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

中文翻译：

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Gerstenhaber-Batalin-Vilkovisky代数的Koszul-Tate类型解析

泰特（Tate）提供了一种明确的方法，可以杀死泰特（Tate）中可交换Noether环R上的可交换差分梯度代数的非平凡同源性类（《国际数学杂志》 1957年1月14日至27日）。本文的目的是将其结果推广到GBV（Gerstenhaber–Batalin–Vilkovisky）代数的情况，以及更普遍的后代\（L_ \ infty \）-代数的情况。更精确地，对于给定的GBV代数\（（\ mathcal {A} = \ oplus _ {m \ ge 0} \ mathcal {A} _m，\ delta，\ ell _2 ^ \ delta} \），我们提供了另一个显式GBV代数\（（\\\\\ {{\ mathcal {A}} = \\ oplus _ {m \ ge 0} \\\\\\\\\ {{\ a}} _ m，\\\\\\\\ {{ \ delta}}）\）使得其总同源性与给定GBV代数\ {（\ mathcal {A}，\ delta，\ ell的同源性\（H_0（\ mathcal {A}，\ delta）\）的度数零部分相同_2 ^ \ delta）\）。