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

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Notes

  1. 1.

    Characteristic zero assumption is needed, since we will deal with \(L_\infty \)-algebras and have to divide by n!.

  2. 2.

    In this article, we use the homology version of the shifted \(L_\infty \)-algebra.

  3. 3.

    For example, we have

    $$\begin{aligned} \delta _0(\theta ^2)= & {} \ell ^1(\theta ,\theta )+r \theta +(-1)^{|\theta |} \theta r, \\ \delta _0(\theta ^3)= & {} \ell ^1(\theta ^2, \theta )+ \delta _0(\theta ^2) \theta + \theta ^2 r =(-1)^{|\theta |}\theta \ell ^1(\theta ,\theta ) + (-1)^{|\theta |^2}\ell ^1(\theta ,\theta ) \theta +\delta _0(\theta ^2) \theta + \theta ^2 r, \end{aligned}$$

    and \(\delta _0:\mathcal {A}_0 \rightarrow \mathcal {A}_1\) is defined inductively by

    $$\begin{aligned} \delta _0(\theta ^m v)=\ell ^1(\theta ^m, v)+\delta _0(\theta ^m) v +(-1)^{m|\theta |}\theta ^m \delta _0(v), \quad v\in \mathcal {B}_{(0)}, m \ge 0, \end{aligned}$$

    where \(\ell (\theta ^m,v) \in \mathcal {B}_{(1)}\) is defined by using (3.3).

  4. 4.

    If we define \([x \bullet y]:=(-1)^{|x|} \ell _2^\delta (x,y)\), then \((\mathcal {A}, Q, \delta , [\bullet ])\) is a dGBV algebra in the sense of [9].

  5. 5.

    If M were 1, then \((\mathcal {A},\delta )\) is a CDGA (commutative differential graded \(k\)-algebra) and we may not need \(U_1, U_2, \ldots \). The set \(U_0\) is enough to kill the cohomology class r.

  6. 6.

    For example, the \(m=3\) case means that

    $$\begin{aligned} K_{k}\ell _3^k(x_1,x_2,x_3)= & {} \ell _2^{k+1}(\ell _2^k(x_1,x_2), x_3) + (-1)^{|x_1|} \ell _2^{k+1}(x_1, \ell _2^k(x_2, x_3)) \\&+(-1)^{(|x_1|+1)|x_2|}\ell _2^{k+1}(x_2, \ell _2^k(x_1,x_3))+\ell _3^{k+1}(K_{k-1}x_1, x_2, x_3)\\&+ (-1)^{|x_1|} \ell _3^{k+1}(x_1, K_{k-1}x_2, x_3) +(-1)^{|x_1|+|x_2|}\ell _3^{k+1}(x_1,x_2,K_{k-1}x_3). \end{aligned}$$

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Acknowledgements

This problem of finding a resolution for GBV algebras is motivated by the work of Jae-Suk Park on algebraic formalisms for quantum field theory. The authors would like to thank him for his encouragement and useful comments. The authors appreciate Gabriel Drummond Cole for providing both useful mathematical comments and numerous english corrections. The authors also thank the referee for his valuable comments and suggestions.

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Correspondence to Donggeon Yhee.

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J. Park was partially supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (2013023108 and NRF-2018R1A4A1023590) and was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013053914). D. Yhee was supported by Basic Science Research Programs through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013053914), funded by Basic Science Research Programs through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2014R1A6A3A03053691), and funded by Korean Government (MSIT-2017R1A5A1015626).

Communicated by Jim Stasheff.

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Park, J., Yhee, D. The Koszul–Tate type resolution for Gerstenhaber–Batalin–Vilkovisky algebras. J. Homotopy Relat. Struct. 14, 455–475 (2019). https://doi.org/10.1007/s40062-018-0218-2

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Keywords

  • BV algebra
  • Descendant functor
  • Koszul–Tate type resolution