Classical and variational Poisson cohomology
Bojko Bakalov, Alberto De Sole, Reimundo Heluani, Victor G. Kac, Veronica Vignoli
aa r X i v : . [ m a t h . R T ] J a n CLASSICAL AND VARIATIONAL POISSON COHOMOLOGY
BOJKO BAKALOV, ALBERTO DE SOLE, REIMUNDO HELUANI, VICTOR G. KAC,AND VERONICA VIGNOLI
Abstract.
We prove that, for a Poisson vertex algebra V , the canonical in-jective homomorphism of the variational cohomology of V to its classical co-homology is an isomorphism, provided that V , viewed as a differential algebra,is an algebra of differential polynomials in finitely many differential variables.This theorem is one of the key ingredients in the computation of vertex algebracohomology. For its proof, we introduce the sesquilinear Hochschild and Har-rison cohomology complexes and prove a vanishing theorem for the symmetricsesquilinear Harrison cohomology of the algebra of differential polynomials infinitely many differential variables. Contents
1. Introduction 12. Variational PVA cohomology 43. Preliminaries on the symmetric group and on graphs 64. Classical PVA cohomology 95. The Main Theorem 146. Sesquilinear Harrison cohomology 157. Relation between symmetric sesquilinear Harrison and classical PVAcohomology complexes 218. Vanishing of the sesquilinear Harrison cohomology 249. Proof of the Main Theorem 5.2 32References 331.
Introduction
In the series of papers [BDSHK19, BDSHK20, BDSK20, BDSKV21, BDSK21],the foundations of cohomology theory of vertex algebras have been developed. Themain tool for the computation of this cohomology is the reduction to the variationalPoisson vertex algebra (PVA) cohomology. The latter is a well-developed theorywith many examples computed explicitly [DSK13, BDSK20]. Its importance stemsfrom the fact that vanishing of the first variational PVA cohomology leads to theconstruction of integrable hierarchies of Hamiltonian PDEs.The reduction of the computation of the vertex algebra cohomology to the vari-ational PVA cohomology is performed via the classical PVA cohomology in threesteps as follows. First, let V be a vertex algebra over a field F , with an increasing Key words and phrases.
Poisson vertex algebra (PVA), classical operad, classical PVA coho-mology, variational PVA cohomology, sesquilinear Hochschild and Harrison cohomology. ltration by F [ ∂ ] -submodules such that V := gr V carries a canonical structure of aPVA. Let ( C ch ( V ) , d ) be the vertex algebra cohomology complex of V . A filtrationon V induces a decreasing filtration on C ch ( V ) , and we have a canonical injectivemap [BDSHK19]: gr C ch ( V ) ֒ → C cl ( V ) , (1.1)where ( C cl ( V ) , gr d ) is the classical PVA cohomology complex of V . Moreover, themap (1.1) is an isomorphism, provided that V ≃ V , as F [ ∂ ] -modules [BDSHK20].Second, in [BDSK21], we constructed a spectral sequence from the classical PVAcohomology of V to the vertex algebra cohomology of V .Third, in [BDSHK19], we constructed a canonical injective map H PV ( V ) ֒ → H cl ( V ) (1.2)from the variational PVA cohomology of V to its classical PVA cohomology, and weconjectured that (1.2) is an isomorphism, provided that V , viewed as a differentialalgebra, is an algebra of differential polynomials in finitely many indeterminates.The main goal of the present paper is to prove this conjecture.Recall that a Poisson vertex algebra (abbreviated PVA) is a differential algebra V with a derivation ∂ , endowed with a bilinear λ -bracket V × V → V [ λ ] , satisfyingthe axioms of a Lie conformal algebra and the Leibniz rules (see (i)–(iii) and (iv)-(iv’), respectively, in Definition 2.1). In order to construct the variational PVAcohomology complex ( C PV ( V ) , d ) , introduce the vector spaces V n = V [ λ , . . . , λ n ] / ( ∂ + λ + · · · + λ n ) V [ λ , . . . , λ n ] , n ≥ , (1.3)where λ , . . . , λ n are indeterminates. Then the space of n -cochains C n PV ( V ) consistsof all linear maps f : V ⊗ n → V n , v f λ ,...,λ n ( v ) , (1.4)satisfying the sesquilinearity conditions (2.2), the skewsymmetry conditions (2.3),and the Leibniz rules (2.4). The variational PVA differential d : C n PV ( V ) → C n +1PV ( V ) is defined by formula (2.5).In order to define the classical PVA cohomology complex ( C cl ( V ) , d ) , denote by G ( n ) the set of oriented graphs with vertices { , . . . , n } and without tadpoles. Thenthe space of n -cochains C n cl ( V ) consists of linear maps (cf. (1.3), (1.4)) Y : F G ( n ) ⊗ V ⊗ n → V n , Γ ⊗ v Y Γ λ ,...,λ n ( v ) , (1.5)satisfying the skewsymmetry conditions (4.3), the cycle relations (4.4), and thesesquilinearity conditions (4.7). The classical PVA differential is defined by formula(4.9).The complexes ( C PV ( V ) , d ) and ( C cl ( V ) , d ) both look similar to the Chevalley–Eilenberg complex for a Lie algebra with coefficients in the adjoint representation.The reason for this similarity is the operadic origin for all these cohomology theories,as explained in [BDSHK19].An important observation is that we have a canonical injective map of complexes ϕ : C PV ( V ) → C cl ( V ) defined by ϕ ( f )(Γ ⊗ ( v ⊗ · · · ⊗ v n )) = δ Γ , [ n ] f ( v ⊗ · · · ⊗ v n ) , (1.6)where [ n ] denotes the graph with n vertices and no edges. It was proved in[BDSHK19] that the map (1.6) induces an injective map in cohomology ϕ ∗ : H PV ( V ) ֒ → H cl ( V ) . (1.7) he main result of the present paper is the following (see Theorem 5.2). Theorem 1.1.
Provided that, as a differential algebra, the PVA V is a finitely-generated algebra of differential polynomials, the map ϕ ∗ is an isomorphism. The proof of this theorem uses the s -sesquilinear Hochschild cohomology com-plex, defined for an associative algebra A with a derivation ∂ and a differential bi-module M over A as follows. For s = 1 , this complex is the differential Hochschildcohomology complex, for which the space of n -cochains is Hom F [ ∂ ] ( A ⊗ n , M ) andthe differential d is defined by the usual Hochschild’s formula ( df )( a ⊗ · · · ⊗ a n +1 ) = a f ( a ⊗ · · · ⊗ a n +1 )+ n X i =1 ( − i f ( a ⊗ · · · ⊗ a i − ⊗ a i a i +1 ⊗ a i +2 ⊗ · · · ⊗ a n +1 )+ ( − n +1 f ( a ⊗ · · · ⊗ a n ) a n +1 . (1.8)For an arbitrary positive integer s , the definition is similar but more complicated.Given k = ( k , . . . , k s ) ∈ Z s ≥ , let K = 0 , K t = k + · · · + k t , t = 1 , . . . , s , and n = K s = k + · · · + k s . Given v , . . . , v n ∈ A , we denote v tk = v K t − +1 ⊗ · · · ⊗ v K t ∈ A ⊗ k t , t = 1 , . . . , s , so that v = v ⊗ · · · ⊗ v n = v k ⊗ · · · ⊗ v sk . Then the space of s -sesquilinear Hochschild n -cochains consists of linear maps (cf.(1.3), (1.4)): F Λ ,..., Λ s : A ⊗ n → M [Λ , . . . , Λ s ] / ( ∂ + Λ + · · · + Λ s ) M [Λ , . . . , Λ s ] , satisfying the sesquilinearity conditions ( t = 1 , . . . , s ), F Λ ,..., Λ s ( v k ⊗ · · · ∂v tk · · · ⊗ v sk ) = − Λ t F Λ ,..., Λ s ( v ) . (1.9)The definition of the differential is similar to (1.8): see formulas (6.12) and (6.14).Note that for s = 1 this coincides with the differential Hochschild complex if weidentify M with M [Λ ] / ( ∂ + Λ ) M [Λ ] .If A is a commutative associative algebra and M is a symmetric bimodule over A , the differential Hochschild complex contains the Harrison subcomplex, definedby the Harrison conditions (6.5). We define a similar s -sesquilinear Harrison sub-complex of the s -sesquilinear Hochschild complex by Proposition 6.6. Moreover, wedefine by (6.15) the action of the symmetric group S s on the s -sesquiinear Harrisoncomplex, and the symmetric s -sesquiinear Harrison complex of S s -invariants, whichwe denote by ( C s sym , Har ( A, M ) , d ) .Our key observation is that the classical PVA complex ( C cl ( V ) , d ) is closely re-lated to the complex ( C s sym , Har ( V , V ) , d ) . Namely, introduce an increasing filtrationof C n cl by letting F s C n cl = (cid:8) Y ∈ C n cl (cid:12)(cid:12) Y Γ = 0 if s > n − e (Γ) (cid:9) , where e (Γ) is the number of edges of the graph Γ . We prove the following (seeTheorem 7.2): heorem 1.2. For a PVA V and s ≥ , we have a canonical isomorphism ofcomplexes: gr s C cl ( V ) ≃ C s sym , Har ( V , V ) , where on the right the first V is viewed as a commutative associative differentialalgebra and the second V as a symmetric bimodule over it. Consequently, Theorem 1.1 follows from Theorem 1.2 and the following vanishingtheorem for the sesquilinear Harrison cohomology (see Theorem 8.7).
Theorem 1.3.
Let V be a finitely-generated algebra of differential polynomials.Then H n ( C s sym , Har ( V , V ) , d ) = 0 for ≤ s < n . In order to simplify the exposition, we restricted to the purely even case. How-ever, the same proofs work in the super case. Namely, Theorem 1.2 holds for anyPoisson vertex superalgebra V , while Theorems 1.1 and 1.3 hold if V is a superal-gebra of differential polynomials in finitely many commuting and anticommutingindeterminates.Throughout the paper, the base field F has characteristic 0, and, unless otherwisespecified, all vector spaces, their tensor products and Homs are over F . Acknowledgments.
This research was partially conducted during the authors’ visitsto the University of Rome La Sapienza, to MIT, and to IHES. The first authorwas supported in part by a Simons Foundation grant 584741. The second authorwas partially supported by the national PRIN fund n. 2015ZWST2C_001 andthe University funds n. RM116154CB35DFD3 and RM11715C7FB74D63. Thethird author was partially supported by CPNq grant 409582/2016-0. The fourthauthor was partially supported by the Bert and Ann Kostant fund and by a SimonsFellowship. We would like to thank Pavel Etingof for providing a proof of thedifferential HKR theorem for the algebra of differential polynomials.2.
Variational PVA cohomology
Poisson vertex algebras.Definition 2.1. A Poisson vertex algebra (PVA) is a differential algebra V , i.e. acommutative associative unital algebra with a derivation ∂ , endowed with a bilinear(over F ) λ -bracket [ · λ · ] : V × V → V [ λ ] satisfying:(i) sesquilinearity: [ ∂a λ b ] = − λ [ a λ b ] , [ a λ ∂b ] = ( λ + ∂ )[ a λ b ] ;(ii) skewsymmetry: [ a λ b ] = − [ b − λ − ∂ a ] , where ∂ is moved to the left to act oncoefficients;(iii) Jacobi identity: [ a λ [ b µ c ]] − [ b µ [ a λ , b ]] = [[ a λ b ] λ + µ c ] .(iv) left Leibniz rule [ a λ bc ] = [ a λ b ] c + [ a λ c ] b .From the skewsymmetry (ii) and left Leibniz rule (iv) we immediately get the(iv’) right Leibniz rule [ ab λ c ] = [ a λ + ∂ c ] → b + [ b λ + ∂ c ] → a ,where the arrow means that ∂ is moved to the right, acting on b in the first term,and on a in the second term. .2. Variational PVA complex.
Given a Poisson vertex algebra V , the corre-sponding variational PVA cohomology complex ( C PV , d ) is constructed as follows[DSK13]; see also [BDSK20]. The space C n PV of n -cochains consists of linear maps f : V ⊗ n −→ V [ λ , . . . , λ n ] / h ∂ + λ + · · · + λ n i , (2.1)where h Φ i denotes the image of the endomorphism Φ , satisfying the sesquilinearity conditions ( ≤ i ≤ n ): f λ ,...,λ n ( v ⊗ · · · ⊗ ( ∂v i ) ⊗ · · · ⊗ v n ) = − λ i f λ ,...,λ n ( v ⊗ · · · ⊗ v n ) , (2.2)the skewsymmetry conditions ( ≤ i < n ): f λ ,...,λ i ,λ i +1 ,...,λ n ( v ⊗ · · · ⊗ v i ⊗ v i +1 ⊗ · · · ⊗ v n )= − f λ ,...,λ i +1 ,λ i ,...,λ n ( v ⊗ · · · ⊗ v i +1 ⊗ v i ⊗ · · · ⊗ v n ) , (2.3)and the Leibniz rules ( ≤ i ≤ n ): f λ ,...,λ n ( v , . . . , u i w i , . . . , v n ) = f λ ,...,λ i + ∂,...,λ n ( v , . . . , u i , . . . , v n ) → w i + f λ ,...,λ i + ∂,...,λ n ( v , . . . , w i , . . . , v n ) → u i . (2.4)For example, C = V /∂ V and C = Der ∂ ( V ) is the space of all derivations of V commuting with ∂ .The variational PVA differential d : C n PV → C n +1PV , for n ≥ , is defined by ( df ) λ ,...,λ n +1 ( v ⊗ · · · ⊗ v n +1 ) = ( − n n +1 X i =1 ( − i (cid:2) v iλ i f λ , i ˇ ...,λ n +1 ( v ⊗ i ˇ . . . ⊗ v n +1 ) (cid:3) + ( − n +1 X ≤ i Shuffles. A permutation σ ∈ S m + n is called an ( m, n ) - shuffle if σ (1) < · · · < σ ( m ) , σ ( m + 1) < · · · < σ ( m + n ) . The subset of ( m, n ) -shuffles is denoted by S m,n ⊂ S m + n . Observe that, by def-inition, S ,n = S n, = { } for every n ≥ . If either m or n is negative, we set S m,n = ∅ by convention.3.2. Monotone permutations. The following notion is due to Harrison [Har62](see also [GS87]), and it will be used in Section 6 to define Harrison cohomology. Definition 3.1. A permutation π ∈ S n is called monotone if, for each i = 1 , . . . , n ,one of the following two conditions holds:(a) π ( j ) < π ( i ) for all j < i ;(b) π ( j ) > π ( i ) for all j < i .(Not necessarily the same condition (a) or (b) holds for every i .) When (b) holds,we call i a drop of π . Also, π (1) = k is called the start of π (and we say that π starts at k ).We denote by M n ⊂ S n the set of monotone permutations, and by M kn ⊂ M n the set of monotone permutations starting at k .Here is a simple description of all monotone permutations starting at k . Let usidentify the permutation π ∈ S n with the n -tuple [ π (1) , . . . , π ( n )] . To construct all π ∈ M kn , we let π (1) = k . Then, for every choice of k − positions in { , . . . , n } we get a monotone permutation π as follows. In the selected positions we put thenumbers to k − in decreasing order from left to right; in the remaining positionswe write the numbers k + 1 to n in increasing order from left to right. (The selectedpositions are the drops of π .) Example 3.2. The only monotone permutation starting at is the identity, whilethe only monotone permutation starting at n is σ n = [ n n − · · · . (3.1) Example 3.3. Let n = 5 and k = 3 . The monotone permutations starting at are [3 2 1 4 5] , [3 2 4 1 5] , [3 2 4 5 1 ] , [3 4 2 1 5] , [3 4 2 5 1 ] , [3 4 5 2 1 ] , where we underlined the positions of the drops.Given a monotone permutation π , we denote by dr( π ) the sum of all the dropswith respect to π . According to the previous description, we can easily see that ( − dr( π ) = ( − k − sign( π ) , (3.2)if k is the start of π . .3. Graphs. For an oriented graph Γ , we denoted by V (Γ) the set of verticesof Γ , and by E (Γ) the set of edges. We call an oriented graph Γ an n - graph if V (Γ) = { , . . . , n } . Denote by G ( n ) the set of all n -graphs without tadpoles, andby G ( n ) the set of all acyclic n -graphs.An n -graph L will be called an n - line , or simply a line , if its set of edges is ofthe form { i → i , i → i , . . . , i n − → i n } , where { i , . . . , i n } is a permutation of { , . . . , n } .We have a natural left action of S n on the set G ( n ) : for the n -graph Γ and thepermutation σ , the new n -graph σ (Γ) is defined to be the same graph as Γ but withthe vertex which was labeled as i relabeled as σ ( i ) , for every i = 1 , . . . , n . So, ifthe n -graph Γ has an oriented edge i → j , then the n -graph σ (Γ) has the orientededge σ ( i ) → σ ( j ) . Obviously, S n permutes the set of n -lines. Example 3.4. Let Γ = σ = (6 5 4) and τ = (cid:18) (cid:19) , we have: σ ! = 1 2 3 4 5 6 ,and τ ! = 1 2 3 4 5 6 .3.4. Graphs of type k and proper k -lines. For s ≥ , let k = ( k , . . . , k s ) ∈ Z s ≥ and n = k + · · · + k s , and denote K = 0 and K t = k + · · · + k t , t = 1 , . . . , s , (3.3)so that K s = n . We denote by Γ k ∈ G ( n ) the standard k - line , union of connectedlines of lengths k , . . . , k s , with the labeling of the vertices ordered from left toright: Γ k = · · · K K +1 · · · K · · · K s − +1 · · · n (3.4) e allow some of the k i ’s to be zero, in which case the corresponding connectedcomponent of Γ k is empty. In the special case s = 1 we recover the standard n - line Γ n = ... n . (3.5)An arbitrary k - line is obtained by permuting the vertices of Γ k : Γ = i i · · · i k i i · · · i k · · · i s i s · · · i sk s (3.6)where the set of indices { i ab } is a permutation of { , . . . , n } . Note that, if Γ is a k -line, then it is a σ ( k ) -line for every permutation σ ∈ S s . Hence, when considering k -lines we can (and we will) assume that k ≤ · · · ≤ k s . We say that a k -line is proper if the following further condition holds on the indices of the vertices: i l = min { i l , . . . , i lk l } ∀ l = 1 , . . . , s . (3.7)We then let L ( n ) = n proper k -lines Γ ∈ G ( n ) with k ∈ Z s ≥ , s ≥ , k + · · · + k s = n o . (3.8)Note that, in order not to have repetitions in the set (3.8), we may assume that k ≤ · · · ≤ k s , and that, if k l = k l +1 , then i l < i l +11 . Obviously, Γ k ∈ L ( n ) forevery k ∈ Z s ≥ , while a permutation of Γ k does not necessarily lie in L ( n ) .Finally, we say that a graph Γ ∈ G ( n ) is of type k if it is disjoint union of s connected components of sizes k ≤ · · · ≤ k s . Obviously, any k -line is of type k .We can extend the definition of Γ k for k ∈ Z s ≥ by removing all ’s from k . Inparticular Γ is the empty graph.3.5. Cycle relations on graphs. Let F G ( n ) be the vector space with basis theset of graphs G ( n ) , and R ( n ) ⊂ F G ( n ) be the subspace spanned by the following cycle relations :(i) all Γ ∈ G ( n ) \ G ( n ) (i.e., graphs containing a cycle);(ii) all linear combinations P e ∈ C Γ \ e , where Γ ∈ G ( n ) and C ⊂ E (Γ) is anoriented cycle.By convention, F G (0) = F and R (0) = 0 .Note that reversing an arrow in a graph Γ ∈ G ( n ) gives us, modulo cycle relations,the element − Γ ∈ F G ( n ) . For example, for n = 3 , a cycle relation of type (ii) is: (3.9) Theorem 3.5 ([BDSHK20, Theorem 4.7]) . The set L ( n ) is a basis for the quotientspace F G ( n ) / R ( n ) . Harrison relations. The following result will be used in Section 7. Lemma 3.6. [BDSKV21, Lemma 4.8] Let Γ n be the standard n -line, as in (3.5) .For every m ∈ { , . . . , n } , the following identity holds: Γ n + ( − m X π ∈M mn π Γ n ∈ R ( n ) , (3.10) here the sum is over all monotone permutations π starting at m and the actionof S n on graphs is described in Section 3.3. Notation for subgraphs and collapsed graphs. Let us introduce the fol-lowing notation. For h ∈ { , . . . , n } and Γ ∈ G ( n ) , we denote by Γ \ h ∈ G ( n − the complete subgraph obtained from Γ by removing the vertex h and all edgesstarting or ending in h , and relabeling the vertices from to n − . Moreover, for i, j ∈ { , . . . , n } , we define the graph π ij (Γ) ∈ G ( n − obtained by collapsing thevertices i and j (and any edges between them) into a single vertex, numbered by , and renumbering the remaining vertices from to n − . Example 3.7. For example, if Γ = Γ \ Γ \ π (Γ) = π (Γ) = Γ ∈ G ( n ) and i ∈ { , . . . , n } , we denote by deg − Γ ( i ) the indegree of i in Γ ,namely the number of edges of Γ incoming to i , by deg +Γ ( i ) the outdegree of i in Γ ,namely the number of edges of Γ outcoming from i , and deg Γ ( i ) := deg − Γ ( i ) + deg +Γ ( i ) , the degree of i in Γ . For i, j ∈ { , . . . , n } , we also let ǫ Γ ( i, j ) := if i → j ∈ E (Γ) , − if i ← j ∈ E (Γ) , otherwise . Note that, since Γ ∈ G ( n ) , i → j and j → i cannot be both in E (Γ) .4. Classical PVA cohomology Space of classical cochains. Let V be a Poisson vertex algebra. The cor-responding classical PVA cohomology complex ( C cl , d ) is constructed as follows[BDSHK19]. The space C n cl of classical n -cochains consists of linear maps Y : F G ( n ) ⊗ V ⊗ n −→ V [ λ , . . . , λ n ] (cid:14) h ∂ + λ + · · · + λ n i , (4.1)mapping the n -graph Γ ∈ G ( n ) and the monomial v ⊗ · · · ⊗ v n ∈ V ⊗ n to thepolynomial Y Γ λ ,...,λ n ( v ⊗ · · · ⊗ v n ) , (4.2)satisfying the skewsymmetry conditions, cycle relations, and sesquilinearity condi-tions described below. he skewsymmetry conditions on Y say that, for each permutation σ ∈ S n , wehave Y σ (Γ) λ ,...,λ n ( v ⊗ · · · ⊗ v n ) = sign( σ ) Y Γ λ σ (1) ,...,λ σ ( n ) ( v σ (1) ⊗ · · · ⊗ v σ ( n ) ) , (4.3)where σ (Γ) is defined in Section 3.3.Recall that R ( n ) ⊂ F G ( n ) is the subspace spanned by the cycle relations (i) and(ii) from Section 3.5. The cycle relations on Y say that Y Γ = 0 for Γ ∈ R ( n ) . (4.4)Hence, Y induces a map on F G ( n ) / R ( n ) . As an example, observe that, by the firstcycle relation (i), changing orientation of a single edge of the n -graph Γ ∈ G ( n ) amounts to the change of sign of Y Γ .Let Γ = Γ ⊔· · ·⊔ Γ s be the decomposition of Γ as a disjoint union of its connectedcomponents, and let I , . . . , I s ⊂ { , . . . , n } be the sets of vertices of these connectedcomponents. For each Γ α we write λ Γ α = X i ∈ I α λ i , ∂ Γ α = X i ∈ I α ∂ i , (4.5)where ∂ i denotes the action of ∂ on the i -th factor in the tensor product V ⊗ n . Then,the sesquilinearity conditions on Y say that, for v ∈ V ⊗ n , Y Γ λ ,...,λ n ( v ) is a polynomial in λ Γ , . . . , λ Γ s , (4.6)(and not in the variables λ , . . . , λ n separately), and, for every α = 1 , . . . , s , Y Γ λ ,...,λ n (( ∂ Γ α + λ Γ α ) v ) = 0 . (4.7)Observe that the second sesquilinearity condition (4.7) implies Y Γ λ ,...,λ n ( ∂v ) = − n X i =1 λ i Y Γ λ ,...,λ n ( v ) = ∂ (cid:0) Y Γ λ ,...,λ n ( v ) (cid:1) , v ∈ V ⊗ n , (4.8)i.e. Y Γ : V ⊗ n → V [ λ , . . . , λ n ] (cid:14) h ∂ + λ + · · · + λ n i is an F [ ∂ ] -module homomorphism. Remark . When the graph Γ is connected, the first sesquilinearity condition(4.6) implies that Y Γ λ ,...,λ n ( v ) is a polynomial of λ + · · · + λ n ≡ − ∂ . Hence, it isan element of V [ λ + · · · + λ n ] (cid:14) h ∂ + λ + · · · + λ n i ≃ V . In this case, we will omit the subscript of Y Γ .By convention, for n = 0 the graph Γ is empty and s = 0 ; hence C = V /∂ V .Note also that C = End F [ ∂ ] V . .2. Differential. The classical PVA cohomology differential d : C n cl → C n +1cl isdefined by the following formula: ( dY ) Γ λ ,...,λ n +1 ( v ⊗ . . . ⊗ v n +1 )= X h : deg Γ ( h )=0 ( − n − h h v hλ h Y Γ \ hλ ,... h g ...,λ n +1 ( v ⊗ . . . h g . . . ⊗ v n +1 ) i + X h : deg Γ ( h )=1 j : ǫ Γ ( j,h ) =0 ( − deg +Γ ( h )+ n − h +1 Y Γ \ hλ ,... h g ...,λ j + x,...,λ n +1 ( v ⊗ . . . h g . . . ⊗ v n +1 ) (cid:0)(cid:12)(cid:12) x = λ h + ∂ v h (cid:1) + X i Formula (4.9) defines a differential on the space of classical cochains C cl = L n ≥ C n cl , i.e. d = 0 .Proof. As we will see in Section 4.3, formula (4.9) corresponds to the differential ofthe classical PVA cohomology defined in [BDSHK19] with an operadic aproach. (cid:3) Remark . The Poisson vertex algebra structure on V defines an element X ∈ C by X •−→• ( a ⊗ b ) = ab , X • • λ, − λ − ∂ ( a ⊗ b ) = [ a λ b ] . (4.10)The skewsymmetry of X is equivalent to the commutativity of ab and the skewsym-metry of [ a λ b ] , while the sesquilinearity of X is equivalent to the sesquilinearity of [ a λ b ] and the fact that ∂ is a derivation of ab . Moreover, the associativity for ab , theJacobi identity for [ a λ b ] and the Leibniz rule relating them, together are equivalentto the condition that dX = 0 , see [BDSHK19, Thm.10.7]. Example 4.4. Consider the completely disconnected graph Γ = • • · · · • . Thenin formula (4.9), all deg Γ ( h ) , ǫ Γ ( i, j ) and X ( i ) vanish, and we obtain ( dY ) • ··· • λ ,...,λ n +1 ( v ⊗ . . . ⊗ v n +1 )= n +1 X h =1 ( − n − h h v hλ h Y • ··· • λ ,... h g ...,λ n +1 ( v ⊗ . . . h g . . . ⊗ v n +1 ) i + X ≤ i Consider the case when Γ = Γ n +1 is the standard ( n + 1) -line (3.5).Then deg Γ ( h ) = 1 for the endpoints h = 1 or n + 1 , deg Γ ( h ) = 2 otherwise, sothat the first sum in (4.9) vanishes. The third sum vanishes as well because, when Γ ( i, j ) = 0 , the graph π ij (Γ) has a cycle. In the fourth sum we only have the termswith j = i + 1 . Thus we obtain ( dY ) Γ n +1 ( v ⊗ · · · ⊗ v n +1 )= ( − n +1 v Y Γ n ( v ⊗ · · · ⊗ v n +1 ) + Y Γ n ( v ⊗ · · · ⊗ v n ) v n +1 + n X i =1 ( − n + i − Y Γ n ( v ⊗ · · · ⊗ v i v i +1 ⊗ · · · ⊗ v n +1 ) . For the last term we used the skewsymmetry of Y to bring the factor v i v i +1 inposition i . This is the formula for the Hochschild differential [Hoc45].4.3. Proof of the formula for the differential. In the present paper, the for-mula (4.9) for the classical PVA cohomology differential d is taken as a definition.Here, we show how that formula is derived from the approach of [BDSHK19]. Thisimplies Theorem 4.2.Recall from [BDSHK19, Sec.10] the classical operad P cl (Π V ) , defined as follows.The space P cl (Π V )( n ) consists of maps (4.1) satisfying the cycle relations (4.4)and the sesquilinearity conditions (4.6)-(4.7). There is a natural action of thesymmetric group S n on P cl (Π V )( n ) defined by simultaneously permuting all the λ i ’s, the vectors v i ’s and the vertices of the graph Γ , and multiplying by the signof the permutation, since all vectors in Π V are odd. Explicitly (see [BDSHK19,Eq.(10.10)]) ( Y σ ) Γ λ ,...,λ n ( v ⊗ · · · ⊗ v n ) = sign( σ ) Y σ (Γ) λ σ − ,...,λ σ − n ) ( v σ − (1) ⊗ · · · ⊗ v σ − ( n ) ) . (4.11)Then the skewsymmetry conditions (4.3) are equivalent to the S n invariance of Y .Therefore C n cl = W n − (Π V ) = (cid:0) P cl (Π V )( n ) (cid:1) S n (4.12)is the space of fixed points under the action of the symmetric group S n in theclassical operad P cl (Π V ) .The composition products in P cl (Π V ) are given by [BDSHK19, Eq.(10.11)]. Herewe need the special case of ◦ -product (see [BDSHK19, Rem.10.3 and Eq.(8.18)]).For A ∈ P cl ( k ) , B ∈ P cl ( m ) and G ∈ G ( m + k − , the ◦ -product A ◦ B ∈P cl ( m + k − is given by ( A ◦ B ) Gλ ,...,λ m + k − ( v ⊗ · · · ⊗ v m + k − )= A ¯ G ′′ λ G ′ ,λ m +1 ,...,λ m + k − (cid:0) B G ′ λ + λ G + ∂ G ,...,λ m + λ Gm + ∂ Gm ( v ⊗ · · ·· · · ⊗ v m ) ⊗ v m +1 ⊗ · · · ⊗ v m + k − (cid:1) . (4.13)Here G ′ is the subgraph of G with vertices , . . . , m and all edges from G amongthese vertices; G ′′ is the subgraph of G that includes all edges of G not in G ′ ; and ¯ G ′′ is the graph with vertices labeled , m + 1 , . . . , m + k − and edges obtainedfrom the edges of G ′′ by replacing any vertex ≤ i ≤ m with , keeping the sameorientation. Finally, the graph G i (1 ≤ i ≤ m ) is the subgraph of G ′′ obtained fromthe connected component of the vertex i in G ′′ by removing from it the vertex i and all edges connected to i .By [BDSHK19, Thm.3.4], W cl (Π V ) = L k ≥− W k cl (Π V ) has the structure of a Z -graded Lie superalgebra. In particular, for X ∈ W (Π V ) and Y ∈ W n − (Π V ) , heir Lie bracket is given by [BDSHK19, Eqs. (3.13), (3.16)]: [ X, Y ] = X σ ∈ S n, ( X ◦ Y ) σ − + ( − n X τ ∈ S ,n − ( Y ◦ X ) τ − , (4.14)where S n, and S ,n − denote the sets of shuffles from Section 3.1.The element X ∈ C = W (Π V ) in (4.10) is odd and satisfies [ X, X ] = 0 ,see [BDSHK19, Thm.10.7]. Hence, (ad X ) = 0 , and d = ad X was taken as thedifferential of the classical PVA cohomology complex in [BDSHK19, Def.10.8]. As aconsequence, the classical PVA cohomology H cl ( V ) has an induced Lie superalgebrastructure. Here we show that the differential d in (4.9) coincides with ad X from(4.14): Proposition 4.6. For Y ∈ C n cl = W n − (Π V ) , we have dY = [ X, Y ] .Proof. Recalling from Section 3.1 the definition of shuffles, we have S n, = { σ h } n +1 h =1 where σ h = · · · · · · · · · n n + 11 · · · h g · · · n + 1 h ! , and S ,n − = { τ i,j } ≤ i To a Poisson vertex algebra V we associate two cohomology complexes: the vari-ational PVA cohomology complex C PV introduced in Section 2.2, and the classicalPVA cohomology complex C cl introduced in Section 4. Recall also from Remark2.2 and Section 4.3, that these complexes have the structure of a Lie superalgebra.It is natural to ask what is the relation between these two cohomology theories. Apartial answer was provided by the following: Theorem 5.1 ([BDSHK19, Theorem 11.4]) . We have a canonical injective homo-morphism of Lie superalgebras H PV ( V ) ֒ → H cl ( V ) (5.1) nduced by the map that sends f ∈ C n PV to Y ∈ C n cl such that Y • ··· • = f and Y Γ = 0 if | E (Γ) | 6 = ∅ . It was left as an open question in [BDSHK19] whether (5.1) is, in fact, an iso-morphism. The main result of this paper will be the proof that this is indeed thecase, under some regularity assumption on V . Theorem 5.2. Assuming that the PVA V , as a differential algebra, is a finitely-generated algebra of differential polynomials, the Lie superalgebra homomorphism (5.1) is an isomorphism. The remainder of the paper will be devoted to the proof of Theorem 5.2. InSection 6, we introduce a new cohomology complex, called the sesquilinear Harrisoncohomology complex. In Section 7, we define a filtration of the classical PVAcohomology complex and we prove that its associated graded is isomorphic to thesesquilinear Harrison cohomology complex. We then show, in Section 8 that thecohomology of the sesquilinear Harrison cohomology complex vanishes in positivedegree. Using that, we complete, in Section 9, the proof of Theorem 5.2.6. Sesquilinear Harrison cohomology In the present Section we introduce the sesequilinear Hochschild and Harrisoncohomology complexes. In order to do so, we first review the differential Hochschildand Harrison cohomology complexes.6.1. Differential Hochschild cohomology complex. Let A be an associative al-gebra over the base field F , and M be an A -bimodule. The corresponding Hochschildcohomology complex of A with coefficients in M is defined as follows [Hoc45]. Thespace of n -cochains is Hom( A ⊗ n , M ) , (6.1)and the differential d : Hom( A ⊗ n , M ) → Hom( A ⊗ n +1 , M ) is defined by ( df )( a ⊗ · · · ⊗ a n +1 ) = a f ( a ⊗ · · · ⊗ a n +1 )+ n X i =1 ( − i f ( a ⊗ · · · ⊗ a i − ⊗ a i a i +1 ⊗ a i +2 ⊗ · · · ⊗ a n +1 )+ ( − n +1 f ( a ⊗ · · · ⊗ a n ) a n +1 . (6.2)If A is an associative algebra with a derivation ∂ : A → A , and M is a differentialbimodule over A (i.e., the action of ∂ is compatible with the bimodule structure), wemay consider the differential Hochschild cohomology complex by taking the subspaceof n -cochains Hom F [ ∂ ] ( A ⊗ n , M ) . (6.3)It is clear by the definition (6.2) that the differential d maps Hom F [ ∂ ] ( A ⊗ n , M ) to Hom F [ ∂ ] ( A ⊗ n +1 , M ) . Hence, we have a cohomology subcomplex. .2. Differential Harrison cohomology complex. Let us now recall Harrison’soriginal definition of his cohomology complex [Har62], see also [GS87, L13]. Let A be a commutative associative algebra, and M be a symmetric A -bimodule, i.e.,such that am = ma , for all a ∈ A and m ∈ M . For every < k ≤ n define thefollowing endomorphism on the space Hom( A ⊗ n , M ) : ( L k F )( a ⊗ · · · ⊗ a n ) := X π ∈M kn ( − dr ( π ) F ( a π (1) ⊗ · · · ⊗ a π ( n ) ) , (6.4)where M kn is the set of monotone permutations starting at k , defined in Section3.2.A Harrison n -cochain is defined as a Hochschild n -cochain F ∈ Hom( A ⊗ n , M ) fixed by all operators L k : L k F = F , for every ≤ k ≤ n . (6.5)We will denote by C n Har ( A, M ) ⊂ Hom( A ⊗ n , M ) (6.6)the space of Harrison n -cochains.Furthermore, if A is a differential algebra with a derivation ∂ : A → A , and M is a symmetric differential bimodule, we may consider the space of differentialHarrison n -cochains C n∂, Har ( A, M ) ⊂ Hom F [ ∂ ] ( A ⊗ n , M ) , (6.7)again defined by Harrison’s conditions (6.5). Proposition 6.1 ([GS87, BDSKV21]) . (a) The Harrison cohomology complex ( C Har ( A, M ) , d ) is a subcomplex of the Hochschildcohomology complex.(b) If A is a differential algebra, with a derivation ∂ : A → A , the differential Har-rison cohomology complex ( C ∂, Har ( A, M ) , d ) is a subcomplex of the differentialHochschild cohomology complex. The cohomology of the complex ( C ∂, Har ( A, M ) , d ) is the differential Harrisoncohomology of A with coefficients in M , and is denoted by H ∂, Har ( A, M ) . Clearly, H ∂, Har ( A, M ) = M and H ∂, Har ( A, M ) = Der ∂ ( A, M ) is the space of all derivationsfrom A to M commuting with ∂ . Remark . It follows from [GS87] that H n∂, Har ( A, M ) is a direct summand of thedifferential Hochschild cohomology, for n ≥ .6.3. The sesquilinear Hochschild cohomology complex. Let V be an asso-ciative differential algebra with derivation ∂ , and let M be a differential bimoduleover V . Fix s ≥ and let, as in Section 3.4, k = ( k , . . . , k s ) ∈ Z s ≥ , K = 0 , K t = k + · · · + k t , t = 1 , . . . , s , and n = K s = k + · · · + k s . Given v , . . . , v n ∈ V , we denote v tk = v K t − +1 ⊗ · · · ⊗ v K t ∈ V ⊗ k t , t = 1 , . . . , s , (6.8)so that v := v ⊗ · · · ⊗ v n = v k ⊗ · · · ⊗ v sk ∈ V ⊗ n . (6.9) ote that we allow k t to be , and in this case v tk = 1 ∈ F .The s - sesquilinear Hochschild cohomology complex ( C s sesq , Hoc ( V , M ) , d ) of V withcoefficients in M , is defined as follows. First we introduce the space C k Hoc of alllinear maps F Λ ,..., Λ s : V ⊗ n → M [Λ , . . . , Λ s ] / h ∂ + Λ + · · · + Λ s i , v F Λ ,..., Λ s ( v ) , (6.10)satisfying the sesquilinearity conditions ( t = 1 , . . . , s ), F Λ ,..., Λ s ( v k ⊗ · · · ∂v tk · · · ⊗ v sk ) = − Λ t F Λ ,..., Λ s ( v ) . (6.11)For every t = 1 , . . . , s , we define the t -th differential d ( t ) : C k Hoc → C k + e t Hoc , where e t is the s -tuple with all except for in position t , given by ( d ( t ) F ) Λ ,..., Λ s ( v ⊗ · · · ⊗ v n +1 )= ( − K t − (cid:0)(cid:12)(cid:12) x = ∂ v K t − +1 (cid:1) F Λ ,..., Λ t + x,..., Λ s ( v ⊗ . . . K t − +1 g · · · ⊗ v n +1 )+ K t X i = K t − +1 ( − i F Λ ,..., Λ s ( v ⊗ · · · ⊗ v i v i +1 ⊗ · · · ⊗ v n +1 )+ ( − K t +1 F Λ ,..., Λ t + x,..., Λ s ( v ⊗ . . . K t +1 g · · · ⊗ v n +1 ) (cid:0)(cid:12)(cid:12) x = ∂ v K t +1 (cid:1) . (6.12)In other words, up to the overall sign ( − K t − and up to the shift by ∂ in thevariable Λ t , this is the Hochschild cohomology differential of F , viewed as a functionof v tk + e t = v K t − +1 ⊗ · · · ⊗ v K t +1 , considering all other vectors v t ′ k + e t with t ′ = t as fixed parameters. In equation (6.12) and throughout the rest of the paper, thesubstitution | x = ∂ means that the polynomial in x is expanded, x is replaced by ∂ , and it is applied, in this case, to the vector v K t − +1 in the first term of theright-hand side, and to the vector v K t +1 in the last term. Remark . Note that M [Λ ] / h ∂ + Λ i ≃ M . Using this, we identify the -sesquilinear Hochschild cohomology complex with the differential Hochschild coho-mology complex, defined in Section 6.1. Remark . Note that, for s > , by the sesquilinearity condition (6.11), we have C k Hoc = 0 if one of the k i ’s is zero. Theorem 6.5. For each k ∈ Z s ≥ , equation (6.12) gives well defined maps d ( t ) : C k Hoc → C k + e t Hoc , t = 1 , . . . , s , which are anticommuting differentials: d ( t ) d ( t ′ ) = − d ( t ′ ) d ( t ) for all t, t ′ = 1 , . . . , s . Hence, we get a Z s ≥ -graded s -complex, (cid:16) M k ∈ Z s ≥ C k Hoc , d (1) , . . . , d ( s ) (cid:17) . (6.13) As a consequence, letting C s,n Hoc = M k : K s = n C k Hoc and d = s X t =1 d ( t ) : C s,n Hoc → C s,n +1Hoc , (6.14) we get a cohomology complex (cid:0) C s sesq , Hoc ( V , M ) = L n ≥ C s,n Hoc , d (cid:1) . roof. In order to prove that d ( t ) is well defined, we first check that, if F Λ ,..., Λ s ( v ) = ( ∂ + Λ + · · · + Λ s ) G Λ ,..., Λ s ( v ) , for every v ∈ V ⊗ n , then the right-hand side of (6.12) lies in h ∂ + Λ + · · · + Λ s i .Indeed, using the fact that ∂ is a derivation of the product in V , the first term ofthe right-hand side is equal to ( − K t − ( ∂ + Λ + · · · + Λ s ) (cid:16) G Λ ,..., Λ t + ∂,..., Λ s ( v ⊗ . . . K t − +1 g · · · ⊗ v n ) → v K t − +1 (cid:17) . The second and third term are similar.Next, we check that d ( t ) F satisfies the sesquilinearity conditions (6.10) for every t ′ ∈ { , . . . , s } in place of t and for k + e t in place of k . Let v = v k + e t ⊗· · ·⊗ v k + e t be the factorization of v ∈ V ⊗ ( n +1) as in (6.9). If ∂ acts on the factor v t ′ k + e t with t ′ = t , then in each term of the right-hand side of (6.12) we get a factor of − Λ t ′ ,by the sesquilinearity of F . In the case when t ′ = t we observe that v tk + e t = v K t − +1 ⊗ w , where w = v K t − +2 ⊗ · · · ⊗ v K t +1 . Then ∂v tk + e t = ∂v K t − +1 ⊗ w + v K t − +1 ⊗ ∂w . Hence, if we replace v tk + e t by ∂v tk + e t in ( d ( t ) F ) Λ ,..., Λ s ( v ) , the first term in theright-hand side of (6.12) becomes, up to the sign ( − K t − , F Λ ,..., Λ t + ∂,..., Λ s ( v ⊗ · · · ∂w · · · ⊗ v n ) → v K t − +1 + F Λ ,..., Λ t + ∂,..., Λ s ( v ⊗ · · · w · · · ⊗ v n ) → ∂v K t − +1 = F Λ ,..., Λ t + ∂,..., Λ s ( v ⊗ · · · w · · · ⊗ v n ) → ( − Λ t − ∂ ) v K t − +1 + F Λ ,..., Λ t + ∂,..., Λ s ( v ⊗ · · · w · · · ⊗ v n ) → ∂v K t − +1 = − Λ t F Λ ,..., Λ t + ∂,..., Λ s ( v ⊗ · · · w · · · ⊗ v n ) → v K t − +1 . The other two terms in (6.12) are similar, proving the sesquilinearity of d ( t ) F .Next, we prove that d ( t ) and d ( t ′ ) anticommute for all t, t ′ . For t ′ = t , d ( t ) and d ( t ′ ) act on a different set of variables, hence, due to the overall signs, theyanticommute. For t ′ = t we need to show that ( d ( t ) ) = 0 , which is similar tothe proof that the square of the Hochschild differential is zero. For simplicity ofnotation, let us check this for t = 1 . Then K = k will be denoted simply as k .Applying formula (6.12) twice, we obtain: ( d (1) ( d (1) F )) Λ ,... ( v ⊗ · · · ⊗ v k +2 ⊗ · · · )= (cid:0)(cid:12)(cid:12) x = ∂ v (cid:1) ( d (1) F ) Λ + x,... ( v ⊗ · · · ⊗ v k +2 ⊗ · · · )+ k +1 X i =1 ( − i ( d (1) F ) Λ ,... ( v ⊗ · · · ⊗ v i v i +1 ⊗ · · · ⊗ v k +2 ⊗ · · · )+ ( − k +2 ( d (1) F ) Λ + x,... ( v ⊗ · · · ⊗ v k +1 ⊗ · · · ) (cid:0)(cid:12)(cid:12) x = ∂ v k +2 (cid:1) = (cid:0)(cid:12)(cid:12) x = ∂ v (cid:1)(cid:0)(cid:12)(cid:12) y = ∂ v (cid:1) F Λ + x + y,... ( v ⊗ · · · ⊗ v k +2 ⊗ · · · )+ k +1 X j =2 ( − j − (cid:0)(cid:12)(cid:12) x = ∂ v (cid:1) F Λ + x,... ( v ⊗ · · · ⊗ v j v j +1 ⊗ · · · ⊗ v k +2 ⊗ · · · )+ ( − k +1 (cid:0)(cid:12)(cid:12) x = ∂ v (cid:1) F Λ + x + y,... ( v ⊗ · · · ⊗ v k +1 ⊗ · · · ) (cid:0)(cid:12)(cid:12) y = ∂ v k +2 (cid:1) (cid:0)(cid:12)(cid:12) x = ∂ ( v v ) (cid:1) F Λ + x,... ( v ⊗ · · · ⊗ v k +2 ⊗ · · · )+ k +1 X i =2 ( − i (cid:0)(cid:12)(cid:12) x = ∂ v (cid:1) F Λ + x,... ( v ⊗ · · · ⊗ v i v i +1 ⊗ · · · ⊗ v k +2 ⊗ · · · )+ k +1 X i =3 i − X j =1 ( − i + j F Λ ,... ( v ⊗ · · · ⊗ v j v j +1 ⊗ · · · ⊗ v i v i +1 ⊗ · · · ⊗ v k +2 ⊗ · · · ) − k +1 X i =2 F Λ ,... ( v ⊗ · · · ⊗ v i − v i v i +1 ⊗ · · · ⊗ v k +2 ⊗ · · · )+ k X i =1 F Λ ,... ( v ⊗ · · · ⊗ v i v i +1 v i +2 ⊗ · · · ⊗ v k +2 ⊗ · · · )+ k − X i =1 k +1 X j = i +2 ( − i + j − F Λ ,... ( v ⊗ · · · ⊗ v i v i +1 ⊗ · · · ⊗ v j v j +1 ⊗ · · · ⊗ v k +2 ⊗ · · · )+ k X i =1 ( − i + k +1 F Λ + x,... ( v ⊗ · · · ⊗ v i v i +1 ⊗ · · · ⊗ v k +1 ⊗ · · · ) (cid:0)(cid:12)(cid:12) x = ∂ v k +2 (cid:1) + F Λ + x,... ( v ⊗ · · · ⊗ v k ⊗ · · · ) (cid:0)(cid:12)(cid:12) x = ∂ ( v k +1 v k +2 ) (cid:1) + ( − k (cid:0)(cid:12)(cid:12) x = ∂ v (cid:1) F Λ + x + y,... ( v ⊗ · · · ⊗ v k +1 ⊗ · · · ) (cid:0)(cid:12)(cid:12) y = ∂ v k +2 (cid:1) + k X i =1 ( − k + i F Λ + x,... ( v ⊗ · · · ⊗ v i v i +1 ⊗ · · · ⊗ v k +1 ⊗ · · · ) (cid:0)(cid:12)(cid:12) x = ∂ v k +2 (cid:1) − F Λ + x + y,... ( v ⊗ · · · ⊗ v k ⊗ · · · ) (cid:0)(cid:12)(cid:12) x = ∂ v k +1 (cid:1)(cid:0)(cid:12)(cid:12) y = ∂ v k +2 (cid:1) . An inspection of the right-hand side shows that all terms pairwise cancel with eachother. The remaining assertions of the theorem are an immediate consequence. (cid:3) The symmetric group S s acts naturally on each C s,n Hoc as follows. A permutation σ ∈ S s maps C k Hoc → C σ ( k )Hoc , where we recall that σ ( k ) = ( k σ − (1) , . . . , k σ − ( s ) ) .Given F ∈ C k Hoc , its image F σ ∈ C σ ( k )Hoc is given by ( F σ ) Λ ,..., Λ s ( v ) = ± F Λ σ − ,..., Λ σ − s ) ( v σ − (1) k ⊗ · · · ⊗ v σ − ( s ) k ) , (6.15)where the sign in the right-hand side is ± = ( − P t Let V be a commuta-tive associative differential algebra, and M be a differential symmetric V -bimodule.We define the s - sesquilinear Harrison cohomology complex ( C s sesq , Har ( V , M ) , d ) asa subcomplex of the sesquilinear Hochschild cohomology complex of V with co-efficients in M . First, let C k Har be the subspace of C k Hoc consisting of all linear aps F Λ ,..., Λ s as in equation (6.10) satisfying, in addition to the sesquilinearityconditions (6.11), the following Harrison conditions ( ≤ t ≤ s, ≤ m ≤ k t ): L ( t ) m F := X π ∈M mkt ( − dr ( π ) F Λ ,..., Λ s ( v k ⊗ · · · π − ( v tk ) · · · ⊗ v sk ) = F Λ ,..., Λ s ( v ) , (6.18)where M mk t is the set of monotone permutations in S k t starting at m , cf. (6.4). Proposition 6.6. For every k ∈ Z ≥ , ≤ t, t ′ ≤ s and ≤ m ≤ k t we have d ( t ) L ( t ′ ) m = L ( t ′ ) m d ( t ) . In particular, we obtain a cohomology subcomplex C s sesq , Har ( V , M ) of C s sesq , Hoc ( V , M ) given by C s,n Har = M k : K s = n C k Har and d = s X t =1 d ( t ) : C s,n Har → C s,n +1Har . (6.19) Proof. For t = t ′ , the operators d ( t ) and L ( t ′ ) m commute because they act on differentsets of variables, v tk and v t ′ k respectively. For t = t ′ , the equation d ( t ) L ( t ) m = L ( t ) m d ( t ) holds by a straightforward computation, which is similar to the proof thatthe Harrison cohomology complex is a subcomplex of the Hochschild complex, see[GS87]. (cid:3) Proposition 6.7. Equation (6.15) gives a well defined action of the symmetricgroup S s on C s,n Har , which maps C k Har to C σ ( k )Har . Moreover, σ commutes with thedifferential d in (6.14) .Proof. Recall from the end of the previous subsection, that we have an action σ which maps C k Hoc to C σ ( k )Hoc . We only need to check that this action preserves theHarrison conditions (6.18). This is true because L ( t ) m acts on the vectors from the t -th group v ( t ) k , while σ permutes the groups. The claim that σ commutes with d follows from (6.17). (cid:3) Thanks to Proposition 6.7, we get a cohomology subcomplex given by the S s -invariants: (cid:16) C s sym , Har ( V , M ) = M n ≥ ( C s,n Har ) S s , d (cid:17) , s ≥ . (6.20)We will call this complex the symmetric s - sesquilinear Harrison cohomology com-plex of V with coefficients in M . The degenerate case s = 0 corresponds to setting k = ∅ , n = K = 0 , v = 1 ∈ V ⊗ = F . In this case, the symmetric ( s = 0) -sesquilinear Harrison cohomology complex C s =0sym , Har ( V , M ) is concentrated in degree n = 0 and it is equal to M/∂M , with thezero differential. Remark . As in Remark 6.3, we have M [Λ ] / h ∂ + Λ i ≃ M , and, using this, weidentify the ( s = 1) -sesquilinear Harrison cohomology complex with the differentialHarrison cohomology complex, defined in Section 6.1. . Relation between symmetric sesquilinear Harrison and classicalPVA cohomology complexes We introduce a filtration of the classical PVA complex ( C cl , d ) defined in Section4.1. For a graph Γ we let s (Γ) be the number of connected components of Γ . Recallthat for an acyclic graph Γ ∈ G ( n ) with n vertices, we have s (Γ) = n − | E (Γ) | . Wethen let, for s ∈ Z , F s C cl = (cid:8) Y ∈ C cl (cid:12)(cid:12) Y Γ = 0 for every graph Γ such that s (Γ) < s (cid:9) . (7.1)This defines a decreasing filtration of vector spaces. Note that F s C cl = C cl for s ≤ , because any graph Γ with n vertices and | E (Γ) | > n has a cycle and therefore Y Γ = 0 by definition. The same argument also gives F C n cl = C n cl for n ≥ , becauseany non-empty graph Γ with n vertices and | E (Γ) | > n − has a cycle. However, F C = 0 , and moreover, F s C n cl = 0 for s > n , since s (Γ) = n − | E (Γ) | ≤ n . Proposition 7.1. The filtration (7.1) is preserved by the action of the differential d defined by (4.9) .Proof. For Y ∈ F s C n cl , we need to prove that dY ∈ F s C n +1cl . This means that, forany Γ ∈ G ( n +1) such that s (Γ) < s , we have ( dY ) Γ = 0 . Let us consider separatelythe four terms in the right-hand side of (4.9).First, if deg Γ ( h ) = 0 , then h is an isolated vertex of Γ and s (Γ \ h ) = s (Γ) − The case s = 0 is obvious, so we shall assume s ≥ . Clearly, for k ∈ Z s ≥ ,we have s (Γ k ) ≤ s . Hence, for Y ∈ F s +1 C n cl , we have Y Γ k = 0 , and the map (7.5)is well defined.Next, we show that Y Γ k ∈ C k Har for each k ∈ Z s ≥ . By the first sesquilinearitycondition (4.6), Y Γ k is a map V ⊗ n → V [Λ , . . . , Λ s ] / h ∂ + Λ + · · · + Λ s i , since Λ t = λ (Γ k ) t . The second sesquilinearity condition (4.7) for Y implies the sesquilin-earity (6.11) of Y Γ k . Moreover, the Harrison conditions (6.18) for Y Γ k followfrom Lemma 3.6, or more precisely from equation (3.10) applied to the t -th con-nected component (Γ k ) t = Γ k t of Γ k , and the cycle relations (4.4) for Y . Hence, Y Γ k ∈ C k Har , as stated.In order to check that the right-hand side of equation (7.5) is invariant under thesymmetric group S s , pick a permutation σ ∈ S s and consider its action on Y Γ k ,for a fixed k ∈ Z s ≥ . Using equation (6.15), we find (( Y Γ k ) σ ) Λ ,..., Λ s ( v ) = ± Y Γ k Λ σ − ,..., Λ σ − s ) ( v σ − (1) k ⊗ · · · ⊗ v σ − ( s ) k ) . (7.6)where ± is as in (6.16). Let e σ ∈ S n be the permutation e σ ( K t − + i ) = k σ − (1) + · · · + k σ − ( σ ( t ) − + i , t = 1 , . . . , s, i = 1 , . . . , k t . (7.7)This permutation is defined so that v e σ − (1) ⊗ · · · ⊗ v e σ − ( n ) = v σ − (1) k ⊗ · · · ⊗ v σ − ( s ) k . (7.8)Indeed, we have, by (6.8), v σ − (1) k ⊗ · · · ⊗ v σ − ( s ) k = ( v K σ − − +1 ⊗ · · · ⊗ v K σ − ) ⊗ ( v K σ − − +1 ⊗ . . . · · · ⊗ v K σ − ) ⊗ · · · ⊗ ( v K σ − s ) − +1 ⊗ · · · ⊗ v K σ − s ) ) . n the other hand, we obviously have v e σ − (1) ⊗ · · · ⊗ v e σ − ( n ) = ( v e σ − (1) ⊗ · · · ⊗ v e σ − ( k σ − ) ) ⊗ ( v e σ − ( k σ − +1) ⊗ . . . · · · ⊗ v e σ − ( k σ − + k σ − ) ) ⊗ · · · ⊗ ( v e σ − ( k σ − + ··· + k σ − s − +1) ⊗ . . . · · · ⊗ v e σ − ( k σ − + ··· + k σ − s ) ) ) . The above two formulas match thanks to the definition (7.7) of e σ with t replacedby σ − ( t ) . Notice also that, for the same reason, (cf. (7.4)) λ e σ − ( k σ − + ··· + k σ − t − +1) + · · · + λ e σ − ( k σ − + ··· + k σ − t ) ) = Λ σ − ( t ) , (7.9)and e σ (Γ k ) = Γ σ ( k ) , (7.10)or, equivalently, e σ − (Γ k ) = Γ σ − ( k ) . We then use the skewsymmetry of Y (4.3)with respect to e σ − evaluated on the graph Γ k : Y e σ − (Γ k ) λ ,...,λ n ( v ⊗ · · · ⊗ v n ) = sign( e σ ) Y Γ k λ e σ − ,...,λ e σ − n ) ( v e σ − (1) ⊗ · · · ⊗ v e σ − ( n ) ) . (7.11)Notice that the ± sign in (7.6) is precisely sign( e σ ) . Hence, combining equations(7.6)–(7.11), we get ( Y Γ k ) σ = Y Γ σ − k ) . (7.12)As a consequence, the sum in the right-hand side of (7.5) is S s -invariant, as claimed.Next, we observe that the map (7.5) is injective. Indeed, if P k ∈ Z s ≥ : K s = n Y Γ k =0 in C s,n Har = L k : K s = n C k Har , then Y Γ k = 0 for every k ∈ Z s ≥ , and therefore, byTheorem 3.5, Y Γ = 0 whenever s (Γ) ≤ s . Hence, Y ∈ F s +1 C n cl , so its image in gr s C n cl is zero.Now we prove that (7.5) is surjective. Take an element F = X k F k ∈ C s,n Har = M k : K s = n C k Har , which is invariant under the action of the symmetric group S s . We want to construct Y ∈ F s C n cl such that Y Γ k = F k for every k ∈ Z s ≥ . Note that, by Remark 6.4,we can restrict to k ∈ Z s> . In the degenerate case s = 1 and k = 0 , we have n = 0 and in this case the claim is obvious. First, we define Y ∈ P cl (Π V )( n ) , seeSection 4.3. Recall by Theorem 3.5 that the proper k -lines Γ ∈ L ( n ) , defined by(3.6), (3.7), form a basis for the vector space F G ( n ) / R ( n ) , if k ≤ · · · ≤ k s and i l < i l +11 whenever k l = k l +1 . Hence, it is enough to define Y Γ for each proper k -line Γ satisfying these conditions. Given such Γ , there is a permutation τ ∈ S n such that Γ = τ (Γ k ) , and we set Y Γ λ ,...,λ n ( v ⊗ · · · ⊗ v n ) = sign( τ ) F k Λ ,..., Λ s ( v τ (1) , . . . , v τ ( n ) ) , (7.13)where the Λ t ’s are as in (7.4). This is well defined, since if τ ∈ S n fixes Γ k , then τ = e σ for some σ ∈ S s fixing k , and in this case the right-hand side of (7.13)equals F k Λ ,..., Λ s ( v , . . . , v n ) by the S s -symmetry of F . The cycle relations (4.4)and the first sesquilinearity condition (4.6) on Y hold by construction. The secondsesquilinearity condition (4.7) follows immediately from the sesquiinearity (6.11) of F . We are left to check that the map Y defined by (7.13) satisfies the skewsymmetry(4.3), or equivalently, the S n -invariance Y = Y σ , σ ∈ S n , with respect to the action s = 1 , for which the sesquiinearHarrison complex is equivalent to the differential Harrison complex, and the claimwas proved in [BDSKV21, Lemm4.9]. In the second case, when the permutation σ permutes the lines, the σ -invariance of Y holds by construction.Finally, we show that the map (7.5) commutes with the action of the differentials(4.9) and (6.14). Recall that the differential (4.9) induces, in the associated gradedcomplex gr s C cl the differential (7.2). Let us evaluate the right-hand side of (7.2) for Γ = Γ k . In the first sum, h is a vertex of degree , hence it must be the beginningor end point of one of the s lines in Γ k , and j is the vertex adjacent to it in the line.Hence, when h is the first vertex of the t -th line, we get the first term of (6.12),while when h is the last vertex of the t -th line, we get the third term of (6.12).Furthermore, in the second sum of the right-hand side of (7.2), the only non-zeroterms have ǫ Γ k ( i, j ) = 1 , which means that i and j are consecutive vertices of thesame line in Γ k . When they are in the t -th line we recover the second term of(6.12). This completes the proof. (cid:3) Vanishing of the sesquilinear Harrison cohomology In this section, we prove a vanishing theorem for the (symmetric) sesquilin-ear Harrison cohomology, introduced in Section 6.4. First, we recall some basicfacts about the Hochschild homology and cohomology, and a weak form of theHochschild–Kostant–Rosenberg (HKR) Theorem. Next, we give the proof by P.Etingof of an analogous statement for the differential Hochschild cohomology. Wegeneralize this to the sesquilinear Hochschild cohomology, introduced in Section6.3, to derive the vanishing theorem for the (symmetric) sesquilinear Harrison co-homology.8.1. The Bar complex. Let A be an associative F -algebra. Its Bar-resolution B • ( A ) is a complex of A - A -bimodules with B k ( A ) = A ⊗ · · · ⊗ A | {z } k +2 –times , k ≥ , (8.1)where the differential d : B k ( A ) → B k − ( A ) is given by d (cid:0) a ⊗ · · · ⊗ a k +1 (cid:1) = k X i =0 ( − i a ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ a k +1 , k ≥ . Let A op be A with the opposite product and A e = A ⊗ A op . Then B ( A ) is a complexof left A e -modules by letting ( a ⊗ b ) · a ⊗ · · · ⊗ a k +2 = a · a ⊗ · · · ⊗ a k +2 · b. Any A - A -bimodule M can be viewed as a right A e -module by letting m · ( a ⊗ b ) = b · m · a . Then B • ( A, M ) := M ⊗ A e B ( A ) is a complex of F -vector spaces. The homology of this complex is known as the Hochschild homology of A with coefficients in M and is denoted by HH • ( A, M ) .Given an A - A -bimodule M , we obtain a complex of F -vector spaces C • ( A, M ) := Hom A - A -bimod (cid:0) B • ( A ) , M (cid:1) . he homology of this complex is known as the Hochschild cohomology of A withcoefficients in M . It is easy to see that this cohomology coincides with the onedefined in Section 6.1.For a unital algebra A , we will use the normalized Hochschild complex ¯ C • ( A, M ) [L13, 1.5.7] consisting on Hochschild cochains f ∈ C • ( A, A ) vanishing on elements ofthe form a ⊗ · · ·⊗ a k , where one of the a j is . The inclusion ¯ C • ( A, A ) ֒ → C • ( A, A ) is a quasi-isomorphism. Indeed the map a ⊗ · · · ⊗ a k +2 → ⊗ a ⊗ · · · ⊗ a k +2 , induces a homotopy between the identity map of C • ( A, A ) and its projection to ¯ C • ( A, A ) . Suppose that the algebra A is unital and augmented, with an augmen-tation ideal A + ; in this case we have ¯ C i ( A, A ) = Hom F ( A ⊗ i + , A ) . (8.2)8.2. Kähler differentials. Let A be an associative commutative F -algebra, and I ⊂ A ⊗ A be the kernel of the multiplication map A ⊗ A → A . The A -module Ω A := I/I is called the module of Kähler differentials of A .For an A -module M , a derivation of A with values in M is a linear map D ∈ Hom F ( A, M ) satisfying D ( a · b ) = a · D ( b ) + b · D ( a ) . The space of all derivations Der( A, M ) is an A -module and we have Der( A, M ) ≃ Hom A (Ω A , M ) . In particular, the identity map of Ω A gives a derivation d ∈ Der( A, Ω A ) ; explicitly, da = a ⊗ − ⊗ a mod I . We define the module of n -forms by Ω nA := ^ nA Ω A , n ≥ . Let V be a F -vector space, and consider the free commutative associative unitalalgebra A = S ( V ) generated by V . In this case, Ω nA ≃ A ⊗ ^ n V, since Ω A is a free A -module of rank = dim V . We view Ω • A = L n ≥ Ω nA as a complexwith zero differential. We have the following map of complexes ε : Ω • A → B • ( A, A ) ,called the antisymmetrization map , defined by ε (cid:0) a ⊗ v ∧ · · · ∧ v n (cid:1) = X σ ∈ S n sign( σ ) a ⊗ v σ − (1) ⊗ · · · ⊗ v σ − ( n ) . (8.3) Theorem 8.1 (HKR Theorem [L13, Thm. 3.2.2]) . Let A = S ( V ) as above. Thenthe antisymmetrization map ε , given by (8.3) , is a quasi-isomorphism. In partic-ular, we have an isomorphism ε ∗ : Ω nA ∼ −→ HH n ( A, A ) for all n ≥ induced inhomology. Its inverse is given by the surjective map π ∗ : HH • ( A, A ) → Ω • A , π ∗ (cid:0) a ⊗ · · · ⊗ a k (cid:1) = a da ∧ · · · ∧ da k . Similarly, we have: heorem 8.2. For A = S ( V ) as above, the inclusion of complexes π ♯ : ^ • A Der( A, A ) ֒ → C • ( A, A ) is a quasi-isomoprhism. Consequently, we have an isomorphism of cohomologygroups ε ♯ ∗ : HH • ( A, A ) → ^ • A Der( A, A ) ≃ A ⊗ ^ • V ∗ , defined as the inverse of the map π ♯ ∗ induced by π ♯ in cohomology. The differential setting. Let now A be a differential associative algebra,that is an associative algebra over F with a derivation ∂ . Then the complex B ( A ) is a complex of F [ ∂ ] -modules. Given a differential A -bimodule M , we have thecomplex of F -vector spaces C • ∂ ( A, M ) := Hom ∂A - A -bimod ( B • ( A ) , M ) , where the Hom is taken in the category of differential A - A -bimodules. The homol-ogy of this complex is the differential Hochschild cohomology of A with coefficientsin M , denoted by HH • ∂ ( A, M ) . It is clear that this definition coincides with thedefinition in Section 6.1.Let A = F [ x ( j ) i | ≤ i ≤ N, j ≥ be a differential polynomial algebra in N variables x i = x (0) i and their derivatives ∂x ( j ) i := x ( j +1) i , j ≥ . Let A + ⊂ A be theaugmentation ideal. We will need the following well known result, whose proof weprovide for completeness Lemma 8.3. A + is free as an F [ ∂ ] -module.Proof. Consider first the case when A is a differential polynomial algebra in onevariable x = x (0) , that is A = F [ x (0) , x (1) , . . . ] . An F -basis of A + is given by themonomials x ( λ ) = x ( λ ) · · · x ( λ k ) , λ = λ ≥ · · · λ k ≥ , k ≥ . (8.4)We have ∂x ( λ ) = k X i =1 x ( λ ) · · · x ( λ i +1) · · · x ( λ k ) . (8.5)The module A + is a graded F [ ∂ ] -module with deg x ( λ ) = k + P ki =1 λ i , and deg ∂ = 1 .Notice that the homogeneous components ( A + ) n of degree n are finite dimensionalover F . We consider the weighted reverse lexicographic order on the set of monomials(8.4): for two partitions λ, µ we let x ( λ ) > x ( µ ) if deg x ( λ ) > deg x ( µ ) or deg x ( λ ) =deg x ( µ ) and there exists i ≥ such that λ i = µ i for ≤ i ≤ i and λ i > µ i .This is a total ordering on the set of monomials (8.4).We construct an F [ ∂ ] -basis of A + as follows. For each homogeneous degreecomponent ( A + ) n , we consider a set of monomials B n ⊂ ( A + ) n such that theirimages in ( A + ) n (cid:14) ∂ ( A + ) n − form an F -basis. This set exists since we have a totalordering of a monomial basis of ( A + ) n over F . We let B = ` n ≥ B n . We claimthat B is an F [ ∂ ] -basis of A + .First, let us prove by induction that B spans A + . Let a ∈ A + be a homogeneouselement of degree n . We prove by induction that a can be written as a linearcombination with coefficients in F [ ∂ ] of elements of B . When n = 1 there is nothingto prove as ( A + ) has as a basis B = { x (0) } . Assume that every homogeneouselement of degree less than n is in the F [ ∂ ] -span of B . We can assume that a is a onomial. By the definition of B n , there exist b ∈ ( A + ) n and c ∈ ( A + ) n − suchthat a = b + ∂c with the property that b is an F -linear combination of elementsof B n . By our induction hypothesis, c (and therefore ∂c ) can be written as an F [ ∂ ] -linear combination of elements of B . Therefore, B spans A + over F [ ∂ ] .Let us now prove that the elements of B are linearly independent over F [ ∂ ] .Suppose we are given b , . . . , b r ∈ B such that α ∂ j b + · · · + α r ∂ j r b r = 0 , α i ∈ F , α i = 0 , j i ≥ . (8.6)We may assume that each summand is homogeneous of degree n and that j ≥· · · ≥ j r . Since ∂ is injective on A + , we may assume that j r = 0 . Let i be theminimum such that j i = 0 . Thus b i ∈ B n for i ≤ i ≤ r . It follows that P ri = i α i b i vanishes modulo ∂ ( A + ) n − , which contradicts our choice of B n . This proves that B is an F [ ∂ ] -basis of A + .For general N , writing A N in place of A and denoting the case N = 1 again by A , we have an isomorphism of F [ ∂ ] -modules A N ≃ A ⊗ N = ( A + ⊕ F ⊗ N . Hence, the augmentation ideal ( A N ) + is a direct sum of tensor products of free F [ ∂ ] -modules, and so is free. (cid:3) Now we introduce the subspace of poly-vector fields P • ⊂ Hom F ( A ⊗• , A ) , i.e., alternating maps that are derivations in each argument. We consider P • as acomplex with the zero differential. Since A is a differential algebra, P • is naturallyan F [ ∂ ] -module; let P • ∂ = Ker ∂ . Theorem 8.4. Let A = F [ x ( j ) i | ≤ i ≤ N, j ≥ be a differential polynomialalgebra in N variables and their derivatives. Then for all k ≥ we have an iso-morphism HH k∂ ( A, A ) ≃ P k∂ . Proof. (P. Etingof) We consider the normalized Hochschild complex ¯ C • ( A, A ) de-fined in (8.2). It follows from Theorem 8.2 that π ♯ : P • ֒ → ¯ C • ( A, A ) is a quasi-isomorphism. Notice also that the inclusion π ♯ commutes with the F [ ∂ ] -action.That is, the complexes P • and ¯ C • ( A, A ) are quasi-isomorphic as complexes of F [ ∂ ] -modules. Considering F as a trivial F [ ∂ ] -module, it follows that we have aquasi-isomorphism of complexes of vector spaces: R Hom F [ ∂ ] ( F , P • ) → R Hom F [ ∂ ] ( F , ¯ C • ( A, A )) , (8.7)where R Hom is the right derived functor of Hom , whose cohomology computesthe Ext groups. To compute the cohomology of these complexes, we consider theresolution F [ ∂ ] ∂ · −→ F [ ∂ ] ։ F . (8.8)We replace F by the two term complex F [ ∂ ] → F [ ∂ ] in (8.7) and therefore the spaceof morphisms of F [ ∂ ] -modules (cid:16) F [ ∂ ] ∂ · −→ F [ ∂ ] (cid:17) → P • , (cid:16) F [ ∂ ] ∂ · −→ F [ ∂ ] (cid:17) → ¯ C • ( A, A ) , re naturally bi-complexes of vector spaces. They consist of complexes with tworows and infinitely many columns. Thus, the cohomology of the complexes in(8.7) are given by the cohomology of the total complexes associated to the two-row bicomplexes P • ∂ −→ P • and ¯ C • ( A, A ) ∂ −→ ¯ C • ( A, A ) . We compute the verticalcohomology of the complex P • ∂ −→ P • first.We claim that the map ∂ : P i → P i is surjective for i ≥ . In fact, if we let T i ⊂ ¯ C ( A, A ) i be the subspace of all maps that are derivations on each argument,we see that P i ֒ → T i is a split injection since T i decomposes as a representationof the symmetric group S i on i elements. It suffices to prove that ∂ : T i → T i is surjective for i ≥ . This is equivalent to showing that Ext F [ ∂ ] ( F , T i ) = 0 for i ≥ . Indeed, we may replace F by (8.8) and computing the Ext groups amountsto computing the cohomology of the complex T i ∂ −→ T i , which vanishes in degree if and only if ∂ is surjective. Notice that T i = Hom F (cid:0) ( A + /A ) ⊗ i , A (cid:1) , since a derivation is determined on A by the Leibniz rule. Also note that A + /A is a free F [ ∂ ] -module M ≃ F [ ∂ ] N , with basis given by { x (0) i } ≤ i ≤ N . Since M is afree F [ ∂ ] -module, we obtain Ext F [ ∂ ] ( F , T i ) = Ext F [ ∂ ] (cid:0) F , Hom F (cid:0) M ⊗ i , A (cid:1)(cid:1) = 0 , i ≥ . Since the horizontal differentials of P • ∂ −→ P • vanish (as the differential of P • vanishes), we obtain that the total cohomology of the bicomplex P • ∂ −→ P • is givenas follows. In degree i ≥ , it is P i∂ , that is the ϕ i ∈ P i such that ∂ϕ i = 0 . In degree , we have P ∂ ⊕ A/∂A , the first summand corresponds to the vertical cohomologyin degree of P while the second is the vertical cohomology of degree of P = A .Finally, in degree , we have P ∂ = F .We now consider the cohomology of the complex ¯ C • ( A, A ) ∂ −→ ¯ C • ( A, A ) whichcomputes the right-hand side of (8.7). It follows from Lemma 8.3 that Ext F [ ∂ ] ( F , ¯ C i ( A, A )) = Ext F [ ∂ ] ( F , Hom F ( A ⊗ i + , A )) = Ext F [ ∂ ] ( A ⊗ i + , A ) = 0 , i ≥ . Thus, the vertical differentials of ¯ C • ( A, A ) ∂ −→ ¯ C • ( A, A ) are also surjective for i ≥ . The vertical cohomology of this bicomplex is therefore ¯ C i∂ ( A, A ) for i ≥ ,while in the first column we have the cohomology ¯ C ∂ ( A, A ) = A ∂ = F in degree and ¯ C ( A, A ) /∂ ¯ C ( A, A ) = A/∂A in degree . Computing now the horizontalcohomology, we obtain that the total cohomology of the bicomplex ¯ C • ( A, A ) → ¯ C • ( A, A ) consists of H i ( ¯ C ∂ ( A, A )) for i ≥ . In degree we have H ( ¯ C ∂ ( A, A )) ⊕ A/∂A , and in degree we have F . We have therefore obtained H i ( ¯ C ∂ ( A, A )) ≃ P i∂ for all i ≥ as claimed. (cid:3) The sesquilinear setting. Let A be an associative differential algebra and s ≥ . Consider the total complex of the s -complex B ( A ) ⊗ s = B ( A ) ⊗ A · · · ⊗ A B ( A ) | {z } s times . This is a complex of A - A -bimodules and of F [ ∂ , . . . , ∂ s ] -modules. Let M be adifferential A - A -bimodule. Define ∆ s M = M ⊗ F [ ∂ ] F [ ∂ , . . . , ∂ s ] , here the left F [ ∂ ] -module structure on F [ ∂ , . . . , ∂ s ] is given by the diagonal map ∂ P ∂ i . Then ∆ s M is an A - A -bimodule and an F [ ∂ , . . . , ∂ s ] -module. We havethe complexes C s, • = Hom A – A –bimod ( B ( A ) ⊗ s , ∆ s M ) , C s, • ∂ = Hom( B ( A ) ⊗ s , ∆ s M ) , the Hom in the right-hand side being taken in the category of A - A -bimodules and F [ ∂ , . . . , ∂ s ] -modules. It is clear from the definition that C s, • ∂ ( A, A ) coincides withthe complex C s, • Hoc from (6.14). Remark . Notice that the complexes C s, • and C s, • ∂ decompose under the actionof products of symmetric groups as follows. For each degree i and a partition k + · · · + k s = i , the complex C s, • has a direct summand consisting of maps B k ( A ) ⊗ · · · ⊗ B k s ( A ) → ∆ s M. The group S k × · · · × S k s acts by permuting the entries on the left-hand side. Thecomplex C s, • is a direct sum of these symmetric group representations for all i andall partitions.Let now A be in addition commutative, let Ω A be the module of Kähler differ-entials of A , and let Ω • A = ^ • A Ω A be the module of differential forms. We consider Ω • A as a complex with zero differ-ential. We have P • = Hom(Ω • , A ) . Note that Ω A and therefore Ω • A are differential A -modules. Hence P • ∂ = Hom A − F [ ∂ ] (Ω • A , A ) , where the Hom is taken in the cate-gory of differential A -modules. Theorem 8.6. Let A = F [ x ( j ) i | ≤ i ≤ N, j ≥ be a differential polynomial alge-bra, and M be its differential module. Then for every i ≥ we have isomorphisms H i ( C s, • ( A, M )) ≃ H i (cid:0) Hom((Ω • A ) ⊗ s , ∆ s M ) (cid:1) ,H i ( C s, • ∂ ( A, M )) ≃ H i (cid:0) Hom A − F [ ∂ ,...,∂ s ] ((Ω • ) ⊗ s , ∆ s M ) (cid:1) . Proof. The first isomorphism is simply a consequence of the HKR Theorem 8.1 for A stating that B ( A ) is quasi-isomorphic to Ω • A . Since the latter is a free A -module,it is flat, and therefore B ( A ) ⊗ s is quasi-isomorphic to (Ω • A ) ⊗ s . The result followsby taking Homs into ∆ s M .The quasi-isomorphism B ( A ) ⊗ s → (Ω • A ) ⊗ s is a quasi-isomorphism of complexesof A -modules and F [ ∂ , . . . , ∂ s ] -modules. It follows that we have a quasi-isomorphismof complexes of A -modules and F [ ∂ , . . . , ∂ s ] -modules C s, • ( A, M ) → Hom A (cid:0) (Ω • A ) ⊗ s , ∆ s M (cid:1) , and hence the following two complexes are quasi-isomorphic R Hom F [ ∂ ,...,∂ s ] (cid:0) F , C s, • ( A, M ) (cid:1) → R Hom F [ ∂ ,...,∂ s ] (cid:0) F , Hom A (cid:0) (Ω • A ) ⊗ s , ∆ s M (cid:1)(cid:1) . (8.9)In order to compute the cohomology of (8.9), we use the Koszul resolution of F as an F [ ∂ , . . . , ∂ s ] -module. We consider the free module Ω F [ ∂ ,...,∂ s ] with a basis d , . . . , d s and the resolution · · · → ^ k Ω F [ ∂ ,...,∂ s ] → ^ k − Ω F [ ∂ ,...,∂ s ] → · · · → Ω F [ ∂ ,...,∂ s ] → F [ ∂ , . . . , ∂ s ] → F . (8.10)This resolution coincides with the two-term resolution (8.8) when s = 1 . he complex (8.10) is non-negatively graded, with F [ ∂ , . . . , ∂ s ] in degree 0. TheKoszul differential is defined by d i ∂ i and extending by the Leibniz rule to aderivation of degree − of the free commutative superalgebra V • Ω F [ ∂ ,...,∂ s ] . Hence,in order to compute the cohomology of (8.9), we need to compute the cohomologyof the total complexes with s + 1 rows ^ • F s ⊗ C s, • ( A, M ) and ^ • F s ⊗ Hom A (cid:0) (Ω • A ) ⊗ s , ∆ s M (cid:1) . (8.11)We compute first the vertical cohomology of the complex on the right. We claimthat for each column i ≥ the vertical cohomology in V • F s ⊗ Hom ((Ω • A ) ⊗ s , ∆ s M ) vanishes in positive degrees. Indeed, let T s, • be the set of all maps in C s, • ( A, M ) that are derivations in each argument. We have a split injection Hom((Ω • A ) ⊗ s , ∆ s M ) ֒ → T i , since the latter splits as a representation of the symmetric group as in Re-mark 8.5. It suffices then to prove that the vertical cohomology of V • F s ⊗ T s, • vanishes in positive degrees. For each partition k + · · · + k s = i ≥ , the corre-sponding summand of T s, • is given by maps ( A + /A ) ⊗ k ⊗ · · · ⊗ ( A + /A ) ⊗ k s → ∆ s M. (8.12)Notice that if some k i = 0 , the corresponding space of maps vanishes since there areno non-trivial derivations of F . So we may assume that all k i > . Since A + /A is free as an F [ ∂ ] -module, it follows that the left-hand side of (8.12) is free as an F [ ∂ , . . . , ∂ s ] -module. Hence Ext j F [ ∂ ,...,∂ s ] ( F , T i ) = 0 , i, j ≥ , proving that the vertical cohomology of the second complex in (8.11) vanishes inpositive degrees for each column i ≥ . The zeroth column is given by the complex V • F ⊗ ∆ s M , where the differential is defined by d i ⊗ m m∂ i extended to aderivation of degree − . Since the horizontal differentials are zero, we obtain thefollowing description of the total cohomology. In each degree i ≥ , we have H i (cid:0) Hom A − F [ ∂ ,...,∂ s ] (cid:0) (Ω • ) ⊗ s , ∆ s M (cid:1)(cid:1) ⊕ H i (cid:16)^ • F s ⊗ ∆ s M (cid:17) , (8.13)where the first summand corresponds to the i -th horizontal cohomology of thezeroth row, while the second is the i -th vertical cohomology of the zeroth column.In degree , we have F .We now analyze the vertical cohomology of the first bicomplex in (8.11). Wenotice that, in the same way as in the proof of Theorem 8.4, for any partition k + · · · + k s = i where all k i > , we have ( A + ) ⊗ P k j is a free F [ ∂ , . . . , ∂ s ] -module.Hence, we obtain Ext j F [ ∂ ,...,∂ s ] (cid:0) F , C s,j (cid:1) = 0 , i, j ≥ . The cohomology is therefore again concentrated in the zeroth row and the zerothcolumn. The zeroth vertical cohomology is given by C s, • ∂ , while the zeroth columnis given by V • ⊗ ∆ s M . We see that the zeroth column contributes the same coho-mology to the second summand of (8.13), while the horizontal cohomology of thezeroth row is now given by H i (cid:0) C s, • ∂ ( A, M ) (cid:1) , proving the theorem. (cid:3) The sesquilinear Hodge decomposition. We recall here the Hodge decom-position of the Hochschild cohomology of a commutative algebra A with coefficientsin its module M ; see [GS87, L13]. The symmetric group S n acts on C n := C n ( A, M ) ≃ Hom( A ⊗ n , M ) y permuting the n factors. Recall the Eulerian idempotents e ( i ) n ∈ Q [ S n ] of thegroup algebra of S n (see [L13, 4.5.2] for an explicit description). They satisfy e (1) n + · · · + e ( n ) n ,e ( i ) n e ( j ) n = 0 , if i = j, and e ( i ) n e ( i ) n = e ( i ) n . It follows from [L13, 4.5.10] that, putting C n ( k ) := e ( k ) n C n , and letting HH n ( k ) ( A, M ) ⊂ HH n ( A, M ) consist of cohomology classes of elements in C n ( k ) , we obtain a directsum decomposition HH n ( A, M ) = HH n (1) ( A, M ) ⊕ · · · ⊕ HH n ( n ) ( A, M ) , n ≥ . The first summand HH n (1) ( A, M ) is identified canonically with the Harrison co-homology H n ( C • Har ( A, M )) by [L13, 4.5.13]. The last summand is identified withpolyvector fields [L13, 4.5.13]: HH n ( n ) ( A, M ) ≃ Hom (cid:16)^ n Ω A , M (cid:17) . (8.14)The above description generalizes to the sesquilinear setting. Recall from theproof of Proposition 6.7 that the complexes C s, • ∂ ( A, M ) are complexes in the cate-gory of representations of symmetric group S s , so that the action of the symmetricgroup S s as described in Section 6.3 commutes with the differential. In addition, itpreserves the Harrison conditions (6.18). For each s and k + · · · + k s = n , we havean action of the product S k × · · ·× S k s on C s, • ∂ ( A, M ) by permuting the entries andcommuting with the differential. Consider the corresponding Eulerian idempotents e ( i ) k j ∈ Q [ S k j ] , for i ≥ and j = 1 , · · · , s . For i = ( i , · · · , i s ) and k = ( k , · · · , k s ) ,we let e ( i ) k = e ( i ) k ⊗ · · · ⊗ e ( i s ) k s ∈ Q [ S k × · · · × S k s ] . For each i , we set C s,n ( i ) ,∂ ( A, M ) = M k + ··· + k s = n e ( i ) k C s,k∂ ( A, M ) . We obtain the corresponding decomposition of the sesquilinear Hochschild coho-mology H n (cid:0) C s, • ∂ ( A, M ) (cid:1) = M i H n (cid:0) C s, • ( i ) ,∂ ( A, M ) (cid:1) . Denote by the s -tuple (1 , . . . , . The summand for i = 1 is identified with thesesquilinear Harrison cohomology in the same way as above: H n (cid:0) C s, • sesq , Har ( A, M ) (cid:1) = H n (cid:0) C s, • (1) ,∂ ( A, M ) (cid:1) . (8.15)In the other extreme case, we obtain from (8.14) the identification of sesquilinearpolyvector fields with the following sum H n (cid:0) Hom A − F [ ∂ ,...,∂ s ] ((Ω • ) ⊗ s , ∆ s M ) (cid:1) ≃ M k + ··· + k s = n H n (cid:0) C s, • ( k ) ,∂ ( A, M ) (cid:1) . (8.16)The main result of this section is the following: Theorem 8.7. Let A = F [ x ( j ) i | ≤ i ≤ N, j ≥ be a differential polynomialalgebra, and M be its differential module. Then for every n > s > the sesquilinearHarrison cohomology of A with coefficients in M vanishes: H n (cid:0) C s, • sesq , Har ( A, M ) (cid:1) = 0 . roof. Let n > s > and consider the sesquilinear Hochschild cohomology of A with coefficients in M , namely H n ( C s, • ∂ ( A, M )) . By Theorem 8.6 and (8.16), wehave an isomorphism H n ( C s, • ∂ ( A, M )) ≃ M k + ··· + k s = n H n (cid:0) C s, • ( k ) ,∂ ( A, M ) (cid:1) . If n > s this implies that in the sum in the right-hand side we must have some k i > , and hence k = 1 . This implies that H n (cid:0) C s, • (1) ,∂ ( A, M ) (cid:1) = 0 , and therefore the theorem follows by (8.15). (cid:3) Corollary 8.8. With the notation of Theorem 8.7, for every n > s > the sym-metric s -sesquilinear Harrison cohomology of A with coefficients in M vanishes: H n (cid:0) C s, • sym , Har ( A, M ) (cid:1) = 0 . Proof. It follows from Proposition 6.7 that the sesquilinear Harrison cohomologycomplex C s, • sesq , Har is a complex of S s -modules. The symmetric s -sesquilinear Har-rison cohomology complex C s, • sym , Har ( A, M ) is defined in (6.20) as its subcomplex of S s -invariants. It follows that the symmetric s -sesquilinear Harrison cohomology isa direct summand of the s -sesquilinear Harrison cohomology. (cid:3) Proof of the Main Theorem 5.2 Recall that by Theorem 5.1 the map (5.1) is injective, and we only need to provethat it is surjective. In other words, we need to show that for every closed element Y ∈ C n cl in the classical complex, dY = 0 , there exist Z ∈ C n − and e Y ∈ C n cl suchthat Y = dZ + e Y , (9.1)and e Y Γ = 0 if | E (Γ) | 6 = 0 . (9.2)Recall the filtration F s C n cl of the classical complex, given by equation (7.1). Clearly, Y ∈ F C n cl = C n cl , and the condition (9.2) on e Y is equivalent to saying that e Y ∈ F n C n cl . Hence, by induction, it suffices to prove that, for ≤ s ≤ n − and Y s ∈ F s C n cl such that dY s = 0 , we can find Z s ∈ C n − and Y s +1 ∈ F s +1 C n cl satisfying Y s = dZ s + Y s +1 . (9.3)Consider the coset Y s + F s +1 C n cl ∈ gr s C n cl . Then, since the differential d of C cl preserves the filtration (7.1), Y s + F s +1 C n cl is a closed element of the complex gr s C cl .By Theorem 7.2, the complex gr s C cl is isomorphic to the complex C s sym , Har , which,by Corollary 8.8, has trivial n -th cohomology, since s ≤ n − . As a consequence,there exists Z s + F s +1 C n − ∈ gr s C n − such that Y s + F s +1 C n cl = d ( Z s + F s +1 C n − ) . This is equivalent to Y s +1 := Y s − dZ s ∈ F s +1 C n cl , proving the theorem. eferences [BDSHK19] B. Bakalov, A. De Sole, R. Heluani and V.G. Kac, An operadic approach to vertexalgebra and Poisson vertex algebra cohomology . Japan. J. Math. (2019), 1–95.[BDSHK20] B. Bakalov, A. De Sole, R. Heluani and V.G. Kac, Chiral versus classical operad .Internat. Math. Res. Not. (2020), n.19, 6463–6488.[BDSK20] B. Bakalov, A. De Sole, and V.G. Kac, Computation of cohomology of Lie conformaland Poisson vertex algebras . Sel. Math. (NS) (2020), no. 4, paper no. 50, 51 pages.[BDSK21] B. Bakalov, A. De Sole, and V.G. Kac, Computation of cohomology of vertex algebras .To appear in Japan. J. Math., preprint arXiv:2002.03612[BDSKV21] B. Bakalov, A. De Sole, V.G. Kac, and V. 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Department of Mathematics, North Carolina State University, Raleigh, NC 27695,USA Email address : [email protected] URL : https://sites.google.com/a/ncsu.edu/bakalov Dipartimento di Matematica, Sapienza Università di Roma, P.le Aldo Moro 2,00185 Rome, Italy Email address : [email protected] URL : ∼ desole IMPA, Rio de Janeiro, Brasil Email address : [email protected] URL : http://w3.impa.br/ ∼ heluani/ Department of Mathematics, MIT, 77 Massachusetts Ave., Cambridge, MA 02139,USA Email address : [email protected] URL : ∼ kac/ Dipartimento di Matematica, Sapienza Università di Roma, P.le Aldo Moro 2,00185 Rome, Italy Email address : [email protected]@mat.uniroma1.it