On the Hochschild homology of involutive algebras
OON THE HOCHSCHILD HOMOLOGY OF INVOLUTIVE ALGEBRAS
RAMS `ES FERN `ANDEZ-VAL `ENCIA AND JEFFREY GIANSIRACUSAA
BSTRACT . We study the homological algebra of bimodules over involutive associativealgebras. We show that Braun’s definition of involutive Hochschild cohomology in terms ofthe complex of involution-preserving derivations is indeed computing a derived functor: the Z / Z /
1. I
NTRODUCTION
Hochschild cohomology is a cohomology theory for associative algebras that describestheir deformation theory. Under mild hypotheses, the groups HH ∗ ( A , A ) can be defined inany of the following equivalent ways:(1) the homology of the usual Hochschild cochain complex of A ,(2) the homology of the complex of coderivations on the tensor coalgebra of Σ A (or equivalently, the complex of continuous derivations on the completed tensoralgebra of Σ − A ∨ ).(3) the derived center of A ,There is a corresponding Hochschild homology theory that can be defined as the derivedabelianization of A or by writing down the usual Hochschild chain complex. The derivedfunctor description is perhaps most fundamental and it is based on the fact that A -bimodulesform an abelian category that can equivalently be described as the category of right modulesover the enveloping algebra A e = A ⊗ A op .In [Bra14], Braun studied the Hochschild theory of involutive algebras (and A ∞ -algebras),meaning algebras equipped with a map a (cid:55)→ a ∗ such that a ∗∗ = a and ( ab ) ∗ = b ∗ a ∗ . He in-troduced an involutive variant of Hochschild cohomology by restricting to the subcomplexof the derivation complex consisting of derivations that commute with the involution. Theordinary Hochschild cohomology of an involutive algebra splits as a sum of this involutiveHochschild cohomology and a skew factor (assuming the characteristic of the ground fieldis not 2).The purpose of this short note is to develop enough homological algebra for bimod-ules over involutive algebras to give a derived functor description of Braun’s involutiveHochschild cohomology. From this perspective we are also able to define the correspond-ing involutive Hochschild homology theory. One key novel feature of the involutive theoryis that it is based on the abelian category of involutive bimodules . In contrast with the D EPARTMENT OF M ATHEMATICS , S
WANSEA U NIVERSITY , S
INGLETON P ARK , S
WANSEA
SA2 8PP,UK
E-mail addresses : [email protected], [email protected] . Date : August 24, 2018. a r X i v : . [ m a t h . A T ] J u l ON THE HOCHSCHILD HOMOLOGY OF INVOLUTIVE ALGEBRAS non-involutive case, involutive bimodules are actually equivalent to modules over a certainsemidirect product of the enveloping algebra with the group ring k [ Z / ] .Our motivation for studying involutive Hochschild theory comes from the first author’swork on unoriented topological conformal field theories. Costello [Cos07] showed that anopen 2d oriented TCFT is essentially a Calabi-Yau A ∞ -algebra, and such a theory admitsa universal extension to an open-closed theory with closed state space (the value of thefunctor on a circle) given by the Hochschild chain complex of the algebra of the opentheory. In [FV15], this picture is extended to Klein (i.e., unoriented) 2d TCFTs: opentheories now correspond to involutive Calabi-Yau A ∞ -algebras, and the closed state spaceof the universal open-closed extension turns out to be the involutive Hochschild chaincomplex of the open state algebra. Acknowledgements.
Both authors were supported by EPSRC grant EP/I003908/2.2. I
NVOLUTIVE ALGEBRAS AND THEIR BIMODULES
Involutive algebras.
Let k be a field. An involutive vector space is a vector space V (assumed to be over k ) equipped with an automorphism of order 2, which we will usuallywrite as v (cid:55)→ v ∗ . I.e., it is a representation of the cyclic group Z /
2. We let i Vect k denote thecategory of involutive k -vector spaces and linear maps that commute with the involutions.An involutive k-algebra is an involutive vector space A equipped with an associative andunital multiplication map A ⊗ k A → A such that ( ab ) ∗ = b ∗ a ∗ for any a , b ∈ A . Note that it follows automatically that 1 ∗ = ∗ = Remark . An associative k -algebra is the same as a monoid in the monoidal categoryof vector spaces with tensor product. Although involutive vector spaces are the same as Z / ⊗ k gives this category a monoidal structure,involutive algebras are not the same as monoids in the monoidal category of Z / ⊗ Z / alsoprovides a monoidal product on the category, but involutive algebras are not monoids forthis structure. Example . (1) Any commutative algebra A becomes an involutive algebra whenequipped with the identity as involution. More generally, any k -algebra map oforder 2 fixing 1 makes A an involutive algebra.(2) Let V be an involutive vector space and let TV = (cid:76) n V ⊗ n be the tensor algebra on V . The tensor algebra becomes an involutive algebra with involution given by ( v ⊗ · · · ⊗ v n ) ∗ = v ∗ n ⊗ · · · ⊗ v ∗ . (3) Let G be a discrete group. The group ring k [ G ] is an involutive k -algebra withinvolution given by g (cid:55)→ g − . N THE HOCHSCHILD HOMOLOGY OF INVOLUTIVE ALGEBRAS 3
Involutive bimodules.
First suppose that A is an associative k -algebra. An A -bimodule M is a k -vector space with left and right multiplication maps A ⊗ k M → M and M ⊗ k A → M that commute: ( a · m ) · b = a · ( m · b ) for all a , b ∈ A and m ∈ M . Incategory-theoretic terms, M is a bimodule for the monoid A in the monoidal category ( Vect k , ⊗ k ) . Equivalently, an A -bimodule is the same as a left module over the envelopingalgebra A e = A ⊗ k A op .Now let A be an involutive k -algebra. An involutive A-bimodule is a bimodule equippedwith an involution satisfying the compatibility condition between the left and right actionsand the involution. ( a · m ) ∗ = m ∗ · a ∗ . Note that, unlike the non-involutive case, here the left and right A -module structuresdetermine each other, so an involutive bimodule is determined by a vector space equippedwith both a left A -module structure and an involution, but there is a compatibility conditionthat these two structures must satisfy coming from the fact that the left and right A -modulestructures on a bimodule commute. This condition is:(2.2.1) b · ( a · m ∗ ) ∗ = b · ( m · a ∗ ) = ( b · m ) · a ∗ = ( a · ( b · m ) ∗ ) ∗ for a , b ∈ A and m in an involutive bimodule M .One can describe the category of involutive bimodules as a category of left modules asfollows. Consider the algebra A ie : = A e ⊗ k [ Z / ] with product defined by ( x ⊗ τ i ) · ( y ⊗ τ j ) = ( x · τ i ( y )) ⊗ τ i + j , where τ is the generator of Z / A e = A ⊗ A op by τ ( a ⊗ b ) = b ∗ ⊗ a ∗ . We call A ie the involutive enveloping algebra of A . Proposition 2.2.1.
There is an equivalence of categoriesA - i Bimod ∼ = A ie - Mod . Proof.
The subalgebra of A ie consisting of elements of the form x ⊗ A e .Given an A ie -module M , the action of A e ⊂ A ie defines an A -bimodule struture on M asusual. The action of the subalgebra k [ Z / ] defines an involution on M , and this in factyields an involutive bimodule since, by the associativity of the A ie -action, multiplying by ( ⊗ ) ⊗ τ · ( a ⊗ b ) ⊗ τ = ( b ∗ ⊗ a ∗ ) ⊗ ( a ⊗ b ) ⊗ τ and then by ( ⊗ ) ⊗ τ . In terms of the inducedbimodule structure and involution on M , this says that ( b ∗ ma ∗ ) is equal to ( am ∗ b ) ∗ , andhence M becomes an involutive bimodule.Conversely, if M is an involutive bimodule then it becomes an A e -module, its involutionmakes it a module over k [ Z / ] , and the compatibility relation ( amb ) ∗ = b ∗ ma ∗ says thatthe involution and bimodule structure combine to define an associative action of A ie . (cid:3) One sees that the forgetful functor A - i Bimod → A - Bimod . is faithful; however, it fails to be conservative (meaning that there are involutive bimodules M and N that are not isomorphic in A - i Bimod , but they become isomorphic in A - Bimod ),and hence it is not full. The following simple example illustrates this.
ON THE HOCHSCHILD HOMOLOGY OF INVOLUTIVE ALGEBRAS
Example . Let A = k with the trivial involution, so A -bimodules are just k -vectorspaces, and involutive A -bimodules are just involutive vector spaces. Let V = k with thetrivial involution, and let W = k with involution ( x , y ) ∗ = ( y , x ) . As bimodules (i.e., vectorspaces), V and W are clearly isomorphic, but as involutive bimodules (i.e., involutive vectorspaces) they are not.If M and N are involutive A -bimodules, then we write Hom A - i Bimod ( M , N ) for the set ofinvolutive A -bimodule homomorphisms from M to N , which is to say the set of bimodulehomomorphisms that commute with the involutions. Both this and the set of bimodulehomomophisms are k -vector spaces and there is a natural linear inclusion mapHom A - i Bimod ( M , N ) (cid:44) → Hom A - Bimod ( M , N ) . However, the vector space Hom A - i Bimod ( M , N ) also carries an involution f (cid:55)→ f ∗ definedby f ∗ ( m ) = f ( m ) ∗ , or equivalently, f ( m ∗ ) . Some functors and adjunctions.
Let A be a k -algebra. We first recall the adjunctionbetween A -bimodules and vector spaces. If M is an A -bimodule then we may consider thefunctor Hom A - Bimod ( M , − ) : A - Bimod → Vect k . Note that in the special case of M = A ⊗ k A with the bimodule structure given by a · ( a ⊗ a ) · a = a a ⊗ a a , the functor Hom A - Bimod ( A ⊗ k A , − ) coincides with the forgetful functor sending a bimoduleto its underlying vector space.If V is a vector space and M is an A -bimodule then the vector spaces M ⊗ k V andHom k ( M , V ) have canonical A -bimodule structures induced from the bimodule structureon M . The functor M ⊗ k ( − ) : Vect k → A - Bimod is left adjoint to Hom A - Bimod ( M , − ) . A free bimodule is a bimodule in the essential imageof ( A ⊗ k A ) ⊗ k ( − ) . When viewed as A e -modules, they are free modules. In section 3.1below we will discuss an analogous notion of free involutive bimodules.We now turn to the involutive variant of the above. First suppose V and W are involutivevector spaces. While V ⊗ W has three involutions to choose from (from the involutionon V , the involution on W , or both at the same time), the quotient V ⊗ Z / W inherits acanonical involution v ⊗ w (cid:55)→ v ∗ ⊗ w = v ⊗ w ∗ (the involution on V is identified with the involution on W , and doing both involutionssimultaneously becomes the identity). This is a special case of the fact that the tensorproduct of R -modules is again R -module when R is a commutative ring; here R is the groupring k [ Z / ] .Now let A be an involutive algebra and M an involutive A -bimodule. We can regardHom A - i Bimod ( M , − ) as a functor A - i Bimod → i Vect k . Given an involutive vector space V ,the involutive vector space M ⊗ Z / V N THE HOCHSCHILD HOMOLOGY OF INVOLUTIVE ALGEBRAS 5 becomes an involutive A -bimodule since ( m ⊗ v ) ∗ · a ∗ = ( m ∗ ⊗ v ) · a ∗ = ( m ∗ · a ∗ ) ⊗ v = ( a · m ) ∗ ⊗ v = (( a · m ) ⊗ v ) ∗ = ( a · ( m ⊗ v )) ∗ . Proposition 2.3.1.
Let A be an involutive algebra and M and involutive A-bimodule. Thereis an adjuction of functorsM ⊗ Z / ( − ) : i Vect (cid:28) A - i Bimod : Hom A - i Bimod ( M , − ) . Proof.
Let L be an involutive A -bimodule and V an involutive vector space. A mor-phism of A -bimodules f : M ⊗ k V → L is adjoint to a morphism of vector spaces g : V → Hom A - Bimod ( M , L ) . Now we claim that f descends to a morphism of involutive bimodules M ⊗ Z / V → L if and only if g factors through a morphism of involutive vector spaces (cid:101) g : V → Hom A - i Bimod ( M , L ) . I.e., we claim that f ( m ⊗ v ∗ ) = f ( m ∗ ⊗ v ) = f ( m ⊗ v ) ∗ if and only if g ( v )( m ) ∗ = g ( v )( m ∗ ) = g ( v ∗ )( m ) . To see this, first suppose that f descends to an involutive morphism. Then we have g ( v )( m ) ∗ = f ( m ⊗ v ) ∗ = f ( m ⊗ v ∗ ) = f ( m ∗ ⊗ v )= g ( v ∗ )( m ) = g ( v )( m ∗ ) , where the equalities on the first and third lines are due to f and g being adjoint, and theequalities on the second line come from the hypotheses on f . The verification of the otherdirection is just a permutation of the above sequence of steps. (cid:3) We now turn our attention to the functor ⊗ A ie . Given involutive bimodules M and N , M ⊗ A ie N is a priori a vector space. It can be described as the quotient of M ⊗ A e N by thevector subspace spanned by the elements m ∗ ⊗ n − m ⊗ n ∗ for m ∈ M and n ∈ N , and so as with M ⊗ Z / N , it carries an involution: ( m ⊗ n ) ∗ = m ∗ ⊗ n = m ⊗ n ∗ . Proposition 2.3.2.
Given involutive bimodules M and N, let Z / act on M ⊗ A N bym ⊗ n (cid:55)→ m ∗ ⊗ n ∗ . Then there is an isomorphism of vector spaces ( M ⊗ A N ) Z / ∼ = M ⊗ A ie N . Proof. In ( M ⊗ A N ) Z / we have [ am ⊗ n ] = [( am ) ∗ ⊗ n ∗ ] = [ m ∗ a ∗ ⊗ n ∗ ] = [ m ∗ ⊗ a ∗ n ∗ ] = [ m ⊗ na ] , so ( M ⊗ A N ) Z / is a quotient of M ⊗ A ie N . On the other hand, M ⊗ A ie N is clearly a quotientof ( M ⊗ A N ) Z / , and so the two are isomorphic. (cid:3) ON THE HOCHSCHILD HOMOLOGY OF INVOLUTIVE ALGEBRAS
Summing up, we have functors ( − ) ⊗ Z / ( − ) : i Vect k × i Vect k → i Vect k or A - i Bimod × i Vect k → A - i Bimod , and ( − ) ⊗ A ie ( − ) : A - i Bimod × A - i Bimod → i Vect k . Center and Abelianization.
We first recall the non-involutive setup. The center ofan A -bimodule M is the vector subspace Z ( M ) : = { m ∈ M | am = ma for all a ∈ A } ⊂ M ;it is a bimodule over the center of A , and is naturally isomorphic to Hom A - Bimod ( A , M ) .The abelianization of M is the quotient vector space Ab ( M ) : = M / ( am ∼ ma | m ∈ M , a ∈ A ) , which canonically has the structure of a A -bimodule and is naturally isomorphic to A ⊗ A e M .We now turn to the case of involutive bimodules. Let A be an involutive algebra and M an involutive A -bimodule. We define the involutive center of M to be the involutive vectorspace iZ ( M ) : = Hom A - i Bimod ( A , M ) , and we define the involutive abelianization of M to be the involutive vector space iAb ( M ) : = A ⊗ A ie M . Proposition 2.4.1.
The involutive center of M is naturally isomorphic to the pullback (i.e.,intersection) of the involutive vector spacesZ ( M ) (cid:44) → M ← (cid:45) M Z / . The involutive abelianization of M is naturally isomorphic to the pushout of the involutivevector spaces Ab ( M ) ← M → M Z / . Proof.
A morphism of involutive A -bimodules f : A → M is entirely determined by f ( ) ,which must be an element in Z ( M ) since a · f ( ) = f ( a ) = f ( ) · a for any a ∈ A , andmust be fixed by the involution since 1 ∈ A is fixed. This shows that iZ ( M ) is containedin Z ( M ) ∩ M Z / . Conversely, sending 1 ∈ A to any element in this intersection uniquelyextends to a well-defined bimodule morphism that clearly commutes with the involutions.Now consider the pushout P of Ab ( M ) ← M → M Z / . First observe that, by the universalproperty of the pushout, there is a natural map P → A ⊗ A ie M sending [ m ] to [ ⊗ m ] . Aninverse to this should send [ a ⊗ m ] to [ am ] , and it remains to check that this is well defined.This formula gives a map f : A ⊗ A e M → P , and it satisfies f ( a ∗ ⊗ m ) = [ a ∗ m ] = [ ma ∗ ] = [ am ∗ ] = f ( a ⊗ m ∗ ) , so it descends to A ⊗ A ie M , giving the desired inverse. (cid:3) N THE HOCHSCHILD HOMOLOGY OF INVOLUTIVE ALGEBRAS 7
3. H
OMOLOGICAL ALGEBRA
Let A be an involutive k -algebra. The categories of involutive vector spaces and involu-tive A -bimodules are abelian categories; this follows immediately from the identificationsas module categories, i Vect k ∼ = k [ Z / ] - Mod and A - i Bimod ∼ = A ie - Mod . Hence we may talk about projective objects, chain complexes, and quasi-isomorphisms ineach of these categories. In this section we will show that if A is projective as an involutivevector space then the usual construction of the bar resolution in fact provides a projective,and hence flat, resolution of A in the category of involutive bimodules.3.1. Flat and projective involutive bimodules.
Projective objects in i Vect k and A - i Bimod are defined by the usual lifting property. As these are module categories, the usual charac-terization holds: an involutive vector space is projective if, viewed as a k [ Z / ] -module, itis a direct sumand of a free module, and an involutive bimodule is projective if and only if,when viewed as a A ie -module, it is a direct summand of a free module. For the purposes ofthis paper we will not need to give a more concrete characterization of projective involutivebimodules. Remark . If the characteristic of k is different from 2 then every finite dimensionalinvolutive vector space (i.e., Z / Z / Z / Proposition 3.1.2.
Let A be an involutive algebra and V and involutive vector space. Con-sidering A ie as a k [ Z / ] -bimodule by the inclusion k [ Z / ] (cid:44) → A ie , we have an isomorphismof vector spaces A ie ⊗ Z / V ∼ = A e ⊗ k V , and under this identification, the involution on the left (coming from the left action of Z / on A ie ) corresponds with ( a ⊗ b ⊗ v ) ∗ = b ∗ ⊗ a ∗ ⊗ v ∗ .Proof. We have an isomorphism of vector spaces, A ie ⊗ Z / V = A e ⊗ k [ Z / ] ⊗ Z / V ∼ = A e ⊗ k V , and one easily checks that this is in fact an isomorphism of A e -modules, i.e., A -bimodules.This isomorphism is defined by sending ( a ⊗ b ⊗ τ ) ⊗ v ∗ = ( a ⊗ b ⊗ ) ⊗ v to ( a ⊗ b ) ⊗ v ,for a ⊗ b ∈ A e and v ∈ V .It remains to examine the involution. The involution on A ie ⊗ Z / V , given by leftmultiplication by τ , is ( a ⊗ b ⊗ τ i ) ⊗ v (cid:55)→ ( b ∗ ⊗ a ∗ ⊗ τ i + ) ⊗ v = ( b ∗ ⊗ a ∗ ⊗ τ i ) ⊗ v ∗ . Thus this corresponds to the involution ( a ⊗ b ) ⊗ v (cid:55)→ ( b ∗ ⊗ a ∗ ) ⊗ v ∗ on A e ⊗ k V . (cid:3) ON THE HOCHSCHILD HOMOLOGY OF INVOLUTIVE ALGEBRAS
Proposition 3.1.3.
Let V be a projective involutive vector space. The involutive bimoduleA ie ⊗ Z / V is projective.Proof.
We make use of the identification from Proposition 3.1.2. Bimodule homomor-phisms f : A e ⊗ k V → N are in bijection with linear maps g : V → N via the correspondence g ( v ) = f ( ⊗ v ⊗ ) , and f ( a ⊗ v ⊗ b ) = ag ( v ) b . Moreover, f commutes with the involutions if and only if g does. Thus, to lift f along asurjection M → N of involutive bimodules, it suffices to produce a lift of g in the categoryof involutive vector spaces, and such a lift exists since V is projective. (cid:3) An involutive bimodule M is flat if it is flat as an A ie -module; equivalently, it is flat ifthe functor M ⊗ A ie : A - i Bimod → i Vect is exact. As usual, if M is a projective involutivebimodule then it is flat.3.2. The bar resolution as an involutive resolution.
First recall the classical bar res-olution of an associative algebra A . We write Bar ( A ) for the chain complex of bi-modules whose degree n part is A ⊗ k ( n + ) . This has the bimodule structure given by a · ( a ⊗ · · · ⊗ a n + ) · b = aa ⊗ · · · ⊗ a n + b and in particular, it is free, and hence projectiveas a bimodule. The differential is defined by the formula d ( a ⊗ · · · ⊗ a n + ) = n ∑ i = ( − ) i a ⊗ · · · ⊗ a i a i + ⊗ · · · ⊗ a n + . The bar resolution of A is augmented by the multiplication map Bar ( A ) = A ⊗ k A → A .Let Σ A denote the graded vector space consisting of A concentrated in degree 1, andwrite T Σ A = (cid:76) n ( Σ A ) ⊗ n for the tensor coalgebra with grading induced from that of Σ A .Regarding T Σ A as a vector space, there is an isomorphism of graded bimodules Bar ( A ) ∼ = → A e ⊗ k T Σ A given by a ⊗ · · · ⊗ a n + (cid:55)→ ( a ⊗ a n + ) ⊗ ( a ⊗ · · · ⊗ a n ) . I.e., Bar ( A ) is the free graded A -bimodule generated by the underlying vector space T Σ A .Now suppose that A is an involutive algebra. In this case T Σ A has an involution givenby ( a ⊗ · · · ⊗ a n ) ∗ = a ∗ n ⊗ · · · ⊗ a ∗ . The bar resolution has an involution given by ( a ⊗ · · · ⊗ a n + ) ∗ = a ∗ n + ⊗ · · · ⊗ a ∗ , and so we see that Bar ( A ) ∼ = A e ⊗ k T Σ A ∼ = A ie ⊗ Z / T Σ A is actually an isomorphismof involutive graded bimodules. One can easily check that the differential in Bar ( A ) commutes with the involutions. Hence we have: Proposition 3.2.1.
If A is an involutive algebra that is projective as an involutive vectorspace then the complex Bar ( A ) is a projective resolution of A as an involutive bimodule.Proof. This follows directly from Proposition 3.1.3. (cid:3)
N THE HOCHSCHILD HOMOLOGY OF INVOLUTIVE ALGEBRAS 9
Involutive Hochschild homology and cohomology.
Involutive Hochschild coho-mology has been defined in [Bra14] as the cohomology of the complex of involutionpreserving coderivations (actually, he dualizes and then works with derivations). Weinstead define involutive Hochschild homology and cohomology as the derived functors ofinvolutive abelianization and involutive center.We propose the following definitions.
Definition . The involutive Hochschild homology iHH ∗ ( A , M ) of an involutive al-gebra A with coefficients in an involutive bimodule M is the left derived functor of iAb : A - i Bimod → i Vect evaluated on M . Similarly, the involutive Hochschild cohomology iHH ∗ ( A , M ) is the right derived functor of iZ evaluated on M .Equivalently, HH ∗ ( A , M ) = Tor A ie ∗ ( A , M ) HH ∗ ( A , M ) = Ext ∗ A ie ( A , M ) . When A is projective as an involutive vector space, then by Proposition 3.2.1 the usualbar resolution in fact provides a resolution in the involutive setting, and so iHH ∗ ( A , M ) and iHH ∗ ( A , M ) can be computed by the complexes Bar ( A ) ⊗ A ie M = iAb ( Bar ( A ) ⊗ A M ) and Hom A - i Bimod ( Bar ( A ) , M ) respectively.The standard Hochschild chain complex, C ∗ ( A , M ) , is the abelianization of the A -bimodule Bar ( A ) ⊗ A M , or equivalently it is Bar ( A ) ⊗ A e M , and this has the familiardescription(3.3.1) C n ( A , M ) ∼ = A ⊗ n ⊗ k M , with differential d : a ⊗ · · · ⊗ a n ⊗ m (cid:55)→ a ⊗ · · · ⊗ a n ⊗ ma + n ∑ i = ( − ) i a ⊗ · · · ⊗ a i a i + ⊗ · · · ⊗ a n + ( − ) n a ⊗ · · · ⊗ a n − ⊗ a n m . If A is an involutive algebra (that is projective as an involutive vector space) and M is aninvolutive bimodule then the involutive Hochschild homology is computed by the complex Bar ( A ) ⊗ A ie M , and by Proposition 2.3.2, this is the Z / Bar ( A ) ⊗ A e M forthe action given by a ⊗ · · · ⊗ a n + ⊗ m (cid:55)→ a ∗ n + ⊗ · · · ⊗ a ∗ n ⊗ m ∗ . Under the identificationin (3.3.1), the Z / Bar ( A ) ⊗ A M corresponds to a ⊗ · · · ⊗ a n ⊗ m (cid:55)→ a ∗ n ⊗ · · · ⊗ a ∗ ⊗ m ∗ . We thus have the following result.
Proposition 3.3.2.
If A is an involutive algebra that is projective as an involutive vectorspace, and M is an involutive bimodule, then iHH ∗ ( A , M ) is computed by the complexiC ∗ ( A , M ) defined asiC n ( A , M ) = A ⊗ n ⊗ M / ( a ⊗ · · · ⊗ a n ⊗ m − a ∗ n ⊗ · · · ⊗ a ∗ ⊗ m ∗ ) ∼ = C n ( A , M ) Z / , with differential induced by the usual Hochschild differential. The Z / C ∗ ( A , M ) induces an action on HH ∗ ( A , M ) . Proposition 3.3.3.
If the characteristic of the ground field k is different from 2 theniHH ∗ ( A , M ) = HH ∗ ( A , M ) Z / . Proof.
This is immediate from the fact that taking Z / (cid:3)
4. C
OMPARISON WITH B RAUN ’ S DEFINITION
We now compare our definition of involutive Hochschild cohomology with Braun’sdefinition and show that they agree when for involutive algebras that are projective asinvolutive vector spaces.4.1.
Derivations and coderivations.
Given a graded algebra A , let Der ( A ) denote thespace of graded derivations of A into itself, and given a coalgebra C , let Coder ( C ) denotethe space of coderivations of C into itself. If A is an involutive algebra then we write i Der ( A ) for the subspace of involution-preserving derivations, and likewise for the notation i Coder ( C ) if C carries an involution. The spaces of derivations and coderivations aregraded Lie algebras with respect to the commutator bracket, and the subspaces i Der ( A ) and i Coder ( A ) are Lie subalgebras.If V is a graded involutive vector space then the tensor algebra TV carries an involutiongiven by ( v ⊗ · · · ⊗ v n ) ∗ = v ∗ n ⊗ · · · ⊗ v ∗ . If A is an associative algebra then the multiplication map induces a coderivation m on T Σ A of degree − m (cid:48) on (cid:98) T Σ − A ∨ also of degree −
1. If A is an involutivealgebra then m and m (cid:48) are both involution-preserving. The commutator [ m , − ] definesa differential on i Coder ( T Σ A ) , and likewise for the space i Der ( (cid:98) T Σ − A ∨ ) of continuousinvolution-preserving derivations on the completed tensor algebra.Braun defines the involutive Hochschild cohomology to be the cohomology computedby the complex Σ − i Der ( (cid:98) T Σ − A ∨ ) . Since T Σ A dualizes to (cid:98) T Σ − A ∨ , algebra derivations on the latter are the same as coalgebracoderivations on the former, and hence there is an isomorphism of complexes Σ − i Der ( (cid:98) T Σ − A ∨ ) ∼ = Σ − i Coder ( T Σ A ) . As we have seen, the bar resolution provides a resolution in the involutive category, andthis next proposition confirms that our derived functor definition of Hochschild cohomologyagrees with Braun’s definition.
N THE HOCHSCHILD HOMOLOGY OF INVOLUTIVE ALGEBRAS 11
Proposition 4.1.1.
There is a canonical isomorphism of complexes,
Hom A - i Bimod ( Bar ( A ) , A ) ∼ = Σ − i Coder ( T Σ A ) . Proof.
Since
Bar ( A ) ∼ = A ⊗ k T Σ A ⊗ k A , the degree n part of Hom A − Bimod ( Bar ( A ) , A ) isthe space of degree − n linear maps T Σ A → A , which is isomorphic to the space of degree ( − n − ) linear maps T Σ A → Σ A . By the universal property of the tensor coalgebra,there is a bijection between degree ( − n − ) linear maps T Σ A → Σ A and degree ( − n − ) coderivations on T Σ A . Hence the degree n part of Hom A − Bimod ( Bar ( A ) , A ) is isomorphicto the degree n part of Σ − Coder ( T Σ A ) . One now checks directly that this isomorphismrestricts to an isomorphism of graded vector spaces i Hom A − Bimod ( Bar ( A ) , A ) ∼ = Σ − i Coder ( T Σ A ) . With a bit of tedious but straightforward algebra one can check that the differentialscoincide under the above isomorphism, cf. [LV12, § (cid:3) R EFERENCES [Bra14] Christopher Braun,
Involutive A-infinity algebras and dihedral cohomology , J. Homotopy Relat.Struct. (2014), 317–337.[Cos07] K. Costello, Topological conformal field theories and Calabi-Yau categories , Adv. Math. (2007), no. 1, 165–214.[FV15] Rams`es Fern`andez-Val`encia,
On the structure of unoriented topological conformal field theories ,arXiv:1503.02465, submitted, 2015.[LV12] Jean-Louis Loday and Bruno Vallette,