky) algebras and, more generally, the descendant \(L_\infty \)-algebras. More precisely, for a given GBV algebra \((\mathcal {A}=\oplus _{m\ge 0}\mathcal {A}_m, \delta , \ell _2^\delta )\), we provide another explicit GBV algebra \((\widetilde{\mathcal {A}}=\oplus _{m\ge 0}\widetilde{\mathcal {A}}_m, \widetilde{\delta }, \ell _2^{\widetilde{\delta }})\) such that its total homology is the same as the degree zero part of the homology \(H_0(\mathcal {A}, \delta )\) of the given GBV algebra \((\mathcal {A}, \delta , \ell _2^\delta )\).