A cup-cap duality in Koszul calculus
aa r X i v : . [ m a t h . R T ] J u l A cup-cap duality in Koszul calculus
Roland Berger and Andrea Solotar ∗ Abstract
We introduce a cup-cap duality in the Koszul calculus of N -homogeneous algebrasdefined by the first author [5]. As an application, we prove that the graded symmetryof the Koszul cap product is a consequence of the graded commutativity of the Koszulcup product. We propose a conceptual approach that may lead to a proof of the gradedcommutativity, based on derived categories in the framework of DG bimodules over DGalgebras. Various enriched structures are developed in a weaker situation correspondingto N > N -homogeneous algebras, N -Koszul algebras, Koszul (co)homology, Hochschild(co)homology, cup and cap products, derived categories, DG algebras, DG bimodules. Just after the appearance of the Koszul calculus on quadratic algebras over a ground field [7],two motivating generalizations have allowed to enlarge the validity of this calculus to widerdomains of applications.The first one concerns a generalization to N -homogeneous algebras, that is, instead of thequadratic relations, homogeneous relations of degree N > N -Koszul algebras and various applications (see [5] for an extendedbibliography).In the second generalization, general quiver algebras with quadratic relations are consid-ered, the quivers having finitely many vertices and arrows. This generalization, applied topreprojective algebras, reveals some interesting interactions with representation theory andPoincar´e duality as presented in [9], with a particular interest on derived categories.In the present paper, we have chosen to focus on N -homogeneous algebras over a groundfield. A Koszul calculus for N -homogeneous relations on quivers is beyond our scope, even ∗ The second mentioned author has been supported by the projects UBACYT 20020130100533BA,pip-conicet 11220150100483CO and PICT 2015-0366. She is a research member of CONICET (Argentina)and a Senior Associate at ICTP.
1f such a calculus deserves to be defined and studied. Actually, in the present paper, we areessentially interested in a question that we detail below and which is already crucial in theone vertex case.A fundamental question in the Koszul calculus defined on N -homogeneous algebras is toknow whether, on Koszul classes, the Koszul cup product ⌣ K is graded commutative and theKoszul cap product ⌢ K is graded symmetric. This question is still open when the algebra isquadratic, that is, when N = 2, although the answer is positive if the quadratic algebra isKoszul [7, Subsection 3.4] and there are no examples up to date neither of non commutativitynor of non symmetry. When N ≥
2, we prove that the graded symmetry of the Koszul capproduct is a consequence of the graded commutativity of the Koszul cup product. For that,we introduce a cup-cap duality in the Koszul calculus of N -homogeneous algebras.For an N -homogeneous algebra A , our cup-cap duality consists in a duality linking a Koszulcup bracket [ α, − ] ⌣ K and a corresponding cap bracket [ α, − ] ⌢ K . In the first argument of thebrackets, α denotes any Koszul cohomology class with coefficients in A , while the secondargument applies to any Koszul cohomology (respectively, homology) class with coefficientsin an A -bimodule. The statement is the following and constitutes the main result of ourpaper (Theorem 4.4 in Section 4). The various ingredients involved in this statement, inparticular the duality isomorphism ζ , will be thoroughly defined and studied in our paper. Theorem 1.1.
Let A be an N -homogeneous algebra over a k -vector space V and let M bean A -bimodule. For any q ≥ p ≥ and any α ∈ HK p ( A ) , there is a commutative diagramin the category of k -vector spaces HK q − p ( A, M ) ∗ [ α, − ] ∗ ⌢K −→ HK q ( A, M ) ∗ ↓ ζ q − p ↓ ζ q (1.1)HK q − p ( A, M ∗ ) − [ α, − ] ⌣K −→ HK q ( A, M ∗ ) where the vertical arrows are isomorphisms. Accordingly, it is immediate that if the bracket [ α, − ] ⌣ K acting on any A -bimodule is zero,then the bracket [ α, − ] ⌢ K acting on any A -bimodule is as well zero. We express this implica-tion by saying that the graded symmetry of ⌢ K is a consequence of the graded commutativityof ⌣ K .We develop a derived category approach -that we hope will lead to a proof of the gradedcommutativity- inspired by the idea of “enriched structures” used in [9]. Unfortunately, wehave to add hypotheses concerning the existence of injective resolutions. However, we thinkthat the material so developed in the framework of derived categories (or possibly in theframework of derived A ∞ categories) is of interest in its own.The contents of the paper are as follows. In Section 2 we recall the basic definitions andresults. In Section 3, we apply a general natural isomorphism to Koszul calculus. In Sec-tion 4, we present and prove Theorem 1.1 where the definition of the duality isomorphism ζ uses the isomorphism of Section 3. As explained above, the graded symmetry of the Koszul2ap product becomes a consequence of the graded commutativity of the Koszul cup product,and we also prove in Corollary 4.5 that both properties hold if the algebra is N -Koszul. Wepresent in Section 5 some examples of non N -Koszul algebras satisfying both properties.In Section 6, we verify the naturality in A of the cup-cap duality isomorphism ζ specializedto M = A , and we combine it for N = 2 with the Koszul duality obtained in [7]. In Section 7,we construct a cup-cap duality in Hochschild calculus for any associative algebra. A classicaltheorem due to Gerstenhaber [11] asserts that the cup product in Hochschild cohomology isgraded commutative. Therefore our Hochschild cup-cap duality allows us to prove that thecap product on Hochschild classes is graded symmetric.In Section 8, we generalize to N -homogeneous algebras the enriched structure obtained onthe Koszul bimodule complex K ( A ) when N = 2 [9]. In this generalization, there is adifficulty coming from the fact that the Koszul cup and cap products are not associative oncochains and chains when N >
2. So in this case we have to work with weak DG algebrasand weak DG bimodules. It is noteworthy that the enriched structures obtained in [9] when N = 2 make sense in the weak situation N > K ( A ) obtained in Section 8, the three functors involved in Proposition 9.1 are enrichedby suitable weak DG structures (Proposition 9.3). We end Section 9 with an application(Theorem 9.4) which constitutes, in terms of derived categories, some evidence to state thegraded commutativity when N = 2.Throughout the article, we use notation and results from [5]. We fix a vector space V overa field k and an integer N ≥
2. The tensor algebra T ( V ) = L m ≥ V ⊗ m is graded by the weight m . We also fix a subspace R of V ⊗ N . The associative algebra A = T ( V ) / ( R ) inheritsthe weight grading. The homogeneous component of weight m of A is denoted by A m . Inparticular, A = k and A m = V ⊗ m if 1 ≤ m ≤ N −
1. The graded algebra A is called an N-homogeneous algebra . If N = 2, A is called a quadratic algebra [7]. In the next section,we recall some objects and results that are necessary in the sequel. A A Let A = T ( V ) / ( R ) be an N -homogeneous algebra, where R is a subspace of V ⊗ N . Thefundamental object of the Koszul calculus of A is the Koszul bimodule complex K ( A ) thatwe recall below. For any A -bimodule M , the Koszul homology space HK • ( A, M ) is definedas the homology of the Koszul chain complex ( M ⊗ W ν ( • ) , b K ). In a similar way, the Koszulcohomology space HK • ( A, M ) is defined as the cohomology of the Koszul cochain complex(Hom( W ν ( • ) , M ) , b K ). See [5, Subsection 2.2] for the definitions of the Koszul differentials b K from the complex K ( A ). 3or the definition of K ( A ), we follow [5]. For any p ≥ N , let W p be the subspace of V ⊗ p defined by W p = \ i + N + j = p V ⊗ i ⊗ R ⊗ V ⊗ j , where i, j ≥ , while W = k and W p = V ⊗ p if 1 ≤ p ≤ N −
1. It is convenient to use the following notation:an arbitrary element of W p will be denoted by a product x . . . x p , where x , . . . , x p are in V . This notation should be thought of as a sum of such products. Moreover, regarding W p as a subspace of V ⊗ q ⊗ W r ⊗ V ⊗ s where q + r + s = p , the element x . . . x p viewed in V ⊗ q ⊗ W r ⊗ V ⊗ s will be denoted in the same way, meaning that x q +1 . . . x q + r is thought ofas a sum belonging to W r and the other x i ’s are arbitrary elements in V .Let ν : N → N be the map such that ν ( p ) = N p ′ if p = 2 p ′ and ν ( p ) = N p ′ + 1 if p = 2 p ′ + 1.The A -bimodule complex K ( A ) is · · · d −→ K p d −→ K p − d −→ · · · d −→ K d −→ K −→ , (2.1)where K p = A ⊗ W ν ( p ) ⊗ A . For any a, a ′ ∈ A and x . . . x ν ( p ) ∈ W ν ( p ) , the differential d isdefined on K p as follows. If p = 2 p ′ + 1, one has d ( a ⊗ x . . . x Np ′ +1 ⊗ a ′ ) = ax ⊗ x . . . x Np ′ +1 ⊗ a ′ − a ⊗ x . . . x Np ′ ⊗ x Np ′ +1 a ′ , and if p = 2 p ′ , one has d ( a ⊗ x . . . x Np ′ ⊗ a ′ ) = X ≤ i ≤ N − ax . . . x i ⊗ x i +1 . . . x i + Np ′ − N +1 ⊗ x i + Np ′ − N +2 . . . x Np ′ a ′ . The homology of K ( A ) is equal to A in degree 0, and to 0 in degree 1.In case N = 2, W p is a subspace of V ⊗ p ⊆ A ⊗ p for all p ≥
0, and K ( A ) is an A -bimodulesubcomplex of the bar resolution B ( A ) of A . However, if N >
2, there is no natural inclusionof K ( A ) in B ( A ).Koszul algebras for N > N = 2, this definition includes Priddy’s definition [16]. Definition 2.1. An N -homogeneous algebra A = T ( V ) / ( R ) is said to be N -Koszul if thehomology of K ( A ) is in any positive degree. K ( A ) into a minimal resolution As recalled in [5, Subsection 2.6], in the category of graded A -bimodules, A has a minimalprojective resolution P ( A ) whose component of homological degree p has the form A ⊗ E p ⊗ A , where E p is a graded space where the degree is still called weight. Moreover,the components of weight less than ν ( p ) in E p are zero while the component of weight ν ( p ) contains W ν ( p ) [3, 12]. So, for any N ≥ K ( A ) can be viewed as a weight gradedsubcomplex of P ( A ), and A is N -Koszul if and only if P ( A ) = K ( A ). Actually, E = k , E = V , E = R and the component of weight ν (3) in E is equal to W ν (3) . However, if N >
2, the component of weight ν (4) in E may strictly contain W ν (4) . It is well-known that,4f N = 2, the component of weight p in E p is equal to W p for any p . See [2, Subsections 2.7and 2.8] and [15, Chapter 1, Proposition 3.1], see also [9, Subsection 2.8] for a generalizationto quadratic quiver algebras.We use now the resolution P ( A ) for computing the Hochschild (co)homology of A and weobtain the following N -analogue of [9, Proposition 2.13] (limited to the one vertex case) bythe same proof. Proposition 2.2.
Let A = T ( V ) / ( R ) be an N -homogeneous algebra. For any A -bimodule M , the inclusion of K ( A ) into P ( A ) induces a linear map HK ( A, M ) → HH ( A, M ) whichis surjective, and a linear map HH ( A, M ) → HK ( A, M ) which is injective. As in [9, Corollary 2.14] (still limited to the one vertex case), we have a more precise resultwhen M = A . In fact, for any N -homogeneous algebra A , we can define a coefficient weightgrading in Koszul (co)homology with coefficients in A . It suffices to extend naturally thedefinition of this grading from the case N = 2 [7, 9] to any N >
2. The grading of HK p ( A )and HK p ( A ) by the coefficient weight r is denoted by HK p ( A ) r and HK p ( A ) r . Unlike theKoszul differentials, the Hochschild differentials are not homogeneous for the coefficientweight, but only for the total weight. The grading of HH p ( A ) and HH p ( A ) for the totalweight t is denoted by HH p ( A ) t and HH p ( A ) t . Then, specializing M = A in Proposition2.2, the linear map HK ( A ) → HH ( A ) is homogeneous from the coefficient weight r to thetotal weight r + N , while the linear map HH ( A ) → HK ( A ) is homogeneous from the totalweight r − N to the coefficient weight r . Corollary 2.3.
Let A = T ( V ) / ( R ) be an N -homogeneous algebra.(i) The linear map HK ( A ) r → HH ( A ) r + N is an isomorphism if r = 0 and r = 1 .(ii) Assume that A is finite dimensional. Let max be the highest m such that A m = 0 . Thelinear map HH ( A ) r − N → HK ( A ) r is an isomorphism if r = max and r = max − . We also recall definitions and properties of the Koszul cup and cap products as stated in [5,Sections 3 and 5].
Definition 2.4.
Let A = T ( V ) / ( R ) be an N -homogeneous algebra. Let P and Q be A -bimodules. Given Koszul cochains f : W ν ( p ) → P and g : W ν ( q ) → Q , we define the Koszul ( p + q ) -cochain f ⌣ K g : W ν ( p + q ) → P ⊗ A Q by1. if p and q are not both odd, so that ν ( p + q ) = ν ( p ) + ν ( q ) , one has ( f ⌣ K g )( x . . . x ν ( p + q ) ) = f ( x . . . x ν ( p ) ) ⊗ A g ( x ν ( p )+1 . . . x ν ( p )+ ν ( q ) ) ,
2. if p and q are both odd, so that ν ( p + q ) = ν ( p ) + ν ( q ) + N − , one has ( f ⌣ K g )( x . . . x ν ( p + q ) ) = − X ≤ i + j ≤ N − x . . . x i f ( x i +1 . . . x i + ν ( p ) ) x i + ν ( p )+1 . . . x ν ( p )+ N − j − (2.2) ⊗ A g ( x ν ( p )+ N − j − . . . x ν ( p )+ ν ( q )+ N − j − ) x ν ( p )+ ν ( q )+ N − j − . . . x ν ( p )+ ν ( q )+ N − . efinition 2.5. Let A = T ( V ) / ( R ) be an N -homogeneous algebra. Let M and P be A -bimodules. For any p -cochain f : W ν ( p ) → P and any q -chain z = m ⊗ x . . . x ν ( q ) in M ⊗ W ν ( q ) , we define the ( q − p ) -chains f ⌢ K z and z ⌢ K f with coefficients in P ⊗ A M and M ⊗ A P respectively, as follows.1. If p and q − p are not both odd, so that ν ( q − p ) = ν ( q ) − ν ( p ) , one has f ⌢ K z = ( f ( x ν ( q − p )+1 . . . x ν ( q ) ) ⊗ A m ) ⊗ x . . . x ν ( q − p ) , (2.3) z ⌢ K f = ( − pq ( m ⊗ A f ( x . . . x ν ( p ) )) ⊗ x ν ( p )+1 . . . x ν ( q ) .
2. If p = 2 p ′ + 1 and q = 2 q ′ , so that ν ( q − p ) = ν ( q ) − ν ( p ) − N + 2 , one has f ⌢ K z = − X ≤ i + j ≤ N − ( x Nq ′ − Np ′ − N + i +2 . . . x Nq ′ − Np ′ − j − f ( x Nq ′ − Np ′ − j . . . x Nq ′ − j ) (2.4) ⊗ A x Nq ′ − j +1 . . . x Nq ′ mx . . . x i ) ⊗ x i +1 . . . x i + Nq ′ − Np ′ − N +1 .z ⌢ K f = X ≤ i + j ≤ N − ( x Nq ′ − j +1 . . . x Nq ′ mx . . . x i ⊗ A f ( x i +1 . . . x Np ′ + i +1 ) (2.5) x Np ′ + i +2 . . . x Np ′ + N − j − ) ⊗ x Np ′ + N − j . . . x Nq ′ − j . The chain f ⌢ K z is called the left Koszul cap product of f and z , while z ⌢ K f is called theirright Koszul cap product. There is an important fact on these products that we recall now. For any Koszul cochains f , g , h and any Koszul chain z , the associativity relations ( f ⌣ K g ) ⌣ K h = f ⌣ K ( g ⌣ K h ) , (2.6) f ⌢ K ( g ⌢ K z ) = ( f ⌣ K g ) ⌢ K z, (2.7)( z ⌢ K g ) ⌢ K f = z ⌢ K ( g ⌣ K f ) , (2.8) f ⌢ K ( z ⌢ K g ) = ( f ⌢ K z ) ⌢ K g, (2.9)hold for N = 2, but they are no longer true in general for N >
2. However these associativityrelations hold on Koszul classes for any N ≥
2. See [5, Sections 3 and 5] for details.For any Koszul cochains f ∈ Hom( W ν ( p ) , P ) and g ∈ Hom( W ν ( q ) , Q ), we have the identity b K ( f ⌣ K g ) = b k ( f ) ⌣ K g + ( − p f ⌣ K b K ( g ) . (2.10)Therefore, specializing P = Q = A , we see that (Hom( W ν ( • ) , A ) , b K , ⌣ K ) is a DG algebrawhen N = 2, but only a weak DG algebra when N >
2. Here “weak” means that the productof the algebra is not necessarily associative. For any
N >
2, following the case of quadraticquiver algebras [9], the weak DG algebra Hom( W ν ( • ) , A ) is denoted by ˜ A . Note that ˜ A is k -central. Moreover, formula (2.10) shows that for any A -bimodule M , Hom( W ν ( • ) , M ) is a6eak DG ˜ A -bimodule for the Koszul cup actions, only weak since the associativity relation(2.6) does not necessarily hold if N >
2. However “weak” can be removed when N = 2.Similarly, for any Koszul p -cochain f and any Koszul q -chain z , the following formulas b K ( f ⌢ K z ) = b K ( f ) ⌢ K z + ( − p f ⌢ K b K ( z ) , (2.11) b K ( z ⌢ K f ) = b K ( z ) ⌢ K f + ( − q z ⌢ K b K ( f ) . (2.12)show that M ⊗ W ν ( • ) is a weak DG ˜ A -bimodule for the Koszul cap actions – again weak sincethe associativity relations (2.7), (2.8) and (2.9) do not necessarily hold if N >
2. However“weak” can be removed when N = 2.For any N ≥ H ( ˜ A ) = HK • ( A ) is a graded associative algebra for the Koszul cup product,so that HK • ( A, M ) and HK • ( A, M ) are graded HK • ( A )-bimodules for the Koszul cup andcap actions respectively. Let A = T ( V ) / ( R ) be an N -homogeneous algebra, where R is a subspace of V ⊗ N . Let f : W ν ( p ) → A be a Koszul p -cocycle with coefficients in A . Denote by α the class of f inHK p ( A ). Recall that the maps f ⌢ K − , − ⌢ K f : M ⊗ W ν ( q ) −→ M ⊗ W ν ( q − p ) define a left and a right action on the complex ( M ⊗ W ν ( • ) , b K ), inducing a left and a rightaction of α on HK • ( A, M ) defined by the maps α ⌢ K − , − ⌢ K α : HK q ( A, M ) −→ HK q − p ( A, M ) , so that HK • ( A, M ) is a graded HK • ( A )-bimodule.Throughout this paper, the (graded) dual space of a (graded) vector space E is denoted by E ∗ , and the transpose map of a (graded) linear map u is denoted by u ∗ . On one hand, wesee HK • ( A, M ) ∗ = M q ≥ HK q ( A, M ) ∗ as a natural graded HK • ( A )-bimodule. In fact, the transpose map R ( α ) := ( α ⌢ K − ) ∗ : HK q − p ( A, M ) ∗ −→ HK q ( A, M ) ∗ defines a right action since it is easy to verify that R ( β ⌣ K α ) = R ( α ) ◦ R ( β ) for any β in HK • ( A ). Similarly L ( α ) = ( − ⌢ K α ) ∗ defines a left action. The associativity formula α ⌢ K ( γ ⌢ K β ) = ( α ⌢ K γ ) ⌢ K β for γ in HK • ( A, M ) shows that R ( α ) ◦ L ( β ) = L ( β ) ◦ R ( α ),so that HK • ( A, M ) ∗ is a graded HK • ( A )-bimodule.7n the other hand, we consider the cochain complex( M ⊗ W ν ( • ) ) ∗ = M q ≥ ( M ⊗ W ν ( q ) ) ∗ endowed with the differential b ∗ K : ( M ⊗ W ν ( q ) ) ∗ → ( M ⊗ W ν ( q +1) ) ∗ . For brevity, we denote this complex by C . Recall that if φ ∈ ( M ⊗ W ν ( q ) ) ∗ , one has b ∗ K ( φ ) = − ( − q φ ◦ b K . Assume that φ is a q -cocycle of C , meaning that φ : M ⊗ W ν ( • ) −→ k ( − q )is a morphism of complexes, where k ( − q ) is the trivial complex k concentrated in homologicaldegree q , so that the class φ in H q ( C ) coincides with the homotopy class of the complexmorphism φ . Then H q ( φ ) : H q ( M ⊗ W ν ( • ) ) −→ H q ( k ( − q )) ∼ = k only depends on φ , allowing us to define a linear map ξ q : H q ( C ) −→ HK q ( A, M ) ∗ by ξ q ( φ ) = H q ( φ ), and a graded linear map ξ : H • ( C ) = H • (( M ⊗ W ν ( • ) ) ∗ ) −→ HK • ( A, M ) ∗ . It is a consequence of a general fact [10, Corollaire 1, p.84] that ξ is an isomorphism. More-over, H • (( M ⊗ W ν ( • ) ) ∗ ) and HK • ( A, M ) ∗ are natural in M and ξ is a natural isomorphism. Proposition 3.1.
The space H • ( C ) is naturally a graded HK • ( A ) -bimodule and the map ξ is a natural isomorphism of graded HK • ( A ) -bimodules.Proof. Let f : W ν ( p ) → A be a Koszul p -cocycle. The linear map( f ⌢ K − ) ∗ : ( M ⊗ W ν ( q − p ) ) ∗ → ( M ⊗ W ν ( q ) ) ∗ sends φ to φ · f := φ ◦ ( f ⌢ K − ). Assume that φ is a ( q − p )-cocycle of C , meaning that φ ◦ b k = 0. Since f is a p -cocycle, we have( f ⌢ K − ) ◦ b K = ( − p b K ◦ ( f ⌢ K − )on M ⊗ W ν ( q +1) . Thus φ ◦ ( f ⌢ K − ) ◦ b K = 0, implying that φ · f is a q -cocycle. Then it iseasy to check that φ · f in H q ( C ) only depends on φ in H q − p ( C ). So we have defined thelinear map H q (( f ⌢ K − ) ∗ ) : H q − p ( C ) → H q ( C )8he reader will easily verify that this map only depends on the class α of f in HK p ( A ), andtherefore it can be written as R ′ ( α ) : H q − p ( C ) → H q ( C ) with R ′ ( α )( φ ) = φ · f also denotedby φ · α . Next it is straightforward to check that ( φ · α ) · β = φ · ( α ⌣ K β ), showing that H • ( C ) endowed with the action R ′ is a graded right HK • ( A )-module.Similarly, the linear map L ′ ( α ) : H q − p ( C ) → H q ( C ) is well-defined by L ′ ( α )( φ ) = f · φ where f · φ = φ ◦ ( − ⌢ K f ), and H • ( C ) endowed with the action L ′ is a graded left HK • ( A )-module. Moreover it is easy to verify that the actions R ′ and L ′ commute. Thus H • ( C ) isa graded HK • ( A )-bimodule.Let us prove now that ξ is right HK • ( A )-linear for the above actions (the left linearity issimilar). Keeping the above notation, we have ξ q ( φ · α ) = H q ( φ · f ) = H q ( φ ◦ ( f ⌢ K − )) = H q − p ( φ ) ◦ H q ( f ⌢ K − ) = H q − p ( φ ) ◦ ( α ⌢ K − )which is equal to ( α ⌢ K − ) ∗ ( H q − p ( φ )) = ξ q − p ( φ ) · α . The naturality in M is clear. Remark 3.2.
We can define the actions φ · f := φ ◦ ( f ⌢ K − ) and f · φ := φ ◦ ( − ⌢ K f ) for any cochain φ of C and any Koszul cochain f ∈ Hom( W ν ( • ) , A ) . Then C is a weakDG ˜ A -bimodule (see the end of Subsection 2.3 for the definition of the weak DG algebra ˜ A ).Passing to cohomologies, we obtain the graded HK • ( A ) -bimodule H • ( C ) defined above. Let us fix an N -homogeneous algebra A = T ( V ) / ( R ) and an A -bimodule M . As usual, M ∗ = Hom( M, k ) is an A -bimodule for the actions defined by ( a.u.a ′ )( m ) = u ( a ′ ma ) for a ∈ A , a ′ ∈ A and m ∈ M . For any p ≥
0, we define the linear map η p : Hom( M ⊗ W ν ( p ) , k ) −→ Hom( W ν ( p ) , M ∗ )by η p ( ϕ )( x . . . x ν ( p ) )( m ) = ϕ ( m ⊗ x . . . x ν ( p ) ) with obvious notations. It is a standard factthat the maps η p are linear isomorphisms. Their direct sum η : ( M ⊗ W ν ( • ) ) ∗ −→ Hom( W ν ( • ) , M ∗ )is a graded linear isomorphism. This isomorphism is natural in the A -bimodule M . Proposition 4.1.
The map η is an isomorphism from the complex C = (( M ⊗ W ν ( • ) ) ∗ , b ∗ K ) to the complex (Hom( W ν ( • ) , M ∗ ) , b K ) , lifting the identity of M ∗ and inducing a graded linearisomorphism H ( η ) : H • ( C ) → HK • ( A, M ∗ ) natural in the A -bimodule M .Proof. It amounts to prove that the diagram( M ⊗ W ν ( p ) ) ∗ η p −→ Hom( W ν ( p ) , M ∗ ) ↓ b ∗ K ↓ b K (4.1)( M ⊗ W ν ( p +1) ) ∗ η p +1 −→ Hom( W ν ( p +1) , M ∗ )9s commutative. Let φ ∈ ( M ⊗ W ν ( p ) ) ∗ and m ∈ M . Assuming that p is even, we have theequality b K ( η p ( φ ))( x . . . x ν ( p +1) )( m ) = η p ( φ )( x . . . x ν ( p ) )( x ν ( p +1) m ) − η p ( φ )( x . . . x ν ( p +1) )( mx ) . The right-hand side is equal to φ ( x ν ( p +1) m ⊗ x . . . x ν ( p ) ) − φ ( mx ⊗ x . . . x ν ( p +1) ) = − φ ◦ b K ( m ⊗ x . . . x ν ( p +1) )while the left-hand side coincides with b ∗ K ( φ )( m ⊗ x . . . x ν ( p +1) ). Thus b K ( η p ( φ )) = η p +1 ( b ∗ K ( φ ))as expected.When p is odd, we use a similar argument. We have the equality b K ( η p ( φ ))( x . . . x ν ( p +1) )( m ) = X ≤ i ≤ N − η p ( φ )( x i +1 . . . x i + ν ( p ) )( x i + ν ( p )+1 . . . x ν ( p +1) mx . . . x i )whose right-hand side is equal to X ≤ i ≤ N − φ ( x i + ν ( p )+1 . . . x ν ( p +1) mx . . . x i ⊗ x i +1 . . . x i + ν ( p ) ) = φ ◦ b K ( m ⊗ x . . . x ν ( p +1) ) . According to Remark 3.2 and the end of Subsection 2.3, the complexes C = ( M ⊗ W ν ( • ) ) ∗ and Hom( W ν ( • ) , M ∗ ) are weak DG ˜ A -bimodules. Proposition 4.2.
The map η is a natural isomorphism of weak DG ˜ A -bimodules, inducinga natural isomorphism H ( η ) of graded HK • ( A ) -bimodules.Proof. Let us prove that η is right ˜ A -linear, meaning that, for any q ≥ p ≥ p -cochain f : W ν ( p ) → A , the diagramHom( M ⊗ W ν ( q − p ) , k ) ( f⌢ K − ) ∗ −→ Hom( M ⊗ W ν ( q ) , k ) ↓ η q − p ↓ η q (4.2)Hom( W ν ( q − p ) , M ∗ ) ± ( − ⌣ K f ) −→ Hom( W ν ( q ) , M ∗ )is commutative, where ± = ( − ( q − p ) p . Note that this sign is a Koszul sign.The vertical arrows are isomorphisms. Define ψ : Hom( W ν ( q − p ) , M ∗ ) → Hom( W ν ( q ) , M ∗ )by ψ = η q ◦ ( f ⌢ K − ) ∗ ◦ ( η q − p ) − . Let g ∈ Hom( M ⊗ W ν ( q − p ) , k ) and h = ( f ⌢ K − ) ∗ ( g ).For brevity, denote η q − p ( g ) by g ′ and η q ( h ) by h ′ , so that ψ ( g ′ ) = h ′ . One has h =( − ( q − p ) p g ◦ ( f ⌢ K − ) so that h ( z ) = ( − ( q − p ) p g ( f ⌢ K z )for every z = m ⊗ x . . . x ν ( q ) ∈ M ⊗ W ν ( q ) . 10ssuming p = 2 p ′ + 1 and q = 2 q ′ , we have ν ( p ) = N p ′ + 1 and ν ( q ) = N q ′ . On one hand, h ′ ( x . . . x Nq ′ )( m ) = h ( z ) with h ( z ) = X ≤ i + j ≤ N − g ( x Nq ′ − Np ′ − N + i +2 . . . x Nq ′ − Np ′ − j − f ( x Nq ′ − Np ′ − j . . . x Nq ′ − j ) (4.3) x Nq ′ − j +1 . . . x Nq ′ mx . . . x i ⊗ x i +1 . . . x i + Nq ′ − Np ′ − N +1 ) . On the other hand, ν ( q − p ) = N q ′ − N p ′ − N + 1 and g ′ : W ν ( q − p ) → M ∗ is defined by g ′ ( x . . . x Nq ′ − Np ′ − N +1 )( m ) = g ( m ⊗ x . . . x Nq ′ − Np ′ − N +1 ), so that( g ′ ⌣ K f )( x . . . x Nq ′ ) = − X ≤ i + j ≤ N − x . . . x i g ′ ( x i +1 . . . x i + Nq ′ − Np ′ − N +1 ) (4.4) x i + Nq ′ − Np ′ − N +2 . . . x Nq ′ − Np ′ − j − f ( x Nq ′ − Np ′ − j . . . x Nq ′ − j ) x Nq ′ − j +1 . . . x Nq ′ provides ( g ′ ⌣ K f )( x . . . x Nq ′ )( m ) = − X ≤ i + j ≤ N − g ( x i + Nq ′ − Np ′ − N +2 . . . x Nq ′ − Np ′ − j − (4.5) f ( x Nq ′ − Np ′ − j . . . x Nq ′ − j ) x Nq ′ − j +1 . . . x Nq ′ mx . . . x i ⊗ x i +1 . . . x i + Nq ′ − Np ′ − N +1 ) . We obtain h ′ = − g ′ ⌣ K f as expected in the case p odd and q even.In the other cases, we have ( − ( q − p ) p = 1 and h ′ ( x . . . x ν ( q ) )( m ) = h ( z ) with h ( z ) = g ( f ( x ν ( q − p )+1 . . . x ν ( q ) ) m ⊗ x . . . x ν ( q − p ) ) , while ( g ′ ⌣ K f )( x . . . x ν ( q ) ) = g ′ ( x . . . x ν ( q − p ) ) f ( x ν ( q − p )+1 . . . x ν ( q ) ) provides( g ′ ⌣ K f )( x . . . x ν ( q ) )( m ) = g ( f ( x ν ( q − p )+1 . . . x ν ( q ) ) m ⊗ x . . . x ν ( q − p ) ) . Therefore h ′ = g ′ ⌣ K f as expected.Let us prove similarly the left linearity of η , that is, the commutativity of the diagramHom( M ⊗ W ν ( q − p ) , k ) ± ( − ⌢ K f ) ∗ −→ Hom( M ⊗ W ν ( q ) , k ) ↓ η q − p ↓ η q (4.6)Hom( W ν ( q − p ) , M ∗ ) f⌣ K − −→ Hom( W ν ( q ) , M ∗ )where ± = ( − pq (a Koszul sign). We use the same notations as above. Now ψ is definedby ψ = ( − pq η q ◦ ( − ⌢ K f ) ∗ ◦ ( η q − p ) − and h ( z ) = ( − pq ( − ( q − p ) p g ( z ⌢ K f ) = ( − p g ( z ⌢ K f ).11irst assume that p = 2 p ′ + 1 and q = 2 q ′ . We have h ( z ) = − X ≤ i + j ≤ N − g ( x Nq ′ − j +1 . . . x Nq ′ mx . . . x i f ( x i +1 . . . x Np ′ + i +1 ) (4.7) x Np ′ + i +2 . . . x Np ′ + N − j − ⊗ x Np ′ + N − j . . . x Nq ′ − j ) . Next from ( f ⌣ K g ′ )( x . . . x Nq ′ ) = − X ≤ i + j ≤ N − x . . . x i f ( x i +1 . . . x i + Np ′ +1 ) (4.8) x i + Np ′ +2 . . . x Np ′ + N − j − g ′ ( x Np ′ + N − j . . . x Nq ′ − j ) x Nq ′ − j +1 . . . x Nq ′ , we deduce that( f ⌣ K g ′ )( x . . . x Nq ′ )( m ) = − X ≤ i + j ≤ N − g ( x Nq ′ − j +1 . . . x Nq ′ mx . . . x i (4.9) f ( x i +1 . . . x i + Np ′ +1 ) x i + Np ′ +2 . . . x Np ′ + N − j − ⊗ x Np ′ + N − j . . . x Nq ′ − j ) , so that h ′ = f ⌣ K g ′ if p odd and q even.In the other cases, h ( z ) = ( − p g ( mf ( x . . . x ν ( p ) ) ⊗ x ν ( p )+1 . . . x ν ( q ) ) while( f ⌣ K g ′ )( x . . . x ν ( q ) ) = ( − pq f ( x . . . x ν ( p ) ) g ′ ( x ν ( p )+1 . . . x ν ( q ) ) , ( f ⌣ K g ′ )( x . . . x ν ( q ) )( m ) = ( − pq g ( mf ( x . . . x ν ( p ) ) ⊗ x ν ( p )+1 . . . x ν ( q ) ) . We conclude that h ′ = f ⌣ K g ′ .From the commutative diagrams (4.2) and (4.6), and from the definitions of the Koszul cupand cap brackets [5, Sections 4 and 6], we deduce immediately the following. Proposition 4.3.
Let A be an N -homogeneous algebra over a k -vector space V , and let M be an A -bimodule. For any q ≥ p ≥ and any Koszul p -cochain f : W ν ( p ) → A , there is acommutative diagram Hom( M ⊗ W ν ( q − p ) , k ) [ f, − ] ∗ ⌢K −→ Hom( M ⊗ W ν ( q ) , k ) ↓ η q − p ↓ η q (4.10)Hom( W ν ( q − p ) , M ∗ ) − [ f, − ] ⌣K −→ Hom( W ν ( q ) , M ∗ )The diagrams (4.2) and (4.6) pass to the respective cohomologies. Moreover we use theisomorphism ξ : H • (( M ⊗ W ν ( • ) ) ∗ ) −→ HK • ( A, M ) ∗ defined in Section 3, and we obtain a natural graded HK • ( A )-bimodule isomorphism ζ : HK • ( A, M ) ∗ −→ HK • ( A, M ∗ )12y ζ = H ( η ) ◦ ξ − . Consequently, for any q ≥ p ≥ α ∈ HK p ( A ), we have thecommutative diagrams HK q − p ( A, M ) ∗ ( α⌢ K − ) ∗ −→ HK q ( A, M ) ∗ ↓ ζ q − p ↓ ζ q (4.11)HK q − p ( A, M ∗ ) ± ( − ⌣ K α ) −→ HK q ( A, M ∗ )and HK q − p ( A, M ) ∗ ± ( − ⌢ K α ) ∗ −→ HK q ( A, M ) ∗ ↓ ζ q − p ↓ ζ q (4.12)HK q − p ( A, M ∗ ) α⌣ K − −→ HK q ( A, M ∗ )from which we deduce the following. Theorem 4.4.
Let A be an N -homogeneous algebra over a k -vector space V and let M bean A -bimodule. For any q ≥ p ≥ and any α ∈ HK p ( A ) , there is a commutative diagramin the category of k -vector spaces HK q − p ( A, M ) ∗ [ α, − ] ∗ ⌢K −→ HK q ( A, M ) ∗ ↓ ζ q − p ↓ ζ q (4.13)HK q − p ( A, M ∗ ) − [ α, − ] ⌣K −→ HK q ( A, M ∗ ) where the vertical arrows are isomorphisms. Corollary 4.5.
For any N -homogeneous algebra A over a k -vector space V , the assertion(i) implies the assertion (ii), where(i) For any A -bimodule M , any α ∈ HK • ( A ) and any β ∈ HK • ( A, M ) , [ α, β ] ⌣ K = 0 .(ii) For any A -bimodule M , any α ∈ HK • ( A ) and γ ∈ HK • ( A, M ) , [ α, γ ] ⌢ K = 0 .In particular, if A is N -Koszul, then (i) and (ii) hold.Proof. The implication ( i ) ⇒ ( ii ) is an immediate consequence of the theorem. As provedby Herscovich [13, Theorem 4.5], if A is N -Koszul, then the graded algebras HH • ( A ) andHK • ( A ) are isomorphic. Moreover, the graded HH • ( A )-bimodule HH • ( A, M ) is isomorphicto the graded HK • ( A )-bimodule HK • ( A, M ). Indeed, this fact is proved by Herscovich foractions on homology but his proof extends similarly to actions on cohomology. Therefore,when A is N -Koszul, (i) holds since it holds in Hochschild calculus by a classical result ofGerstenhaber [11].If we know that the A -bimodule M is Z -graded with respect to the weight grading of A , then M ∗ and more generally the functor Hom( M, − ) can be replaced by their graded versions,13ncluding a Koszul sign in the graded actions as usual. So we obtain graded versions w.r.t Z -gradings of bimodules of the maps ξ , η and ζ , and graded versions of the commutativediagrams (4.2)-(4.13). If moreover M is locally finite – that is, each component of M isfinite-dimensional – and if M is replaced by M ∗ , then M ∗ can be replaced by M , whichgives a true duality. When the A -bimodules involved in Corollary 4.5 are all assumed to belocally finite Z -graded, the assertions (i) and (ii) are equivalent. The aim of this section is to present some examples of non N -Koszul algebras satisfyingboth assertions (i) and (ii) of Corollary 4.5. Let us recall the following result from [5, Corollary 4.6].
Proposition 5.1.
Let A be an N -homogeneous algebra over a k -vector space V , and let M be an A -bimodule. For any α ∈ HK p ( A, M ) with p = 0 or p = 1 and any β ∈ HK q ( A ) with q ≥ , one has [ α, β ] ⌣ K = 0 . (5.1) Corollary 5.2. If A = T ( V ) / ( R ) is an N -homogeneous algebra such that the Koszul coho-mology algebra HK • ( A ) is generated in degrees and , then for any α ∈ HK • ( A ) and any β ∈ HK • ( A ) , one has [ α, β ] ⌣ K = 0 .Proof. Given α ∈ HK p ( A ) and β ∈ HK q ( A ), by hypothesis we can write α = X a j ⌣ K a j ⌣ K . . . ⌣ K a j p where the sum is finite, a j belongs to HK ( A ) and all the a j l ’s belong to HK ( A ). Thusthe result follows from Proposition 5.1 and from the fact that the cup bracket [ − , − ] ⌣ K is aderivation in the first argument on cohomology classes.As the next example shows, the condition assumed in Corollary 5.2 is sufficient but notnecessary.Let A = k h x, y i / ( yx, y − xy ). Let us denote by r the relation yx = 0 and by r the relation y − xy = 0. The Koszul complex K ( A ) is0 −→ A ⊗ W ⊗ A d −→ A ⊗ R ⊗ A d −→ A ⊗ V ⊗ A d −→ A ⊗ A −→ W has one generator that we will call w and the differentials are: • d (1 ⊗ v ⊗
1) = v ⊗ − ⊗ v for v ∈ V , as usual,14 d (1 ⊗ r ⊗
1) = y ⊗ x ⊗ ⊗ y ⊗ x , and d (1 ⊗ r ⊗
1) = y ⊗ y ⊗ ⊗ y ⊗ y − ⊗ x ⊗ y − x ⊗ y ⊗ • d (1 ⊗ w ⊗
1) = y ⊗ r ⊗ − x ⊗ r ⊗ − ⊗ r ⊗ x .The algebra A is not Koszul, since K ( A ) is not exact in homological degree 2. Indeed, theelement y ⊗ r ⊗ x − x ⊗ r ⊗ x − x ⊗ r ⊗ d but not in the image of d .Computing the Koszul cohomology, we obtain the following results. • HK ( A ) = HH ( A ) = Z ( A ) = { a + a x + a xy + P ni =3 a i x i , with n ≥ a i ∈ k } . • HK ( A ) = HH ( A ) is generated by the classes of the elements x ∗ ⊗ x + y ∗ ⊗ y and x ∗ ⊗ x i , for all i ≥ • HK ( A ) is 1-dimensional, generated by the class of r ∗ ⊗ y . Note that HH ( A ) is also1-dimensional. • HK ( A ) is 1-dimensional, generated by the class of w ∗ ⊗ xy . We note that HH ( A ) is2-dimensional. • Since the Koszul complex is zero in degrees greater or equal to 4, we have thatHK i ( A ) = 0 for i ≥ • ( A ) cannot be generated in degrees 0and 1. Moreover, this algebra is graded commutative by Proposition 5.1 with M = A . If the Koszul bimodule complex K ( A ) = A ⊗ W ν ( • ) ⊗ A has length at most 2, that is, if W ν ( p ) = 0 whenever p ≥
3, then HK p ( A, M ) = 0 for all p ≥ q ( A ) on HK p ( A, M ) are zero whenever p + q ≥
3. So Proposition 5.1 and Corollary4.5 imply the following.
Proposition 5.3.
Let A be an N -homogeneous algebra over a k -vector space V . If thelength of the Koszul bimodule complex K ( A ) is at most , then both assertions (i) and (ii)of Corollary 4.5 hold.Proof. It suffices to prove the equality (5.1) when p = 2 and q = 0. But HK ( A ) = Z ( A )the center of A , and Z ( A ) acts symmetrically on HK • ( A, M ).We apply now this proposition to the class of N -homogeneous algebras defined by a singlemonomial relation, as introduced in [4, Proposition 4.2]. Proposition 5.4.
Let V be a non-zero k -vector space of finite dimension n . Fix a basis ( x , . . . , x n ) of V . Let f = x i . . . x i N be a monomial of degree N ≥ , and let R be thesubspace of V ⊗ N generated by f . Then the N -homogeneous algebra A = T ( V ) / ( R ) satisfiesboth assertions (i) and (ii) of Corollary 4.5.Proof. It is proved in [4, Lemma 4.3] that W p = 0 for every p ≥ N + 1, except in the case f = x Ni for some i , 1 ≤ i ≤ n . If f = x Ni , then K ( A ) has length 2 and we conclude by15roposition 5.3. If f = x Ni , then A is N -Koszul (see Proposition below), and we concludeusing Corollary 4.5.Within the class of N -homogeneous algebras A defined by a single monomial relation, thenon N -Koszul algebras are characterized as follows [4, Proposition 4.2]. Proposition 5.5.
Let V be a non-zero k -vector space of finite dimension n . Fix a basis ( x , . . . , x n ) of V . Let f = x i . . . x i N be a monomial of degree N ≥ , and let R be thesubspace of V ⊗ N generated by f . Then A = T ( V ) / ( R ) is not N -Koszul if and only if thereexists m in { , . . . , N − } such that f = ( x i . . . x i m ) q x i . . . x i r , where N = mq + r with ≤ r < m , and where i , . . . , i m are not all equal. For example, if n = 2 and ( x, y ) is a basis of V , then A = k h x, y i / ( xyx ) is not 3-Koszul.Note that the algebras A considered in Proposition 5.5 are always Koszul when N = 2.In Subsection 2.2, we have recalled how the Koszul complex K ( A ) can be embedded into aminimal resolution P ( A ). We want to examine this embedding in the situation of Proposition5.5 when A is not N -Koszul. In this case, the Koszul complex is zero in degrees greater orequal to 3. This is not the case for the minimal resolution P ( A ). For monomial algebras, P ( A ) is isomorphic to Bardzell’s resolution [1]. Recall that Bardzell’s resolution can bewritten down · · · −→ A ⊗ R ⊗ A d −→ A ⊗ R ⊗ A d −→ A ⊗ V ⊗ A d −→ A ⊗ A −→ R is a generating set of the monomial relations and for all i ≥ R i is the set of( i − A = k h x, y i / ( xyx ), one has R i = { ( xy ) i − x } for all i ≥
2. In general, for A monomial, W is the subset of R of diagonal elements [1]. Denote by C the generalized Manin category of N -homogeneous k -algebras [6, 5] and by E the category of graded k -vector spaces. Recall that in C , the objects are the N -homogeneousalgebras and the morphisms are the morphisms of graded algebras. The A -bimodule A ∗ =Hom( A, k ) is defined by the actions ( a.u.a ′ )( x ) = u ( a ′ xa ) for any linear map u : A → k , and x , a , a ′ in A . We have the following result [5, Proposition 2.3]. Notice that in this statement, A ∗ can be replaced by the graded dual – including a Koszul sign in the left action – of the A -bimodule A endowed with the weight grading. Proposition 6.1.
The rules A HK • ( A ) and A HK • ( A, A ∗ ) respectively define acovariant functor X and a contravariant functor Y from C to E . Let us specialize the isomorphism ζ defined just before Theorem 4.4 to M = A . We obtaina graded HK • ( A )-bimodule isomorphism ζ : HK • ( A ) ∗ −→ HK • ( A, A ∗ ) , A . In other words, if we denote by L theendofunctor E E ∗ (graded dual) of E , then ζ defines an isomorphism of functors L ◦ X ∼ = Y from C to E . When A ∗ is replaced by the graded dual, one has a graded version of ζ satisfyingan analogous natural property.We want to combine the graded version of ζ with the Koszul duality developed for thequadratic Koszul calculus in [7, Section 8]. In particular, A ∗ denotes the graded dual of theweight-graded A -bimodule A , as in [7, Section 8]. Using the same notations and hypothesesas in [7, Section 8], we suppose that V is finite-dimensional, N = 2 and A = T ( V ) / ( R )is a quadratic algebra. We have the endofunctor D : A A ! of the Manin category C of quadratic k -algebras over finite-dimensional vector spaces. Recall that we have anisomorphism θ : HK • ( A ) ∼ = ˜HK • ( A ! , A ! ∗ )from the (HK • ( A ) , ⌣ K )-bimodule HK • ( A ) with actions ⌢ K , N × N -graded by the biweight,to the ( ˜HK • ( A ! ) , ˜ ⌣ K )-bimodule ˜HK • ( A ! , A ! ∗ ) with actions ˜ ⌣ K , N × N -graded by the in-verse biweight. This statement uses the fact that the bigraded algebras (HK • ( A ) , ⌣ K ) and( ˜HK • ( A ! ) , ˜ ⌣ K ) are isomorphic. See [7, Section 8] for details and proofs. For any p ≥ m ≥
0, one has a linear isomorphism θ p,m : HK p ( A ) m ∼ = ˜HK m ( A ! , A ! ∗ ) p . (6.1)As noted in [7, Remark 8.10], θ defines an isomorphism of functors X ∼ = ˜ Y ◦ D from C to E ,where ˜ Y is the contravariant functor A ˜HK • ( A, A ∗ ).Under the above assumptions, the maps ξ : H • (( A ⊗ W ν ( • ) ) ∗ ) −→ HK • ( A ) ∗ and η : ( A ⊗ W ν ( • ) ) ∗ −→ Hom( W ν ( • ) , A ∗ ) are homogeneous w.r.t. the biweight ( p, m ). Therefore thesame is true for the map ζ : HK • ( A ) ∗ −→ HK • ( A, A ∗ ). Then ζ is the direct sum of linearisomorphisms ζ p,m : HK p ( A ) ∗ m −→ HK p ( A, A ∗ ) m . Since the vector spaces involved in ζ p,m are finite-dimensional, we can consider the gradedHK • ( A )-bimodule isomorphism ζ ∗ : HK • ( A, A ∗ ) ∗ −→ HK • ( A ) , direct sum of the linear isomorphisms ζ ∗ p,m : HK p ( A, A ∗ ) ∗ m −→ HK p ( A ) m . Combining cup-cap duality and Koszul duality, we define an isomorphismΘ := θ ◦ ζ ∗ : HK • ( A, A ∗ ) ∗ −→ ˜HK • ( A ! , A ! ∗ )from the (HK • ( A ) , ⌣ K )-bimodule HK • ( A, A ∗ ) ∗ with transpose actions of ⌣ K , N × N -gradedby the biweight, to the ( ˜HK • ( A ! ) , ˜ ⌣ K )-bimodule ˜HK • ( A ! , A ! ∗ ) with actions ˜ ⌣ K , N × N -gradedby the inverse biweight. For any p ≥ m ≥
0, one has a linear isomorphismΘ p,m : HK p ( A, A ∗ ) ∗ m −→ ˜HK m ( A ! , A ! ∗ ) p .
17n conclusion, Θ defines an isomorphism of functors L ◦ Y ∼ = ˜ Y ◦ D from C to E .Similarly, define an isomorphismΘ ′ := ˜ ζ − ◦ θ ◦ D : HK • ( A ! ) −→ ˜HK • ( A ) ∗ from the (HK • ( A ! ) , ⌣ K )-bimodule HK • ( A ! ) to the ( ˜HK • ( A ) , ˜ ⌣ K )-bimodule ˜HK • ( A ) ∗ . For any p ≥ m ≥
0, one has a linear isomorphismΘ ′ p,m : HK p ( A ! ) m −→ ˜HK m ( A ) ∗ p . Then Θ ′ defines an isomorphism of functors X ◦ D ∼ = L ◦ ˜ X from C to E , where ˜ X : A ˜HK • ( A ). It is known by the experts that the cap product is graded symmetric on Hochschild homologyclasses. Because of lack of a suitable reference, we include here a proof of this result, thatwe present as a consequence of a Hochschild cup-cap duality. In fact the cup-cap dualityin Koszul calculus makes sense in a similar manner in Hochschild calculus for any nonnecessarily graded algebra.Let A be a unital associative k -algebra. In Hochschild calculus, the definition of the cupproduct ⌣ and the definition of the left and right cap product ⌢ coincide with the definitionsin Koszul calculus when N = 2 and when the elements x i of V involved in the spaces W p arechosen arbitrarily in A . The defining formulas of the cup and cap products for a Hochschild p -cochain f : A ⊗ p → P , a Hochschild q -cochain g : A ⊗ q → Q and a Hochschild q -chain z = m ⊗ a . . . a q ∈ M ⊗ A ⊗ q with coefficients in A -bimodules P , Q and M respectively, arethe following. ( f ⌣ g )( a . . . a p + q ) = ( − pq f ( a . . . a p ) ⊗ A g ( a p +1 . . . a p + q ) , (7.1) f ⌢ z = ( − ( q − p ) p ( f ( a q − p +1 . . . a q ) ⊗ A m ) ⊗ a . . . a q − p , (7.2) z ⌢ f = ( − pq ( m ⊗ A f ( a . . . a p )) ⊗ a p +1 . . . a q . (7.3)Denote by b the Hochschild differentials. For any A -bimodule M , we still have an isomor-phism of complexes η H : (Hom( M ⊗ A ⊗• , k ) , b ∗ ) → (Hom( A ⊗• , M ∗ ) , b )defined in the same way than η , that is, η Hq ( ϕ )( a . . . a q )( m ) = ϕ ( m ⊗ a . . . a q )18here ϕ : M ⊗ A ⊗ q → k , the a i ’s are in A and m ∈ M . Then it is easy to prove that, for q ≥ p ≥ p -cochain f : A ⊗ p → A , the diagramsHom( M ⊗ A ⊗ q − p , k ) ( f⌢ − ) ∗ −→ Hom( M ⊗ A ⊗ q , k ) ↓ η Hq − p ↓ η Hq (7.4)Hom( A ⊗ q − p , M ∗ ) ± ( − ⌣f ) −→ Hom( A ⊗ q , M ∗ )Hom( M ⊗ A ⊗ q − p , k ) ± ( − ⌢f ) ∗ −→ Hom( M ⊗ A ⊗ q , k ) ↓ η Hq − p ↓ η Hq (7.5)Hom( A ⊗ q − p , M ∗ ) f⌣ − −→ Hom( A ⊗ q , M ∗ )commute, where ± = ( − ( q − p ) p in (7.4) and ± = ( − pq in (7.5). In fact, following theproof of the commutativity of the diagrams (4.2) and (4.6), it suffices to assume that N = 2in this proof (no cases to distinguish) and that the elements x i are chosen arbitrarily in A .We leave the details to the reader.We also have a natural graded HH • ( A )-bimodule isomorphism θ H : H • (( M ⊗ A • ) ∗ ) −→ HH • ( A, M ) ∗ defined as in Section 3 and we introduce a natural graded HH • ( A )-bimodule isomorphism ζ H : HH • ( A, M ) ∗ −→ HH • ( A, M ∗ )by ζ H = H ( η H ) ◦ ( θ H ) − . The commutative diagrams (7.4) and (7.5) pass to cohomology,and for any q ≥ p ≥ α ∈ HH p ( A ), we have the commutative diagramHH q − p ( A, M ) ∗ [ α, − ] ∗ ⌢ −→ HH q ( A, M ) ∗ ↓ ζ Hq − p ↓ ζ Hq (7.6)HH q − p ( A, M ∗ ) − [ α, − ] ⌣ −→ HH q ( A, M ∗ )We know from Gerstenhaber [11] that [ α, β ] ⌣ = 0 for any A -bimodule M , any α ∈ HH • ( A )and any β ∈ HH • ( A, M ). From the commutative diagram (7.6), we thus obtain the expectedresult.
Theorem 7.1.
Let A be a unital associative algebra. Then the cap product is graded sym-metric on Hochschild homology classes, that is, for any A -bimodule M , any α ∈ HH • ( A ) and any γ ∈ HH • ( A, M ) , we have [ α, γ ] ⌢ = 0 . The application to functoriality developed in Section 6 follows along the same lines. Denoteby A the category of unital associative k -algebras. Recall that E is the category of graded k -vector spaces. We have functors X H : A HH • ( A ) and Y H : A HH • ( A, A ∗ ) from A to E [14]. Then the graded HH • ( A )-bimodule isomorphism ζ H : HH • ( A ) ∗ −→ HH • ( A, A ∗ )defines an isomorphism of functors L ◦ X H ∼ = Y H from A to E .19 An enriched structure on K ( A ) The motivation of this section comes from a general strategy, developed in the next section,for proving the graded commutativity of the Koszul cup product, that is, property (i) inCorollary 4.5. This general strategy is inspired by a similar strategy used in [9], where aKoszul complex Calabi-Yau property is introduced for any quadratic quiver algebra A , thisproperty implying a Poincar´e Van den Bergh duality for the Koszul homology/cohomologyof A with coefficients in any A -bimodule M . In order to obtain a more precise dualityexpressed as a cap product by a fundamental class, a stronger version of the Koszul complexCalabi-Yau property is defined in [9, Section 5]. The stronger definition consists in enrichingthe DG ˜ A -bimodules Hom( W • , M ) and M ⊗ W • by a compatible right action of an additionalassociative algebra B . Limiting us to the one vertex case, the DG ˜ A -bimodules Hom( W • , M )and M ⊗ W • are the same as those defined in Subsection 2.3 when N = 2.Our aim is now to generalize the enriched structure of the DG ˜ A -bimodule M ⊗ W • obtainedin the quadratic case [9, Section 3], to M ⊗ W ν ( • ) associated with any N -homogeneousalgebra A – we leave to the reader the analogous generalization for Hom( W ν ( • ) , M ). Next,following [9, Section 3], we will apply this construction to B = M = A e and then we willshow that K ( A ) has an enriched structure for any N -homogeneous algebra A .Throughout this section, A = T ( V ) / ( R ) denotes an N -homogeneous algebra over a field k ,where N ≥ R is a subspace of V ⊗ N . As usual, we set A e = A ⊗ A op and for any A -bimodule M , k is assumed to act centrally on M . So M can be considered as a left (right) A e -module.The additional data is the following: B is a unital associative k -algebra and the A -bimodule M is a right B -module such that the actions of k induced on M by A and by B areequal. Moreover, the bimodule actions of A and the right action of B on M are assumed tocommute, that is, for any m ∈ M , one has ( a.m.a ′ ) .b = a. ( m.b ) .a ′ for a , a ′ in A and b in B . Both properties are equivalent to saying that M is an A e - B -bimodule. We also say thatthe right B -module structure is compatible with the A -bimodule structure. For example, M = A e is an A e - A e -bimodule for the product of the algebra A e (here B = A e ).In the sequel, Vect k denotes the category of k -vector spaces, A - Bimod the category of A -bimodules, and Mod - B the category of right- B -modules. We have to work with DG modulesin abelian categories, following the general framework used by Yekutieli in his recent bookon derived categories [18]. Let us recall from [18, Definition 3.8.1] the definition of a DG˜ A -bimodule in the abelian category Mod - B . Actually, we extend in an obvious manner thisdefinition to weak DG bimodules over the weak DG algebra ˜ A . Definition 8.1.
A weak chain (cochain) DG ˜ A -bimodule C in Mod - B is a chain (cochain)complex in Mod - B endowed with a weak DG ˜ A -bimodule structure such that the bimoduleactions of ˜ A and the right action of B are compatible. Remark that if the abelian category
Mod - B is replaced by Vect k in this definition, a weakDG ˜ A -bimodule in Vect k is just a weak DG ˜ A -bimodule. Remark also that if the weak DGalgebra ˜ A is replaced by the ground field k (viewed as a trivial DG algebra), we just obtaincomplexes of right B -modules. 20e start with the weak chain DG ˜ A -bimodule M ⊗ W ν ( • ) as recalled at the end of Subsection2.3 for any A -bimodule M . Now M is an A e - B -bimodule. We have to define a right actionof B on M ⊗ W ν ( • ) . For that, we define a right action of B on each space M ⊗ W p , p ≥
0, asfollows. Fix a basis ( u α ) α of the space W p . Any element z of M ⊗ W p uniquely decomposesas a finitely supported sum z = P α m α ⊗ u α . For any b ∈ B , we set z.b = X α ( m α .b ) ⊗ u α . The following lemma shows that the element z.b of M ⊗ W p is well-defined, that is, it doesnot depend on the choice of the basis ( u α ) α . Lemma 8.2.
For any finite sum z = P i m i ⊗ w i with m i ∈ M and w i ∈ W p , one has X α ( m α .b ) ⊗ u α = X i ( m i .b ) ⊗ w i . (8.1) Proof.
Fix a basis ( e λ ) λ of the space V . Any u α uniquely decomposes as a finitely supportedsum u α = X λ ...λ p c αλ ...λ p ( e λ ⊗ · · · ⊗ e λ p )where the coefficients are in k , so that we have uniquely z = X α,λ ...λ p m α c αλ ...λ p ⊗ ( e λ ⊗ · · · ⊗ e λ p ) . (8.2)Similarly w i uniquely decomposes as a finitely supported sum w i = X λ ...λ p d iλ ...λ p ( e λ ⊗ · · · ⊗ e λ p )with coefficients in k , so we have uniquely z = X i,λ ...λ p m i d iλ ...λ p ⊗ ( e λ ⊗ · · · ⊗ e λ p ) . (8.3)Comparing (8.2) and (8.3), we have for any λ . . . λ p , X α m α c αλ ...λ p = X i m i d iλ ...λ p . If we act with b on this equality and use the compatibility of the actions, we get X α ( m α .b ) c αλ ...λ p = X i ( m i .b ) d iλ ...λ p . Tensoring by ( e λ ⊗ · · · ⊗ e λ p ) and summing the so-obtained equalities over the indices λ , . . . λ p , we obtain formula (8.1). 21s commonly used in Koszul calculus [7, 5, 9], we write down the element z of M ⊗ W p bythe compact notation z = m ⊗ x . . . x p recalled at the beginning of Subsection 2.1. Lemma8.2 shows that it makes sense to define z.b by the compact notation z.b = ( m.b ) ⊗ x . . . x p .We systematically use this compact notation in all the sequel. For example, it is clear fromthis notation that ( z.b ) .b ′ = z. ( bb ′ ). Notice that the action of k induced by B on M ⊗ W p is equal to the action defined by the k -vector space M ⊗ W p .Next, the differential b K of M ⊗ W ν ( • ) is right B -linear. It suffices to recall [5, Subsection2.2] that, for any q -chain z = m ⊗ x . . . x ν ( q ) , one has b K ( z ) = mx ⊗ x . . . x Nq ′ +1 − x Nq ′ +1 m ⊗ x . . . x Nq ′ (8.4)if q = 2 q ′ + 1, and b K ( z ) = X ≤ i ≤ N − x i + Nq ′ − N +2 . . . x Nq ′ mx . . . x i ⊗ x i +1 . . . x i + Nq ′ − N +1 (8.5)if q = 2 q ′ . The compatibility formula ( m.b ) .a = ( m.a ) .b valid for m ∈ M , a ∈ A and b ∈ B shows that b K ( z.b ) = b k ( z ) .b .Similarly, for any Koszul p -cochain f : W ν ( p ) → A , we easily verify that f ⌢ K ( z.b ) = ( f ⌢ K z ) .b and ( z.b ) ⌢ K f = ( z ⌢ K f ) .b from the defining expressions of f ⌢ K z and z ⌢ K f recalled in Definition 2.5. Therefore thebimodule actions of ˜ A and the right action of B are compatible. We can conclude by thefollowing. Proposition 8.3.
Let A = T ( V ) / ( R ) be an N -homogeneous algebra over a field k . Fixa unital associative k -algebra B and an A -bimodule M . We assume that M is a right B -module compatible with the A -bimodule structure. Then the chain complex M ⊗ W ν ( • ) is aweak DG ˜ A -bimodule in Mod - B . Let us specialize this result to B = M = A e as in [9, Section 3]. The left A e -module A e is identified with A o ⊗ A , that is, A ⊗ A endowed with the outer action ( a ⊗ a ′ ) . ( α ⊗ β ) =( aα ) ⊗ ( βa ′ ), where a , a ′ , α and β are in A . Similarly, the right A e -module is identifiedwith A i ⊗ A endowed with the inner action ( α ⊗ β ) . ( a ⊗ a ′ ) = ( αa ) ⊗ ( a ′ β ). Our aim is toidentify the A -bimodule complex K ( A ) with the complex (( A o ⊗ A ) ⊗ W • , b K ) endowed withthe right action of A e . The following proposition is an N -generalization of Proposition 3.5of [9] limited to the one vertex case. Proposition 8.4.
Let A = T ( V ) / ( R ) be an N -homogeneous algebra over a field k .(i) For any q > , the linear map ϕ q : ( A o ⊗ A ) ⊗ W q → A ⊗ W q ⊗ A defined by ϕ q (( α ⊗ β ) ⊗ x . . . x q ) = β ⊗ ( x . . . x q ) ⊗ α ia an isomorphism.(ii) The direct sum Φ of the maps ϕ ν ( q ) for q ≥ defines an isomorphism Φ from thecomplex (( A o ⊗ A ) ⊗ W ν ( • ) , b K ) to the Koszul complex ( K ( A ) , d ) . iii) The isomorphism Φ is right A e -linear.Proof. The assertion (i) is clear. Let us show that Φ is a morphism of complexes. Take z = ( α ⊗ β ) ⊗ x . . . x ν ( q ) . If q = 2 q ′ + 1, we deduce from b K ( z ) = ( α ⊗ βx ) ⊗ x . . . x Nq ′ +1 − ( x Nq ′ +1 α ⊗ β ) ⊗ x . . . x Nq ′ , the following Φ( b K ( z )) = βx ⊗ x . . . x Nq ′ +1 ⊗ α − β ⊗ x . . . x Nq ′ ⊗ x Nq ′ +1 α whose right-hand side is equal to d ( β ⊗ x . . . x Nq ′ +1 ⊗ α ).Now if q = 2 q ′ , from b K ( z ) = X ≤ i ≤ N − ( x i + Nq ′ − N +2 . . . x Nq ′ α ⊗ βx . . . x i ) ⊗ x i +1 . . . x i + Nq ′ − N +1 we get Φ( b K ( z )) = X ≤ i ≤ N − βx . . . x i ⊗ x i +1 . . . x i + Nq ′ − N +1 ⊗ x i + Nq ′ − N +2 . . . x Nq ′ α whose right-hand side is equal to d ( β ⊗ x . . . x Nq ′ ⊗ α ).Let us prove (iii) . Here the A -bimodule A ⊗ W q ⊗ A is seen as a right A e -module. For z = ( α ⊗ β ) ⊗ x . . . x q and a , a ′ in A , we have ϕ q ( z. ( a ⊗ a ′ )) = ϕ q (( αa ⊗ a ′ β ) ⊗ x . . . x q )= a ′ β ⊗ ( x . . . x q ) ⊗ αa = ϕ q ( z ) . ( a ⊗ a ′ ) , therefore ϕ q is right A e -linear for any q ≥
0. In particular, Φ q = ϕ ν ( q ) is right A e -linear forany q ≥ A o ⊗ A ) ⊗ W • is a weak DG ˜ A -bimodule in A - Bimod .We transport this structure via the chain complex isomorphism Φ and we obtain.
Proposition 8.5.
Let A = T ( V ) / ( R ) be an N -homogeneous algebra over a field k . Thenthe Koszul bimodule complex K ( A ) is a weak DG ˜ A -bimodule in A - Bimod . Let us give explicitly the underlying weak ˜ A -bimodule structure of the weak DG ˜ A -bimodule K ( A ). Consider z = ( α ⊗ β ) ⊗ x . . . x ν ( q ) in ( A o ⊗ A ) ⊗ W ν ( q ) and f in Hom( W ν ( p ) , A ). Set z ′ = Φ q ( z ) = β ⊗ x . . . x ν ( q ) ⊗ α .If p and q − p are not both odd, we draw from the formulas giving f ⌢ K z and z ⌢ K f inDefinition 2.5 that the actions of f on K ( A ) are defined by f ⌢ K z ′ = ( − ( q − p ) p β ⊗ x . . . x ν ( q − p ) ⊗ f ( x ν ( q − p )+1 . . . x ν ( q ) ) α, (8.6) z ′ ⌢ K f = ( − pq βf ( x . . . x ν ( p ) ) ⊗ x ν ( p )+1 . . . x ν ( q ) ⊗ α. (8.7)23f p = 2 p ′ + 1 and q = 2 q ′ , we use again the formulas giving f ⌢ K z and z ⌢ K f in Definition2.5, and we easily deduce f ⌢ K z ′ = − X ≤ i + j ≤ N − ( βx . . . x i ) ⊗ ( x i +1 . . . x i + Nq ′ − Np ′ − N +1 ) (8.8) ⊗ ( x Nq ′ − Np ′ − N + i +2 . . . x Nq ′ − Np ′ − j − f ( x Nq ′ − Np ′ − j . . . x Nq ′ − j ) x Nq ′ − j +1 . . . x Nq ′ α ) ,z ′ ⌢ K f = X ≤ i + j ≤ N − ( βx . . . x i f ( x i +1 . . . x Np ′ + i +1 ) x Np ′ + i +2 . . . x Np ′ + N − j − ) (8.9) ⊗ ( x Np ′ + N − j . . . x Nq ′ − j ) ⊗ ( x Nq ′ − j +1 . . . x Nq ′ α ) . We would like to obtain a proof of the graded commutativity of the Koszul cup product forany N -homogeneous algebra A , that is, a proof of property (i) in Corollary 4.5. Our idea isto divide such a proof in two steps as follows.1) First Step: use the fact that the Koszul cohomology is isomorphic to a Hochschild hy-percohomology. This fact was obtained when N = 2 in [7, Subsection 2.3], and extendedwithout detailed proof to any N in [5, Subsection 2.2].2) Second Step: show that the isomorphism of the First Step sends the Koszul cup productto a Hochschild cup product. Then we could conclude by using the graded commutativityof the Hochschild cup product [11].The First Step is stated in Proposition 9.1 below, including a detailed proof. The SecondStep lies on the general strategy presented in the previous section. The enriched structureswill be explicitly described for any N -homogeneous algebra A in Subsection 9.3 below. Anextra hypothesis (H) is needed even if N = 2 in order to make use of derived categories(Theorem 9.4). Moreover we obtain a product on the image of the isomorphism that wewould still need to relate to a usual Hochschild cup product. Proposition 9.1.
Let A = T ( V ) / ( R ) be an N -homogeneous algebra over a field k . For any A -bimodule M , the Koszul cohomology HK • ( A, M ) is isomorphic, as a graded vector space,to the Hochschild hypercohomology HH • ( A, Hom A ( K ( A ) , M )) .Proof. For the definition of the derived category D ( C ) of an abelian category C , we referto [17, Chapter 10]. We will make precise below the boundedness conditions which wewill use. Denote by Vect k the category of k -vector spaces and by A - Bimod the category of A -bimodules. 24ur proof is based on an isomorphism of functors depending on a fixed A -bimodule M that we will consider as left A e -module. In general, unless the contrary is explicitly stated, A -bimodules will always be identified to left A e -modules. Let F : A - Bimod → Vect k bethe contravariant functor F : P Hom A e ( P, M ). Let G : A - Bimod → A - Bimod be thecontravariant functor G : P Hom A ( P, M ) where Hom A ( P, M ) denotes the space of left A -module morphisms u : P → M , endowed with the following A -bimodule structure( a.u.a ′ )( x ) = u ( xa ) a ′ , for a, a ′ ∈ A, x ∈ P. (9.1)Finally, let H : A - Bimod → Vect k be the covariant functor H : P Hom A e ( A, P ).Denote by C + ( A - Bimod ) the category of chain complexes C = ( C n ) of A -bimodules whichare bounded below, that is, such that C n = 0 for all n <<
0. Denote by C + ( A - Bimod ) and C + ( Vect k ) the categories of the cochain complexes C = ( C n ) of A -bimodules –respectively,vector spaces– which are bounded below, that is, such that C n = 0 for all n <<
0. Thenwe have natural functors that we deduce from F , G , H and we still denote as before F : C + ( A - Bimod ) → C + ( Vect k ), G : C + ( A - Bimod ) → C + ( A - Bimod ) and H : C + ( A - Bimod ) →C + ( Vect k ).Following [17, Chapter 10], we consider the right derived functors RF : D + ( A - Bimod ) →D + ( Vect k ), RG : D + ( A - Bimod ) → D + ( A - Bimod ) and RH : D + ( A - Bimod ) → D + ( Vect k ).We define a natural transformation of functors ρ M : F → H ◦ G as follows. For any A -bimodule P , we first define the map ρ PM : Hom A e ( P, M ) → Hom A e ( A, Hom A ( P, M ))by ρ PM ( u )( a )( x ) = u ( xa ) for any A -bimodule morphism u : P → M , a in A and x in P . Itis easy to check that ρ PM ( u ) belongs to Hom A e ( A, Hom A ( P, M )). Since ρ PM ( u )(1) = u , itfollows that ρ PM is linear. For any v in Hom A e ( A, Hom A ( P, M )), define ρ ′ PM ( v ) = v (1). Then ρ ′ PM ( v ) belongs to Hom A e ( P, M ) and the map ρ ′ PM is linear. Consequently, the maps ρ PM and ρ ′ PM are inverse to each other. So ρ PM is a linear isomorphism.Moreover ρ PM is functorial in P , defining thus a natural isomorphism ρ M : F ∼ = H ◦ G from C + ( A - Bimod ) to C + ( Vect k ). Therefore, R ( ρ M ) is an isomorphism of functors RF ∼ = RH ◦ RG from D + ( A - Bimod ) to D + ( Vect k ). In particular, for any bounded below chain complex C ofprojective A -bimodules, one has an isomorphism R ( ρ M )( C ) : RHom A e ( C, M ) ∼ = RHom A e ( A, RHom A ( C, M )) (9.2)in D + ( Vect k ). Applying this isomorphism to C = K ( A ), passing to cohomology and usingthe definition of the hypercohomology [17, Chapter 10], we obtain a graded linear isomor-phism HK • ( A, M ) ∼ = HH • ( A, Hom A ( K ( A ) , M )) . (9.3)Remark that, if A is N -Koszul, then K ( A ) ∼ = A in D + ( A - Bimod ) so that we recoverHK • ( A, M ) ∼ = HH • ( A, M )), see [5, Subsection 2.2].25 .3 Koszul cohomology and Hochschild hypercohomology, with en-riched structures
Let us present precisely our strategy for proving property (i) in Corollary 4.5 for any N -homogeneous algebra A = T ( V ) / ( R ). We have to work with weak DG ˜ A -bimodules in theabelian category A - Bimod . As seen in Proposition 8.5, K ( A ) is such a weak DG ˜ A -bimodulein A - Bimod . Recall Definition 8.1 in the particular case B = A e . Definition 9.2.
A weak chain (cochain) DG ˜ A -bimodule C in A - Bimod is a chain (cochain)complex in A - Bimod endowed with a weak DG ˜ A -bimodule structure such that the bimoduleactions of ˜ A and A are compatible. Denote by C w + ( ˜ A, A - Bimod ) the category of weak bounded below chain DG ˜ A -bimodulesin A - Bimod . Denote by C + w ( ˜ A, A - Bimod ) and C + w ( ˜ A, Vect k ) the category of weak boundedbelow cochain DG ˜ A -bimodules in A - Bimod and
Vect k respectively. Notice that C + w ( ˜ A, Vect k )coincides with the category C + w ( ˜ A ) of weak bounded below cochain DG ˜ A -bimodules.In case N = 2, K ( A ) is a DG ˜ A -bimodule in A - Bimod , so that in the above notations we candrop the subscript w and we have that C + ( ˜ A, A - Bimod ) is the category of bounded belowchain DG ˜ A -bimodules in A - Bimod , while C + ( ˜ A, A - Bimod ) and C + ( ˜ A, Vect k ) = C + ( ˜ A ) arethe categories of bounded below cochain DG ˜ A -bimodules in A - Bimod and
Vect k respectively.We begin with the general case N ≥
2. Fix an A -bimodule M and a weak chain DG ˜ A -bimodule C in A - Bimod . It is easy to check that Hom A e ( C, M ), where Hom A e ( C, M ) q :=Hom A e ( C q , M ), is a weak cochain DG ˜ A -bimodule in Vect k for the actions( f.u )( x ) = ( − p u ( x.f ) , ( u.f )( x ) = u ( f.x ) , (9.4)where f : A ⊗ k W p ⊗ k A → A , u : C q → M are morphisms of A -bimodules, and x ∈ C p + q .Note that x.f and f.x are in C q by the graded actions of ˜ A on C . Recall also that if d : C q +1 → C q is the differential of C , the differentialHom A e ( d, M ) : Hom A e ( C q , M ) → Hom A e ( C q +1 , M )of Hom A e ( C, M ) is defined by Hom A e ( d, M )( u ) = − ( − q u ◦ d for u ∈ Hom A e ( C q , M ). Sowe have defined a functorHom A e ( − , M ) : C w + ( ˜ A, A - Bimod ) → C + w ( ˜ A, Vect k ) . Note that if C = K ( A ), then x.f = x ⌢ K f and f.x = f ⌢ K x are explicitly expressed justafter Proposition 8.5, so that we recover the Koszul cup actions, that is, f.u = f ⌣ K u and u.f = u ⌣ K f . Remark also that, if N = 2 and C is a chain DG ˜ A -bimodule in A - Bimod ,then Hom A e ( C, M ) is a cochain DG ˜ A -bimodule in Vect k , as seen in [9, Subsection 5.3].Keeping the same notation but now u : C q → M is only left A -linear, we assert thatHom A ( C, M ), where Hom A ( C, M ) q := Hom A ( C q , M ), is a weak cochain DG ˜ A -bimodule in A - Bimod . Actually, we just explain the structures involved in Definition 9.2 and we leavethe verifications of their properties to the reader. Firstly the differentialHom A ( d, M ) : Hom A ( C q , M ) → Hom A ( C q +1 , M )26f Hom A ( C, M ) is defined by Hom A ( d, M )( u ) = − ( − q u ◦ d for u ∈ Hom A ( C q , M ). It is adifferential of A -bimodules for the actions defined in (9.1). Secondly, the actions of ˜ A beingdefined by the same formulas (9.4), it is easy to verify that f.u and u.f are left A -linear.Finally it is routine to check that the actions of A and ˜ A on Hom A ( C, M ) are compatible,and that the following formulas holdHom A ( d, M )( f.u ) = b K ( f ) .u + ( − p f. Hom A ( d, M )( u ) , (9.5)Hom A ( d, M )( u.f ) = Hom A ( d, M )( u ) .f + ( − q u.b K ( f ) . (9.6)In conclusion, we have defined a functorHom A ( − , M ) : C w + ( ˜ A, A - Bimod ) → C + w ( ˜ A, A - Bimod ) . Note that if C = K ( A ) and u is only left A -linear, we have only f.u = f ⌣ K u . Remarkagain that, if N = 2 and C is a chain DG ˜ A -bimodule in A - Bimod , we can remove “weak”everywhere as well as the subscript w (we leave the verifications to the reader).Similarly, for any weak cochain DG ˜ A -bimodule ( C, d ) in A - Bimod , Hom A e ( A, C ), whereHom A e ( A, C ) q := Hom A e ( A, C q ), is a weak cochain DG ˜ A -bimodule in Vect k . The differen-tial Hom A e ( A, d ) is defined by Hom A e ( A, d )( v ) = d ◦ v for v : A → C . The actions of f : A ⊗ k W p ⊗ k A → A on v : A → C p + q are defined by( f.v )( a ) = f.v ( a ) , ( v.f )( a ) = v ( a ) .f, (9.7)for any a ∈ A . It is straightforward to show that we have so obtained a functorHom A e ( A, − ) : C + w ( ˜ A, A - Bimod ) → C + w ( ˜ A, Vect k ) . Here again, if N = 2 and C is a cochain DG ˜ A -bimodule in A - Bimod , we can remove “weak”and the subscript w everywhere. Proposition 9.3.
Let A = T ( V ) / ( R ) be an N -homogeneous algebra over a field k and M be an A -bimodule. The isomorphism of functors ρ M : F ∼ = H ◦ G defined in the proof ofProposition 9.1 has an enriched version ρ M : Hom A e ( − , M ) ∼ = Hom A e ( A, − ) ◦ Hom A ( − , M ) which is an isomorphism of functors from C w + ( ˜ A, A - Bimod ) to C + w ( ˜ A ) . If N = 2 , one has anenriched version ρ M : Hom A e ( − , M ) ∼ = Hom A e ( A, − ) ◦ Hom A ( − , M ) which is an isomorphism of functors from C + ( ˜ A, A - Bimod ) to C + ( ˜ A ) .Proof. Fix a weak chain DG ˜ A -bimodule ( C, d ) in A - Bimod . It is easy to check that ρ CM : Hom A e ( C, M ) ∼ = Hom A e ( A, Hom A ( C, M ))27s a morphism of complexes. It remains to verify that ρ CM is a morphism of weak ˜ A -bimodules.We use the structures of weak ˜ A -bimodules introduced above. Consider the following mor-phisms of A -bimodules: f : A ⊗ k W p ⊗ k A → A , f ′ : A ⊗ k W p ′ ⊗ k A → A and u : C q → M .Then f.u.f ′ ∈ Hom A e ( C p + p ′ + q , M ) and we take a ∈ A and x ∈ C p + p ′ + q .On one hand, one has ρ CM ( f.u.f ′ )( a )( x ) = ( − p u ( f ′ . ( xa ) .f ). On the other hand, one has( f.ρ CM ( u ) .f ′ )( a ) = f.ρ CM ( u )( a ) .f ′ and( f.ρ CM ( u )( a ) .f ′ )( x ) = ( − p ρ CM ( u )( a )( f ′ .x.f ) = ( − p u (( f ′ .x.f ) a ) . But the bimodule actions of ˜ A and of A on C are compatible. In particular, f ′ . ( xa ) .f =( f ′ .x.f ) a , so that we conclude that ρ CM ( f.u.f ′ ) = f.ρ CM ( u ) .f ′ .If N = 2, we can remove “weak” and the subscript w everywhere.We do not know whether the corresponding derived categories make sense if N >
2. How-ever, if N = 2, we can remove “weak” and the subscript w , and so we recover the frameworkof Yekutieli [18]. Consequently, we assume that N = 2 throughout the remainder of thissection. In this situation, the derived categories D + ( ˜ A, A - Bimod ), D + ( ˜ A, A - Bimod ) and D + ( ˜ A, Vect k ) = D + ( ˜ A ) are defined in [18]. We are now ready to introduce the followinghypothesis (H). Hypothesis (H): The right derived functors RHom A e ( − , M ) : D + ( ˜ A, A - Bimod ) → D + ( ˜ A )and RHom A ( − , M ) : D + ( ˜ A, A - Bimod ) → D + ( ˜ A, A - Bimod ) exist for any A -bimodule M , aswell as the right derived functor RHom A e ( A, − ) : D + ( ˜ A, A - Bimod ) → D + ( ˜ A ).In his book, Yekutieli shows that if suitable resolutions exist, then the functors betweencategories of DG modules in an abelian category can be derived [18, Theorem 10.1.20,Theorem 10.2.15, Theorem 10.4.8, Theorem 10.4.9]. We do not know whether these generalresults by Yekutieli can be applied, in the present context, to the existence of injectiveresolutions for proving the assertion (H). However, from Proposition 9.3 and from generalproperties of abstract derived functors [18, Subsection 8.3], we obtain the following. Theorem 9.4.
Let A = T ( V ) / ( R ) be a quadratic algebra over a field k and M be an A -bimodule. Under the hypothesis (H), the isomorphism of functors ρ M : Hom A e ( − , M ) ∼ =Hom A e ( A, − ) ◦ Hom A ( − , M ) from C + ( ˜ A, A - Bimod ) to C + ( ˜ A ) induces an isomorphism offunctors R ( ρ M ) : RHom A e ( − , M ) ∼ = RHom A e ( A, − ) ◦ RHom A ( − , M ) from D + ( ˜ A, A - Bimod ) to D + ( ˜ A ) . In particular, for any DG ˜ A -bimodule C in A - Bimod which is a bounded below chain complexof projective A -bimodules, we obtain an isomorphismRHom A e ( C, M ) ∼ = RHom A e ( A, RHom A ( C, M )) (9.8)in D + ( ˜ A ). Applying this isomorphism to C = K ( A ) and passing to cohomology, we get thatthe graded linear isomorphism (9.3), that is,HK • ( A, M ) ∼ = HH • ( A, Hom A ( K ( A ) , M ))28s now an isomorphism of graded HK • ( A )-bimodules.Therefore, in order to prove that the graded HK • ( A )-bimodule HK • ( A, M ) is commutative,it suffices to prove that the graded HK • ( A )-bimodule HH • ( A, Hom A ( K ( A ) , M )) is commu-tative. Unfortunately, we have not succeeded to relate the so-obtained graded action ofHK • ( A ) on HH • ( A, Hom A ( K ( A ) , M )) to the usual Hochschild cup product which is knownto be graded commutative.Finally, pursuing the same strategy used in this section, we would like to suggest a proof ofthe graded commutativity in the nonquadratic case N >
2. In this case, the first author hasconjectured in [5] that the weak DG algebra ˜ A is actually an A ∞ algebra and that the weakDG ˜ A -bimodule Hom( W ν ( • ) , M ) is an A ∞ bimodule over this A ∞ algebra (Herscovich hasdefined such structures when A is N -Koszul [13]). Taking this conjecture as granted, theisomorphism ρ M in Proposition 9.3 should be an isomorphism of A ∞ ˜ A -bimodules. Thenthe isomorphism R ( ρ M ) in Theorem 9.4 could be generalized to N > ∞ categories. References [1] M. Bardzell, The alternating syzygy behavior of monomial algebras,
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Roland Berger: Univ Lyon, UJM-Saint-´Etienne, CNRS UMR 5208, Institut Camille Jordan, F-42023, Saint-´Etienne, France [email protected]
Andrea Solotar: IMAS and Dto de Matem´atica, Facultad de Ciencias Exactas y Naturales,Universidad de Buenos Aires, Ciudad Universitaria, Pabell`on 1, (1428) Buenos Aires, Argentina [email protected]@dm.uba.ar