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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 ∞ -algebras. More precisely, for a given GBV algebra (A = ⊕ m ≥ 0 A m , δ , ℓ 2 δ) , we provide another explicit GBV algebra (A ~ = ⊕ m ≥ 0 A ~ m , δ ~ , ℓ 2 δ ~) such that its total homology is the same as the degree zero part of the homology H 0 (A , δ) of the given GBV algebra (A , δ , ℓ 2 δ) . [ABSTRACT FROM AUTHOR]