THE KOSZUL-TATE TYPE RESOLUTION FOR GERSTENHABER-BATALIN-VILKOVISKY ALGEBRAS

THE KOSZUL-TATE TYPE RESOLUTION FOR GERSTENHABER-BATALIN-VILKOVISKY ALGEBRAS JEEHOON PARK AND DONGGEON YHEE Abstract. J. Tate provided an explicit way to kill a nontrivial homology class of a commutative differential graded algebra over a commutative noetherian ring R in [9]. 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 δ 2 ), we provide another explicit GBV algebra (Ã = m 0Ãm, δ, l δ 2 ) 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 δ 2 ). Contents 1. Introduction 2 1.1. Tate s result and the main theorem 2 1.2. Motivation and an idea of a proof 4 2. Descendant functor 6 2.1. Why does Tate s method fail in the generalized situation? 6 2.2. Descendant L -algebra 6 3. The main results 8 3.1. Outline of the proof 8 3.2. The case of GBV algebras 8 3.3. The general case 12 4. Applications 15 References 16 J. P. 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 was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2013053914). D. Y. was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2013053914) was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2014R1A6A3A03053691). 1

2 JEEHOON PARK AND DONGGEON YHEE 1. Introduction 1.1. Tate s result and the main theorem. Throughout this paper, R will be a commutative noetherian k-algebra where k is a field of characteristic zero. 1 Let M be an ideal of R. In [9], J. Tate proved that there exists an acyclic commutative differential graded algebra (so called CDGA) X over R, whose underlying graded algebra is free, such that its zeroth homology H 0 (X) = R/M by applying the explicit technique of killing nontrivial homology classes to the Koszul chain complex of R/M. In other words, there exists a free resolution of a given CDGA (R/M,0) (the graded algebra R/M equipped with the zero differential) in the category of commutative differential graded algebras over R. The goal of this paper is to generalize his technique of killing nontrivial homology classes, to the case when M is not an ideal of R but satisfies some more general condition; in particular, we generalize this to the case of GBV (Gerstenhaber-Batalin-Vilkovisky) algebras which plays an interesting role in mirror symmetry (e.g. see p. 212, [1]). The GBV algebra is a triple (A, δ, l 2 ) where a graded algebra A = m 0 A m equipped with a differential δ : A m A m 1 and a graded Lie bracket (1.1) l 2 (x, y) = l δ 2(x, y) := δ(x y) δx y ( 1) x x δy. See section 4.1, [3] for more details on GBV algebras. In the case of GBV algebras, our result can be formulated as the following theorem: Theorem 1.1. Let (A = m 0 A m, δ, l 2 ) be a GBV algebra. Then there is an explicit GBV algebra (Ã = m 0Ãm, δ, l 2 ) such that its total homology is the same as the degree zero part H 0 (A, δ) = A 0 / (δ(a 1 ) A 0 ) of the given GBV algebra (A, δ, l 2 ). In Theorem 1.1, the abstract existence of (Ã, δ, l 2 ) follows from the general theory of Koszul duality for the GBV operads (see [2] and [6], for instance). The homotopy transfer theorem equips H (A, δ) with a homotopy GBV algebra structure and the general Koszul duality will provide an existence of a homotopy GBV algebra whose total homology is isomorphic to H 0 (A, δ). The general theory allows us to rectify this homotopy GBV algebra to a quasi-isomorphic GBV algebra. But the point of Theorem 1.1 is to provide an explicit quasi-free resolution of a GBV algebra. In order to explain the condition for M in the more general case, we reformulate Tate s result using a Lie algebra representation. Since R is noetherian, the ideal M is generated by f 1,, f m R with m 1. Consider the abelian Lie algebra g M over k with dimension m. Denote its k-basis by e 1,, e m. Then we define a Lie algebra representation (1.2) ρ M : g M End k (R), e i m fi where m fi is the multiplication by f i for each i = 1,, m. Let us recall the Chevalley-Eilenberg complex which computes the Lie algebra homology: we consider a Z-graded vector space E(g M ; ρ M ) = p 0 E p (g M ; ρ M ), where E p (g M ; ρ M ) := R k Λ p g M, and equip it with the differential δ ρ defined by n δ ρ (r (x 1 x n )) = ( 1) i+1 ρ M (x i )(r) (x 1 ˆx i, x n ) (1.3) i=1 + ( 1) i+j r ([x i, x j ] x 1 ˆx i ˆx j x n ). 1 i<j n 1 Characteristic zero assumption is needed, since we will deal with L -algebras and have to divide by n!.

THE KOSZUL-TATE TYPE RESOLUTION FOR GBV ALGEBRAS 3 Note that [x i, x j ] = 0, since g M is abelian. The 0 th Lie algebra homology H 0 (g M, R) is isomorphic to the residue class ring R/M. We put a Z-graded k-algebra structure by (1.4) (r 1 λ 1 ) (r 2 λ 2 ) := (r 1 r 2 λ 1 λ 2 ) R k Λ p+q g M for r 1 λ 1 R k Λ p g M and r 2 λ 2 R k Λ q g M. Since M is assumed to be an ideal of R, (E(g M ; ρ M ), δ ρ ) becomes a commutative differential graded algebra. Then Tate s theorem is tantamount to saying that one can kill all the homologies of the CDGA (E(g M ; ρ M ), δ ρ ) except for the zeroth homology and construct a free CDGA over R which is quasi-isomorphic to (H 0 (g M, R), 0). Now we explain how to generalize the Tate theorem by suitably weakening the condition that M is an ideal of R. We need two notions for our generalization; the order of a differential operator and the category C of chain complexes over k equipped with a super-commutative product. Let A be a unital Z-graded k-algebra and π End k (A). We call a non-zero operator π a differential operator of order n, if n is the smallest non-negative integer such that l π n 0 and l π n+1 = 0, where l π n(x 1, x 2,, x n ) = [[ [[π, L x1 ], L x2 ], ], L xn ](1 A ), l π 0 (x 1 ) = π(x 1 ) for x 1,, x n A. Here L x : A A is left multiplication by x, the commutator [L, L ] := L L ( 1) L L L L End k (A), and 1 A is the identity element of A. Then the condition that M is an ideal of R corresponds to the condition that ρ M (e i ) is a differential operator of order 0 for each i. Let g be a finite dimensional Lie algebra over k and let ρ be a Lie algebra representation of g on a commutative noetherian k-algebra R. Assume that there exist an integer m such that ρ(g) is a differential operator of order m for each g g. If m is strictly bigger than 0, then Im (δ ρ ) R is not an ideal of R and (E(g M ; ρ M ), δ ρ ) is not a CDGA under the multiplication (1.4) because δ ρ is not a derivation. If the order of ρ(g) is 0 then (E(g M ; ρ M ), δ ρ ) is a CDGA and Im (δ ρ ) R corresponds to M in our terminology. We need to enlarge the category of (unital) CDGAs to the category C = C k of chain complexes over a field k equipped with a super-commutative product. An object of C is a unital Z-graded associative and super-commutative k-algebra A with differential δ, denoted (A, δ) such that δ(1 A ) = 0. A morphism in C k is a unit preserving chain map of degree zero (note that a morphism is not necessarily a ring homomorphism). If (A, δ) is an object of C, we let A m denote the k-submodule of A consisting of homogeneous elements of degree m, and write the Z-graded decomposition as A = m Z A m. We use a to denote the degree of a homogeneous element a A. The degree is additive ab = a + b, which implies that the product is a degree zero k-bilinear map. This category C can be thought of as a generalization of the category of CDGAs. Recall that (A, δ) is called a CDGA if and only if (A, δ) Ob(C) and δ is a derivation of the product, δ(ab) = δ(a)b + ( 1) a aδ(b), a, b A. There is a canonical faithful functor from the category of CDGAs to C. Note that the Chevalley- Eilenberg chain complex (E p (g; ρ), δ ρ ) is an object of C under the multiplication (1.4). Now we state the main theorem. Theorem 1.2. Let g be a finite dimensional Lie algebra over k and ρ be a Lie algebra representation of g on a commutative noetherian k-algebra R. Assume that ρ(g) is a differential operator of a fixed finite order µ for each g g and the total homology H(g, R) is finite dimensional over k. Then we explicitly construct an object (B ρ, δ ρ ) of the category C such that B ρ = m 0 Bρ m, the differential K ρ has order µ + 1, and the total homology of (B ρ, δ ρ ) is naturally isomorphic to the vector space H 0 (g, R) := R/(Im δ ρ R)

4 JEEHOON PARK AND DONGGEON YHEE In other words, we have an exact sequence of R-modules δ ρ B ρ 2 δ ρ B ρ 1 such that δ ρ End k (B) has order µ + 1. δ ρ B ρ 0 = R R/Im δ ρ R Remark 1.3. (a) If ρ(g) is a differential operator of order 0 for each g g, i.e. µ = 0, then Theorem 1.2 is equivalent to the theorem of J. Tate. (b) If µ = 1, then Theorem 1.2 reduces to Theorem 1.1. We can reformulate Theorem 1.2 in ring theoretic language in the following way; Theorem 1.4. Let M be a k-subspace of R such that M = n i=1 Im (o i) for certain elements o 1,, o n End k (R). Assume that there exist elements x i1,, x iµ R for each i = 1,, n such that the R-submodule generated by the image of [ [[o i, L xi1 ], L xi2 ], ], L xiµ ] End k (R) for every i = 1,, n becomes an ideal of R. Then there exists a resolution (B, K) of R/M which is an object of the category C and the order (as a differential operator) of K is less than or equal to µ + 1. If µ = 0, i.e. M is an ideal of R, then Theorem 1.4 is again equivalent to the theorem of J. Tate. 1.2. Motivation and an idea of a proof. In [8], the authors found a GBV algebra for smooth projective hypersurfaces. This result was generalized to smooth projective complete intersections in [5]. They computed the total homology of the GBV algebra BV X = (A X = m 0 A X,m, δ X, l 2 ) of a smooth projective complete intersection X and proved that the zeroth part of its homology is naturally isomorphic to the primitive middle dimensional cohomology H prim (X, C) of X. Though the homology groups in other degrees are closely related to the lower degree singular cohomology of X, they do not match precisely. In the context of homological mirror symmetry for X, it would be nice to construct a formal Frobenius manifold structure on H prim (X, C) in an explicit manner. For such a purpose, it turns out that a key technical step is to construct another GBV algebra BV X whose total homology is isomorphic to H prim (X, C), which is our main motivation for the current article. We can apply Theorem 1.1 to BV X to obtain BV X. We also explain a motivation for a more general result (Theorem 1.2), which is why we consider the above generalization of the Koszul-Tate resolution in the context of the category C. This category C seems odd at first glance (since the two parts of the structure, the differential and the product, are not compatible), but it turns out that C provides a good framework to study period integrals of algebraic varieties. Our basic principle is that period integrals of any sort of interesting mathematical objects can be understood as a homotopy class of a morphism in C. The first two authors, in [8], analyzed the period integrals of a smooth projective hypersurface X using the framework of C, and consequently constructed a formal semi-frobenius manifold structure on the middle dimensional primitive cohomology of X. Furthermore they used the deformation functor attached to an L - algebra to make an extended formal deformation space of X and a formal flat connection on its tangent bundle which generalizes the Gauss-Manin connection. One of the important technical tools in [8] is the descendant functor Des from the category C to the category of L -algebras, which is also a main technical tool of the current article. The descendant functor measures the failures of compatibility between two mathematical structures. More precisely, for a given object (A, δ) in C k, define Des (A, δ) = (A, l δ ), where l δ = l δ 1, l δ 2,, is the family of linear maps l δ n : S n (A) A of

THE KOSZUL-TATE TYPE RESOLUTION FOR GBV ALGEBRAS 5 degree -1, inductively defined by the formula l δ 1 = δ, l δ n(x 1,, x n 1, x n ) = l δ n 1(x 1,, x n 2, x n 1 x n ) l δ n 1(x 1,, x n 1 ) x n ( 1) xn 1 (1+ x1 + + xn 2 ) x n 1 l δ n 1(x 1,, x n 2, x n ), for n 2 and any homogeneous elements x 1, x 2,, x n A. Then it is proven in [8] that l δ n is well-defined on S n (A), where S n (A) the n-th symmetric product of A, and (A, l δ ) is a shifted L - algebra in the sense of Definition (2.1). Note that l δ 2 measures the failure of δ being a derivation of the product. This implies that (A, δ = l δ 1) is a CDGA if and only if l δ 2 = l δ 3 = = 0. In fact, the notion of the descendant functor is intimately related to homotopy probability theory, as developed in [4]. We will briefly review the definition of the shifted L -algebra and the descendant L -algebra in subsection 2.1. If one tries to extend the results of [8] to a more general class of algebraic varieties (e.g. projective or toric complete intersections), then it seems technically important to find a resolution of the zeroth homology of a certain Lie algebra representation inside the category C. This leads us to consider the problem of generalizing the Koszul-Tate resolution inside the category of CDGAs to the situation of chain complexes with a binary product (i.e. a resolution inside the category C). The main idea to prove Theorem 1.2 (and Theorem 1.1) is to use an explicit description of the descendant L -algebra (A, l δ ) associated to (A, δ). We summarize a correspondence between Tate s situation and our situation in terms of a Lie algebra representation ρ : g End k (R) and (A, δ) = (E(g; ρ), δ ρ ) in Table 1. The Koszul-Tate resolution ρ(g) is an order 0 differential operator for every g g (E(g; ρ), δ ρ ) is a CDGA M = Im (δ ρ ) R is an ideal R The enhanced Koszul-Tate resolution ρ(g) is a differential operator of order m for some g g, m > 0 (E(g; ρ), δ ρ ) is an object of the category C M = Im (δ ρ ) R is not an ideal of R R/M H 0 (g, R) is not a ring R/M is a commutative ring which is isomorphic to H 0 (g, R) the L -descendants l δ 2 = l δ 3 = = the L -descendants l δ M+1 = 0 l δ M+2 = = 0 for some M 2 (A, δ) = (E(g; ρ), δ ρ ) admits an (A, δ) = (E(g; ρ), δ ρ ) admits an acyclic resolution in the category of acyclic resolution in the category C CDGAs Table 1. We briefly explain the contents of each section. In subsection 2.1, we explain why Tate s method fails in our generalized setting and indicate how to remedy the situation. In subsection 2.2, we briefly review the descendant L -algebras which are important technical tools for our construction. Section 3 is the main section. In subsection 3.1, we briefly explain an outline of the proof. In subsection 3.2, we prove the main theorem in the special case of GBV (Gerstenhaber-Batalin-Vilkovisky)

6 JEEHOON PARK AND DONGGEON YHEE algebras (i.e. Theorem 1.1, or equivalently, µ = 1 in Theorem 1.2). In subsection 3.3, we deal with the general case and finish the proof. In Section 4, we apply the main theorem to a GBV algebra associated to a smooth projective complete intersection X to obtain a GBV algebra whose total homology is naturally isomorphic to the middledimensional primitive cohomology of X. 2. Descendant functor 2.1. Why does Tate s method fail in the generalized situation? In this subsection we explain why the Koszul-Tate type resolution fails by using a simple example, when (A, δ) is not a CDGA and indicate how to salvage the situation. For simplicity, assume that (A, δ) has a GBV(Gerstenhaber- Batalin-Vilkovisky) algebra structure and has homology only inn degree 0 and 1. We further assume that H 1 (A, δ) k and let r be a homology representative of a nonzero class in H 1 (A, δ). Tate s method suggests to adjoin a formal variable θ to A to consider A[θ] and deform the differential δ by δ + r θ. He proved that (A[θ], δ + θ ) is a resolution of (A, δ), if (A, δ) is a CDGA. But if (A, δ) is not a CDGA, then a serious problem occurs; namely, δ + r θ is not a differential. Let θn b A[θ]. Let us compute (δ + r θ )2 (θ n b) = (δ + r ((δ θ ) + r ) θ )(θn b) = (δ + r θ ) ( θ n δ(b) + rnθ n 1 b ) = (δ + r θ ) ( θ n δ(b) + nθ n 1 rb ) = nθ n 1 r δ(b) + nθ n 1 δ(rb) = nθ n 1 l δ 2(r, b), where l δ 2 was defined in (1.1) and we used r 2 = 0 (by super-commutativity), δ(r) = 0, and δ 2 = 0. Thus if l δ 2 0 (which corresponds to the fact that (A, δ) is not a CDGA), then Tate s method does not work and we need a new idea. Our idea relies on the notion of descendant L -algebras, which we review in the next subsection. 2.2. Descendant L -algebra. We briefly recall from [8] the formalism of the descendant functor, which is the main technical tool we use to construct resolutions. In order to fix our convention on L - algebras, we set up notations for partitions. A partition π = B 1 B 2 of the set [n] = {1, 2,, n} is a decomposition of [n] into a pairwise disjoint non-empty subsets B i, called blocks. Blocks are ordered by the minimum element of each block and each block is ordered by the ordering induced from the ordering of natural numbers. The notation π means the number of blocks in a partition π and B means the size of the block B. If k and k belong to the same block in π, then we use the notation k π k. Otherwise, we use k π k. Let P (n) be the set of all partitions of [n]. For a permutation σ of [n], we define a map ˆσ : T n (V ) T n (V ) by ˆσ(x 1 x n ) = ɛ(σ)x σ(1) x σ(n) for homogeneous elements x 1,, x n V, where ɛ(σ) = ɛ(σ, {x 1,, x n }) = ±1 is the Koszul sign determined by decomposing ˆσ as composition of adjacent transpositions ˆτ, where ˆτ : x i x j ( 1) xi xj x j x i = ɛ(τ, {x i, x j }) x j x i, i.e. the Koszul sign is defined as the product of the signs ɛ(τ, {x i, x j }) of such adjacent transpositions. Note that ɛ(σ) depends on the degrees of x 1,, x n but we omit this dependence in the notation for simplicity. The Koszul sign ɛ(π) for a partition

THE KOSZUL-TATE TYPE RESOLUTION FOR GBV ALGEBRAS 7 π = B 1 B 2 B π P (n) is defined to be the Koszul sign ɛ(σ) of the permutation σ determined by x B1 x B π = x σ(1) x σ(n), where x B = x j1 x jr if B = {j 1,, j r }. Let x B = x j1 + + x jr. Definition 2.1 (shifted L -algebra). A shifted L -algebra is a Z-graded vector space V = m Z V m over a field k with a family l = l 1, l 2,, where l k : S k (V ) V is a linear map of degree -1 on the super-commutative k-th symmetric product S k (V ) for each k 1 such that (2.1) ɛ(π, i)l π (x B1,, x Bi 1, l(x Bi ), x Bi+1,, x B π ) = 0. π P (n) B i =n π +1 Here we use the following notation; l(x B ) = l r (x j1,, x jr ) if B = {j 1,, j r } ɛ(π, i) = ɛ(π)( 1) x B 1 + + x Bi 1. An L -algebra with unity (or a unital L -algebra) is a triple (V, l, 1 V ), where (V, l) is an L -algebra and 1 V is a distinguished element in V 0 such that for all n 1 and every x 1,, x n 1 V. l n (x 1,, x n 1, 1 V ) = 0 If we let π = B 1 B i 1 B i B i+1 B π P (n), then the condition B i = n π + 1 in the summation implies that the set B 1,, B i 1, B i+1,, B n are singletons. Let l 1 = δ. For n = 1, the relation (2.1) says that δ 2 = 0. For n = 2, the relation (2.1) says that (2.2) For n = 3, we have δl 2 (x 1, x 2 ) = l 2 (δx 1, x 2 ) ( 1) x1 l 2 (x 1, δx 2 ). δl 3 (x 1, x 2, x 3 ) = l 2 (l 2 (x 1, x 2 ), x 3 ) + ( 1) x1 l 2 (x 1, l 2 (x 2, x 3 )) + ( 1) ( x1 +1) x2 l 2 (x 2, l 2 (x 1, x 3 )) +l 3 (δx 1, x 2, x 3 ) + ( 1) x1 l 3 (x 1, δx 2, x 3 ) + ( 1) x1 + x2 l 3 (x 1, x 2, δx 3 ). Definition 2.2. For a given object (A, δ) in C k, we define Des (A, δ) = (A, l δ ), where l δ = l δ 1, l δ 2,, is the family of linear maps 2 l δ n : S n (A) A of degree -1, inductively defined by the formula (2.3) l δ 1 = δ, l δ n(x 1,, x n 1, x n ) = l δ n 1(x 1,, x n 2, x n 1 x n ) l δ n 1(x 1,, x n 1 ) x n ( 1) xn 1 (1+ x1 + + xn 2 ) x n 1 l δ n 1(x 1,, x n 2, x n ), for n 2 and any homogeneous elements x 1, x 2,, x n A. Since A has a unit 1 A and δ(1 A ) = 0, one can also check that l δ n(x 1, x 2,, x n ) = [[ [[δ, L x1 ], L x2 ], ], L xn ](1 A ) for any homogeneous elements x 1, x 2,, x n A. Here L x : A A is left multiplication by x and [L, L ] := L L ( 1) L L L L End k (A), where L means the degree of L. In particular, l δ 2 : S 2 (A) A is given by l δ 2(x, y) = δ(x y) δx y ( 1) x x δy. 2 In this article, we use the homology version of the shifted L -algebra.

8 JEEHOON PARK AND DONGGEON YHEE so that l δ 2 measures the failure of δ being a derivation of the product. In [8], the authors proved that (A, 1 A, l δ ) is a unital L -algebra (see Definition (2.1)), which we call a descendant L -algebra. One can easily check from the inductive definition that if l δ m = 0 then l δ m+1 = l δ m+2 = = 0. For a given morphism f : (A, δ) (A, δ ) in C k, the authors of [8] defined an L -morphism Des(f), that is, family φ f = φ f 1, φf 2,, of linear maps φf n : S n (A) A of degree 0, and proved that the assignment Des is a functor from C k to the category of unital L -algebras. We call this functor Des a descendant functor. 3. The main results 3.1. Outline of the proof. When there is a non-trivial homology class r, the main idea of the proof is killing a homology class r by employing a new formal variable θ so that δ(θ) = r, as Tate did in [9]. However, our case is very different from [9]. In [9], δ is a derivation so that δ(θ n ) = nθ n 1 δ(θ) and δ(θ v) = θδ(v) ± δ(θ)v up to sign. Then one can easily prove that if the new formal variable θ does not generate a non-trivial homology class, then neither do the elements like δ(θ n ) and δ(θ v). But, in our case, we need to add additional new formal variables to kill δ(θ n ) and δ(θ v), since δ in this paper is not a derivation. Our main tool of the proof is to use the descendant L -structure. The construction of adding variables has to be coherent to the descendant L -structure of (A, l δ ). In order to kill r, we need to add infinitely many new variables to A inductively on the degree of the homology classes so that the extended algebra à is explicitly computable and remains to be a descendant L -algebra. More precisely, for a given generating (as a k-algebra) set S of A, we extend S inductively to S U 0, S U 0 U 1, and so on, where U n is a finite set of formal variables, and define à to be the k-algebra (modulo the obvious super-commutative relation) generated by S U 1 U 2 (see A θ in (3.2) in the GBV case and (3.8) in the general case for precise definitions). Then we construct a differential δ on à using the descendant L -relation (see (3.4) in the GBV case and (3.10) in the general case for details) and show that δ 2 = 0 and the non-trivial homology class r we wanted to kill only vanishes. Note that à is always an infinitely generated k-algebra, even if we start from a finitely generated k-algebra A and like to kill only one nontrivial homology class (in positive degree) of A. 3.2. The case of GBV algebras. Let (A, δ) be an object of the category C such that A is nonnegatively graded i.e. A = m 0 A m. We assume that A is a finitely generated Z-graded k-algebra whose generators are 1, q 1,, q N and that the chain complex (A, δ) is a GBV (Gerstenhaber-Batalin- Vilkovisky) algebra containing a non-trivial homology class r of degree 1. Note that (A, δ, l δ 2) is a GBV algebra if and only if the L -descendants satisfy l δ 3 = l δ 4 = = 0. The goal is to kill the nontrivial homology class r in the category of GBV algebras. We will do this by adding elements of higher degree than the degree of r. Theorem 1.1 follows by applying the following lemma repeatedly. Lemma 3.1. We can explicitly construct a GBV algebra (A θ, δ θ ) such that A A θ and as k-vector spaces. H(A θ, δ θ ) H(A, δ)/k r, Proof. We introduce a formal variable θ corresponding to r such that θ 1 = r. Let (3.1) U 1 = {q 1,, q N } U 0 = {θ}.

THE KOSZUL-TATE TYPE RESOLUTION FOR GBV ALGEBRAS 9 For each n 1, we define a finite set U n of formal variables l n (u, v) of degree u + v 1 inductively as follows: U n = {l n (u, v) : u U n 1, v V n 1 := U n 1 U 1 }, for each n 1. For example, we have that U 1 = {l 1 (u, v) : u U 0, v V 0 = U 0 U 1 }, U 2 = {l 2 (u, v) : u U 1, v V 1 = U 1 U 0 U 1 }. The notation l n (u, v) suggests that later we will use the formula l 2 (u, v) = l n (u, v) to put a descendant L -structure on the newly extended algebra. (see (3.3) below). We define an associative and supercommutative k-algebra A (n) := A k k[u n U n 1 U 0 ] = A[U n U n 1 U 0 ], i.e. A (n) is the symmetric tensor product of A and the (super-)symmetric k-tensor algebra of the k-vector space whose basis is given by U n U n 1 U 0. This clearly defines an inductive system A (n) A (n+1) of associative and super-commutative k-algebras. Let A ( ) := lim A n (n). Let I n (respectively, I ) be the ideal of A (n) (respectively, A ( ) ) generated by l k+1 (u k, v k ) ( 1) u k v k l k+1 (v k, u k ) for u k, v k U k, k = 0,, n 1 (respectively, k = 0, 1, ). Let us define (3.2) B (n) := A (n) /I n, A θ := B ( ) := A ( ) /I. Now we would like to define a k-linear operator δ θ : A θ A θ such that δ θ δ θ = 0 and δ θ A = δ; we do this job by constructing δ k : B (k) B (k+1), k = 1, 0, 1,, inductively. Let B ( 1) := A, δ 1 = K : B ( 1) B ( 1) B (0). The formal element l k (x, y) is supposed to satisfy all the properties of l 2 as Gerstenhaber bracket. In other words, l k measures the failure of δ k 2 to be a derivation, l k is super-commutative, l k (, z) is a derivation with respect to the product, and l k is bilinear and is trivial if one of the components is 1. This suggests the following definition: (3.3) l k (x, y) := l δ k 2 2 (x, y) = δ k 2 (xy) δ k 2 (x)y ( 1) x xδ k 2 (y), x, y V k 2, l k (z, x) := ( 1) x z l k (x, z), x, z V k 1, l k (xy, z) := ( 1) x xl k (y, z) + ( 1) y z l k (x, z)y, x, y, z V k 1, l k (1, x) := 0, l k (ax + by, z) := al k (x, z) + bl k (y, z), a, b k, x, y, z B (k 1), for each k 1. This definition (3.3) implies that if δ k 2 : B (k 2) B (k 1) is defined then we can extend the definition of l k (u, v) in B (k) for any u, v B (k 1) not only for u U k 1, v V k 1. Let us give a construction of δ k : first, we define and δ 0 (θ) := r, δ 1 (r) = 0, δ 0 (v) := δ(v), v A, (3.4) δ k (l k (u, v)) := l k+1 (δ k 1 (u), v) ( 1) u l k+1 (u, δ k 1 (v)), l k (u, v) U k, k 1.

10 JEEHOON PARK AND DONGGEON YHEE This definition (3.4) is motivated by the binary L -relation (2.2). Then we use the following relations (with the help of notation (3.3)) inductively (3.5) δ k (uv) := l k+1 (u, v) + δ k (u)v + ( 1) u uδ k 1 (v), u U k, v B (k 1), δ k (uu ) := l k+1 (u, u ) + δ k (u)u + ( 1) u uδ k (u ), u, u U k = V k \ V k 1. to extend (3.4) and obtain a k-linear map K k : B (k) B (k+1) for each k 0. This relation (3.5) is motivated by the definition of l δ 2-descendant. 3 This clearly induces a k-linear operator such that δ θ A = δ. δ θ : A θ = B ( ) A θ = B ( ), δ θ := lim δ k k Now we show that δ θ δ θ = 0. We use an induction over the index k for B (k). If we assume that δ k+1 (δ k (u)) = δ k+1 (δ k (v)) = 0 for u, v B (k), then we have ) δ k+1 (δ k (uv)) = δ k+1 (l k+1 (u, v) + δ k (u)v + ( 1) u uδ k (v) (3.6) = l k+2 (δ k (u), v) ( 1) u l k+2 (u, δ k (v)) + δ k+1 (δ k (u)v) + ( 1) u δ k+1 (uδ k (v)) = δ k+1 (δ k (u))v + ( 1) u +1 δ k (u)δ k (v) = 0. + ( 1) u δ k+1 (u)δ k (v) + ( 1) u uδ k+1 (δ k (v)) Becasue δ 2 = δ 2 1 = 0, (δ 1 δ 0 )(θ) = δ 1 (r) = 0, the above computation (3.6) shows that (δ 1 δ 0 )(x) = 0 for all x B (0). If we assume that δ k δ k 1 = 0, a similar computation using the inductive l 2 -relation (3.4) (in the L -algebra relations) shows that δ k+1 δ k = 0. We conclude δ θ δ θ = 0. It is clear that δ θ A = δ and we thus have a differential δ θ. Moreover, condition (3.3) implies that (by setting n = 3 in (2.3)) l δθ 3 (x, y, z) = 0, x, y, z A θ, which implies that (A θ, δ θ, l δθ 2 ) is a GBV algebra such that A A θ. It remains to show that H(A θ, δ θ ) H(A, δ)/k r as k-vector spaces. First, note that r = δ θ (θ) and r is no longer a nontrivial homology class of (A θ, δ θ ). What we have to show is that the introduction of the new formal variables in U n (n = 0, 1, ) do not generate any nontrivial homology class. We begin the argument by showing that if δ 0 (θ m v) = 0, v A, m 1, then v = 0. We divide into two cases according to the parity of the degree of θ. Assume that θ is odd so that r is even. Then θ 2 = 0 and l 1 (θ, θ) = 0. We only have to consider 3 For example, we have δ 0 (θ 2 ) = l 1 (θ, θ) + rθ + ( 1) θ θr, δ 0 (θv) = l 1 (θ, v) + rv θδ(v) = 0, δ 0 (θ 3 ) = l 1 (θ 2, θ) + δ 0 (θ 2 )θ + θ 2 r = ( 1) θ θl 1 (θ, θ) + ( 1) θ 2 l 1 (θ, θ)θ + δ 0 (θ 2 )θ + θ 2 r, and δ 0 : A 0 A 1 is defined inductively by where l(θ m, v) B (1) is defined by using (3.3). δ 0 (θ m v) = l 1 (θ m, v) + δ 0 (θ m )v + ( 1) m θ θ m δ 0 (v), v B (0), m 0,

THE KOSZUL-TATE TYPE RESOLUTION FOR GBV ALGEBRAS 11 which implies that l 1 (θ, v) = 0. We thus have v = 0. Now let us assume that θ is even so that r is odd. Then a simple computation confirms that m δ 0 (θ) = r, δ 0 (θ m ) = ( (k 1)) θ m 2 l 1 (θ, θ) + m rθ m 1, m 2. Then the condition k=2 δ 0 (θ m v) = mθ m 1 l 1 (θ, v) + δ 0 (θ m )v + θ m δ(v) = 0, m 1, implies that l 1 (θ, v) = 0 and δ 0 (θ m )v = 0, which in turn implies that v = 0. In general, we need to show that if δ m (uv) = 0, 0 u U m, v B (m 1), m 1 (where U m means the ideal of B (m) generated by U m ), then v = 0. We use an induction. Assume that δ k 1 (uv) = 0 for 0 u U k 1 and v B (k 2) implies that v = 0. Consider the condition (3.7) δ k (uv) = l k+1 (u, v) + δ k (u)v + ( 1) u uδ k 1 (v) = 0, u U k, v B (k 1), Since δ k (u) = δ k (l k (u k, v k )) = l k+1 (δ k 1 (u k ), v k ) ( 1) u k l k+1 (u k, δ k 1 (v k )), condition (3.7) implies that uδ k 1 (v) = 0. We thus have δ k 1 (v) = 0. The induction hypothesis implies that v = 0, unless v B (k 2). If v B (k 2), then δ (k 1) (v) = δ (k 2) (v)=0. We can continue this way to conclude that v A Ker (δ). Then (3.7) says that l k+1 (u, v) + δ k (u)v = 0, u U k, v A = B ( 1), which implies that δ k (u)v = 0. Thus v = 0. We therefore conclude that δ m -closed elements in A θ for any m 0 should be zero, which is exactly what we want. Let Q be a derivation with respect to the multiplication of a GBV-algebra (A, δ, l δ 2) such that (1) (A, Q) is a CDGA, (2) Qδ + δq = 0, (3) (ker Q, δ) (A, δ) induces an isomorphism H (ker Q, δ) = H (A, δ). If a GBV algebra (A, δ, l δ 2) is equipped with a derivation Q which satisfies the first two conditions above, then (A, Q, δ) is called a dgbv algebra. 4 The third condition is used to construct a formal Frobenius manifold structure from a dgbv algebra (see the section 6 of [7]). Proposition 3.2. In the case of GBV algebras, we can extend Q to Q θ so that (1) (A θ, Q θ ) is a CDGA, (2) Q θ δ θ + δ θ Q θ = 0, (3) (ker Q θ, δ θ ) (A θ, δ θ ) induces an isomorphism H (ker Q θ, δ θ ) = H (A θ, δ θ ). Proof. The extension (A θ, δ θ ) is given by Lemma 3.1. We may assume that r ker Q via the isomorphism H (ker Q, δ) = H (A, δ). Then we define Q θ on A θ by the following rules Q θ (θ) = 0, Q θ (a b) = Q θ (a) b + ( 1) a aq θ (b), Q θ l δθ 2 (a, b) = l δθ 2 (Q θ (a), b) ( 1) a l δθ 2 (a, Q θ (b)). It is obvious that Q θ and δ θ satisfy the condition 1 and 2. Since ker Q θ ker δ θ contains ker Q ker δ, the induced morphism H (ker Q θ, δ θ ) H (A θ, δ θ ) is surjective. Since ker Q θ is a subalgebra of A θ containing θ and equipped with δ θ, Lemma 3.1 implies that H (ker Q θ, δ θ ) = H (ker Q, δ)/k r = H (A, δ)/k r. 4 If we define [x y] := ( 1) x l δ 2 (x, y), then (A, Q, δ, [ ]) is a dgbv algebra in the sense of [7].

12 JEEHOON PARK AND DONGGEON YHEE Remark 3.3. Compare Lemma 3.1 to Tate s theorem (Theorem 2, [9]). Before killing a nontrivial (co)homology class r, Tate assumed that r was a skew non-zerodivisor in his (co)homology algebra. In this paper, such an assumption is not needed for two reasons. 1. H(A, δ) does not inherit an algebra structure from A : for a, b ker δ, l δ 2(a, b) = δ(ab) δ(a)b ± aδ(b) 0 and ab ker δ in general. This is why we treat H(A, δ) and H(A θ, δ θ ) as vector spaces. 2. In [9], the assumption is required to show that new variables do not generate new (co)cycles. In this paper, this annihilation property follows from the failure of Leibniz s rule. This is explained in the final step in the proof of Lemma 3.1. 3.3. The general case. The goal here is to prove Theorem 1.2 and we will achieve this via the key lemma 3.4. Let (A, δ) be an object of the category C such that A is non-negatively graded i.e. A = m 0 A m, and l δ M+1 = 0 for some M 1. The M = 2 case was done in subsection 3.2. In order to describe an explicit formula for the general case, we use the notion of (a homology version of) a shifted L -algebra using partitions (see definition 2.1). Assume that A is a finitely generated Z-graded k-algebra whose generators are 1, q 1,, q N and (A, δ) has a non-trivial cohomology class r of degree 1. We introduce a formal variable θ corresponding to r such that θ 1 = r. Let U 1 = {q 1,, q N }, U 0 = {θ}. We define a set of formal variables l n m(u 1,, u m ) of degree m i=1 u i 1 in an inductive manner as follows: U n = {l n m(u 1, u 2,, u m ) : u 1 U n 1, u 2,, u m V n 1 := U n 1 U 1, 2 m M}, for M 2 and for each n 1. 5 For example, we have We define a k-algebra U 1 = {l 1 m(u 1, u 2,, u m ) : u 1 U 0, u 2, u m U 0 U 1, 2 m M}, U 2 = {l 2 m(u 1, u 2,, u m ) : u 1 U 1, u 2,, u m U 1 U 0 U 1, 2 m M}. A (n) := A k k[u n U n 1 U 0 ] = A[U n U n 1 U 0 ], i.e. A (n) is the symmetric tensor product of A and the (super-)symmetric k-tensor algebra of the k-vector space whose basis is given by U n U n 1 U 0. This clearly defines an inductive system A (n) A (n+1) of associative and super-commutative k-algebras. Let A ( ) := lim A (n). n Let I n (respectively, I ) be the ideal of A (n) (respectively, A ( ) ) generated by l k+1 m (x σ(1),, x σ(m) ) sgn(σ(x 1,, x m ))l k+1 m (x 1,, x m ), x 1,, x m U k, for k = 0,, n 1, (respectively, k = 0, 1,,) and 2 m M. Let us define (3.8) B (n) := A (n) /I n, A θ := B ( ) := A ( ) /I. 5 If M were 1, then (A, δ) is a CDGA (commutative differential graded k-algebra) and we may not need U1, U 2,. The set U 0 is enough to kill the cohomology class r.

THE KOSZUL-TATE TYPE RESOLUTION FOR GBV ALGEBRAS 13 We define l k m(x 1,, x m ) := l δ k 2 m (x 1,, x m ) for x 1,, x m V k 2, and use the following definition: (3.9) l k m(x σ(1),, x σ(m) ) := sgn(σ(x 1,, x m ))l k m(x 1,, x m ), l k m 1(x 1,, x m 2, x m 1 x m ) := l k m(x 1,, x m 1, x m ) + l k m 1(x 1,, x m 1 ) x m + ( 1) xm 1 (1+ x1 + + xm 2 ) x m 1 l k m 1(x 1,, x m 2, x m ), for x 1,, x m V k 1, and l k m(ax 1 + bx 1, x 2,, x m ) := al k m(x 1, x 2,, x m ) + bl k m(x 1, x 2,, x m ), l k 2(1, x 1,, x m ) := 0, for a, b k and x 1, x 1, x 2,, x m B (k 1). This definition (3.9) implies that if δ k 2 : B (k 2) B (k 1) is defined then we can extend the definition of l k m to cover l k m(u 1,, u m ) in B (k) for any u 1,, u m B (k 1) not only for u 1 U k 1, u 2,, u m V k 1. For example, we have that l k 2(xy, z) := l k 3(x, y, z) + ( 1) x xl k 2(y, z) + ( 1) y z l k 2(x, z)y, l k 3(xy, z, u) := l k 4(x, y, z, u) + ( 1) x xl k 3(y, z, u) + ( 1) y z + y u l k 3(x, z, u)y. Let B ( 1) := A, δ 1 = δ : B ( 1) B ( 1) B (0). Let us give a construction of a k-linear operator δ k : B (k) B (k+1) (we also denote δ k by l k+1 1 below); first, we define δ 0 (θ) := r, δ 1 (r) = 0, δ 0 (v) := δ(v), v A, and (3.10) δ k (l k m(x 1,, x m )) := ɛ(π, i)l k+1 π (x B1,, x Bi 1, l k (x Bi ), x Bi+1,, x B π ), π P (n) B i =n π +1 n for m 0 and k = 0, 1, 2,. Here we use the the following notation: l k (x B ) = l k r(x j1,, x jr ) if B = {j 1,, j r }, l k 1 = K k 1, ɛ(π, i) = ɛ(π)( 1) x B 1 + + x Bi 1, The equality (3.10) is simply the L -algebra relation in Definition 2.1. 6 Then we use the following relations (with a help of the definition (3.9)) inductively δ k (uv) := l k+1 2 (u, v) + δ k (u)v + ( 1) u uδ k 1 (v), u U k, v B (k 1), δ k (uu ) := l k+1 2 (u, u ) + δ k (u)u + ( 1) u uδ k (u ), u, u U k = V k \ V k 1. 6 For example, the m = 3 case means that K k l k 3 (x 1, x 2, x 3 ) = l k+1 2 (l k 2 (x 1, x 2 ), x 3 ) + ( 1) x 1 l k+1 2 (x 1, l k 2 (x 2, x 3 )) + ( 1) ( x 1 +1) x 2 l k+1 2 (x 2, l k 2 (x 1, x 3 )) +l k+1 3 (K k 1 x 1, x 2, x 3 ) + ( 1) x 1 l k+1 3 (x 1, K k 1 x 2, x 3 ) + ( 1) x 1 + x 2 l k+1 3 (x 1, x 2, K k 1 x 3 ).

14 JEEHOON PARK AND DONGGEON YHEE to extend (3.10) and obtain a k-linear map δ k : B (k) B (k+1) for each k 0. This induces a k-linear operator such that δ θ A = K. δ θ : A θ = B ( ) A θ = B ( ), δ θ := lim δ k k Lemma 3.4. The tuple (A θ, δ θ ) := (A, δ ) is an object of C such that A A θ, and as k-vector spaces. l δθ M+1(x 1,, x M+1 ) = 0, x 1,, x M+1 A θ, H(A θ, δ θ ) H(A, δ)/k r, Proof. The first thing to show is that δ θ is a differential of degree 1. As in the GBV case, we can use an induction over the index k for B (k). If we assume that δ k+1 (δ k (u)) = δ k+1 (δ k (v)) = 0 for u, v B (k), then the same computation (the term l k (u, v) replaced by l k 2(u, v)) as (3.6) implies that δ k+1 (δ k (uv)) = 0 (the higer L -terms like l k 3, l k 4,, do not appear in (3.6)). Because δ 2 = δ 2 1 = 0, (δ 1 δ 0 )(θ) = δ 1 (r) := δ(r) = 0, the equality (3.6) shows that (δ 1 δ 0 )(x) = 0 for all x B (0). On the other hand, if we assume that δ k δ k 1 = 0, a standard computation using the inductive L -relation (3.10) shows that δ k+1 δ k = 0. Hence we have δ θ δ θ = 0. It is clear that δ θ A = δ and we thus have a differential δ θ. Moreover, condition (3.9) implies that (setting n = M + 1 in (2.3)) l δθ M+1(x 1,, x M+1 ) = 0, x 1,, x M+1 A θ. It remains to show that H(A θ, δ θ ) H(A, δ)/k r as k-vector spaces. First, note that r = δ θ (θ) and r is no longer a nontrivial homology class of (A θ, δ θ ). As in the GBV case, what we have to show is that new formal variables U n (n = 0, 1, ) do not generate any nontrivial homology classes. The arguments are very similar to the GBV case. We begin the argument by showing that if δ 0 (θ m v) = 0, v A, m 1, then v = 0. We divide into two cases according to the parity of the degree of θ. Assume that θ is odd so that r is even. Then θ 2 = 0 and l 1 (θ, θ) = 0. We only have to consider δ 0 (θv) = l 1 2(θ, v) + rv θδ(v) = 0, which implies that l 1 (θ, v) = 0. We thus have v = 0. Now let us assume that θ is even so that r is odd. Then a simple computation confirms that δ 0 (θ) = r δ 0 (θ m ) = l 1 m(θ,, θ) + a 1 θl 1 m 1(θ,, θ) + +a m 2 θ m 2 l 1 2(θ, θ) + m rθ m 1, m 2, for some a 1,, a m 2 Z (set l 1 m(θ,, θ) = 0 if m M + 1). Then the condition δ 0 (θ m v) = l 1 2(θ m, v) + δ 0 (θ m )v + θ m δ(v) = 0, m 1, implies that l 1 2(θ m, v) = 0 and δ 0 (θ m )v = 0, which in turn implies that v = 0. In general, we need to show that if δ m (uv) = 0, 0 u U m, v B (m 1), m 1 (where U m means the ideal of B (m) generated by U m ), then v = 0. The argument for this is exactly the same as the

THE KOSZUL-TATE TYPE RESOLUTION FOR GBV ALGEBRAS 15 GBV case (just replace l 2 (u, v) by l k+1 2 (u, v)). We therefore conclude that δ m -closed elements in A θ for any m 0 should be zero. Theorem 1.2 follows by applying Lemma 3.4, in the case (A, δ) = (E(g M ; ρ M ), δ ρ ), inductively for any non-trivial homology class r of degree bigger than or equal to 1. 4. Applications We assume that k = C from now on. Let P n be the projective n-space over k. Let V be a projective algebraic variety of dimension m in P n. Definition 4.1. Let H k (V, k) be k-th singular cohomology of V. An object (A, δ) Ob(C) is called a homotopy enhancement of H k (V, k), if H(A, δ) is concentrated in degree 0 and is isomorphic to H k (V, k). Moreover, if (A, δ, l δ 2) is a GBV algebra, then we call (A, δ) a BV homotopy enhancement. Note that the binary product is responsible for the word enhancement. Such a homotopy enhancement plays an important role in analyzing the period matrices of deformations of a projective algebraic variety. In [8], the authors found a homotopy enhancement of the primitive middle-dimensional cohomology H n 1 prim (V, k), when V is a smooth projective hypersurface in P n. This result was generalized to smooth projective complete intersections in [5]. The main application of the current article is the following. Corollary 4.2. Let X be a smooth projective complete intersection of dimension m. Then there is a BV homotopy enhancement of the primitive middle-dimensional cohomology Hprim m (X, k) of X. Our method to achieve such a resolution of Hprim m (X, k) can be summarized as follows: Attach to X a certain Lie algebra representation ρ X originating from quantum mechanics. Construct the Chevalley-Eilenberg chain complex (A, δ) X associated to ρ X. Apply Theorem 1.2 to get a resolution (Ã, δ) X which provides a GBV algebra. Let X P n be a smooth projective complete intersection defined by G 1,, G k so that m = n k. We briefly review the construction of ρ X. We introduce formal variables y 1,, y k corresponding to G 1,, G k. Let N = n + k + 1 and A := C[q µ ] µ=1,,n where q 1 = y 1,, q k = y k and q k+1 = x 0,, q N = x n. We consider the Dwork potential S(q) := k y l G l (x). l=1 Let g be an abelian Lie algebra of dimension N. Let α 1, α 2,, α N be a C-basis of g. We associate a Lie algebra representation ρ X on A of g as follows: ρ i := ρ X (α i ) := + S(q), for i = 1, 2,, N q i q i We extend this C-linearly to get a map ρ X : g End C (A). This is clearly a Lie algebra representation of g. Then we consider the Chevalley-Eilenberg complex (A ρx, δ ρx ) associated to ρ X, which computes the Lie algebra homology of ρ X, i.e. H i (A ρx, δ ρx ) H i (ρ X, A). The Z-graded super-commutative

16 JEEHOON PARK AND DONGGEON YHEE algebra A ρx and differentials δ ρx and Q ρx are given explicitly as follows: A := A ρx = C[q][η] = C[q 1, q 2,, q N ][η 1, η 2,, η N ], δ := δ ρx = N ( S(q) + ) : A A, q i q i η i Q := Q ρx = i=1 N S(q) : A A. q i η i i=1 Note that η i η j = η j η i, which implies that ηi 2 S(y,x) = 0. Since q i η i is a differential operator of order 1, the differential Q is a derivation of the product of A. Thus (A, Q) is a CDGA (commutative differential graded algebra). But K is not a derivation of the product, because the differential operator has order 2. We also introduce the C-linear map q i η i := δ Q = N i=1 q i η i : A A. Note that is a also a differential of degree -1, i.e. 2 = 0. Since Q 2 = 2 = δ 2 = 0, we get We have Q + Q = 0. 0 A N δ A N 1 δ δ A 0 0. The triple (A, δ, l δ 2) is a GBV (Gerstenhaber-Batalin-Vilkovisky) algebra over C, i.e. (A, δ) satisfies all the assumptions of Theorem 1.1. In particular, we have l δ 3 = l δ 4 = = 0. Thus we apply our main theorem 1.1 to get a resolution (Ã, δ) X Ob(C) whose cohomology vanishes except for H 0 (Ã, δ) = H 0 (A, δ) H n k prim (X, C). Acknowledgement. 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. References [1] Barannikov, S.; Kontsevich, M.: Frobenius manifolds and formality of Lie algebras of polyvector fields, Internat. Math. Res. Notices, 4 (1998), 201-215. [2] Imma Gálvez-Carrillo; Andy Tonks; Bruno Vallette, Homotopy Batalin-Vilkovisky algebras, arxiv:0907.2246. [3] Cao, Huai-Dong; Zhou, Jian: DGBV algebras and mirror symmetry, First International Congress of Chinese Mathematicians (Beijing, 1998), 279-289, AMS/IP Stud. Adv. Math., 20, Amer. Math. Soc., Providence, RI, 2001. [4] Drummond-Cole, G.-C., Park, J.-S., Terilla, J.: Homotopy probability theory I. J. Homotopy Relat. Struct. 1 11 (2013). [5] Kim, Yesule; Park, Jeehoon: Deformations for period matrices of smooth projective complete intersections, preprint. [6] Loday, J-L.; Vallete, B.: Algebraic operads. Spinger Verlag. [7] Manin, Yu. I.: Three constructions of Frobenius manifolds: a comparative study. Sir Michael Atiyah: a great mathematician of the twentieth century. Asian J. Math. 3 (1999), no. 1, 179 220. [8] Park, Jae-Suk; Park, Jeehoon: Enhanced homotopy theory for period Integrals of smooth projective hypersurfaces, Communications in Number theory and Physics, Volume 10 (2016), Number 2, Pages 235-337. [9] Tate, John: Homology of Noetherian rings and local rings. Illinois J. Math. 1 (1957), 14-27.

THE KOSZUL-TATE TYPE RESOLUTION FOR GBV ALGEBRAS 17 Department of Mathematics, Pohang University of Science and Technology (POSTECH), 77 Cheongam-ro, Nam-gu, Pohang, Gyeongbuk, Korea 790-784 E-mail address: jeehoonpark@postech.ac.kr Department of Mathematical Sciences, Seoul National University, Seoul, Korea 151-747 E-mail address: dgyhee@gmail.com