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 $$(\mathcal {A}, \delta , \ell _2^\delta )$$ .