Differential forms on smooth operadic algebras
DDifferential forms on smoothoperadic algebras R ICARDO C AMPOS P EDRO T AMAROFF
The classical Hochschild–Kostant–Rosenberg (HKR) theorem computes the Hochschild homo-logy and cohomology of smooth commutative algebras. In this paper, we generalise this resultto other kinds of algebraic structures. Our main insight is that producing HKR isomorphismsfor other types of algebras is directly related to computing quasi-free resolutions in the categoryof left modules over an operad; we establish that an HKR-type result follows as soon as thisresolution is diagonally pure.As examples we obtain a permutative and a pre-Lie HKR theorem for smooth commutative andsmooth brace algebras, respectively. We also prove an HKR theorem for operads obtained froma filtered distributive law, which recovers, in particular, all the aspects of the classical HKR the-orem. Finally, we show that this property is Koszul dual to the operadic PBW property definedby V. Dotsenko and the second author.
MSC 2020:
Hochschild homology is a classical homology theory for associative algebras[29] dating back to1945. Originally conceived by Hochschild to obtain a cohomological proof of Wedderburn’s the-orem [32], this cohomology theory plays nowadays important roles in representation theory [2],deformation theory [17, 20, 24], derived geometry [44], factorisation homology [23], and formalityresults [28], among others.While Hochschild homology of an associative k -algebra A is in general difficult to compute, in thecase where k is a field of characteristic zero and A is commutative and smooth (for example if it isthe coordinate ring of a smooth algebraic variety) the celebrated Hochschild–Kostant–Rosenberg(HKR) theorem [30] identifies the Hochschild homology of A with its module Ω ∗ A of algebraicdifferential forms, which is nothing but a free commutative algebra over the module Ω A of Kählerdifferentials of A . In fact, this result is also used the other way around: it provides us with away to generalise geometrical results, usually stated in terms of differential forms and fields onmanifolds, to non-commutative or non-smooth algebras by replacing these geometrical objects a r X i v : . [ m a t h . K T ] N ov Differential forms on smooth operadic algebras with Hochschild homology and cohomology. This philosophy falls under the general theory ofnon-commutative geometry; see [7, 9, 25, 43].The HKR theorem depends on two hypothesis on the underlying algebra that are of very differentflavours: while smoothness is a property that concerns certain geometric regularity of the algebraitself, and is thus intrinsic to the category of commutative algebras, the constraint that the associ-ative algebra be commutative for one to obtain a description of its cohomology involves, perhapsin a more mysterious way, the interplay between the category of commutative algebras and the oneof associative algebras.Recently, V. Dotsenko and the second author have shown in [16] that one can produce, using thelanguage of operads, what they consider the ‘bare-bones’ framework for Poincaré–Birkhoff–Witt(PBW) type theorems about universal enveloping algebras of types of algebras. There, they haveshown that one can understand PBW-type results by way of studying a homological propertybetween morphisms of operads: the universal enveloping algebra functor associated to a map ofalgebraic operads satisfies a PBW-type property if and only if it makes its codomain a free rightmodule.We pursue this philosophy here, by considering the question of the existence of an HKR-liketheorem for operadic algebras. Since the ingredients we will need are slightly more involved thanthose in [16], let us first recall these.Given an operad P and an algebra A over P , we can consider its cotangent homology [41], whichwe will write H P ∗ ( A , M ) , and which corresponds to the Hochschild homology when P is the operadgoverning associative algebras, to Chevalley–Eilenberg homology when P is the operad governingLie algebras, and to Harrison homology when P is the operad governing commutative algebrasover a field of characteristic zero. Any map of algebraic operads f : P −! Q induces a restrictionfunctor f ∗ : Q - Alg −! P - Alg and, in turn, a map H ∗ ( f ) : H Q ∗ ( A , A ) −! H P ∗ ( A , A ) , and an HKR theorem can be seen as a way to promote this map to an isomorphism, by applying anappropriate functor to the codomain; the resulting object in the codomain deserves to be thoughtas “differential forms” on A . With this in mind, our first step towards obtaining an HKR-typeformalism is the following result. It says that promoting the map H ∗ ( f ) to a possible candidatefor an HKR isomorphism can be done as soon as one produces a quasi-free resolution of Q in left P -modules. This resolution will have the form ( P ◦ Y , d ) for some graded symmetric sequence Y ofgenerators, and these will play the central role of the “functor of differential forms for f ”. We willsee it is convenient to phrase our result in terms of the complexes Def P ∗ ( A ) and Def Q ∗ ( A ) computingthe homology groups above. Theorem.
Let F = ( P ◦ Y , d ) be a quasi-free resolution of Q in left P -modules. Then there exists afunctorial complex Ω ∗ F , A of ‘differential forms’ on A associated to f , depending on F and Def Q ∗ ( A ) ,and a morphism of complexes HKR F , A : Def Q ∗ ( A ) −! Ω ∗ F , A . . Campos and P. Tamaroff There is a well developed theory of Kähler differentials Ω A for operadic algebras [39], that is,algebraic 1-forms, which we use to construct the module of differential forms Ω ∗ F , A . The construc-tion Ω A Ω ∗ F , A is not intrinsic to P -algebras but depends on the homotopical properties of themorphism P −! Q and, up to quasi-isomorphism, on a choice of resolution F of Q , as we explainin Section 3.4. With this construction at hand, we consider the notion of smoothness in Section 3.2as a generalisation of one of the equivalent notions of smoothness in the commutative case, whichwe recall from the excellent monograph [36]. This allows us to make the following definition,central to our paper: Definition.
The map f : P −! Q has the Hochschild–Kostant–Rosenberg property if for everysmooth Q -algebra A , the map HKR F , A is a quasi-isomorphism.The main result of this paper is that the PBW property is, in the sense made precise below, Koszuldual to the HKR property, as we record in Corollary 3.23. We cannot avoid to note this follows the‘mantra’ pursued by B. Ward in [45], that it is desirable to consider Koszul duality not as an aspectof categories separately, but rather as a construction which intertwines functors between them. Theorem.
Let f : P −! Q be a map of Koszul operads. Then f has the HKR property if themorphism of Koszul dual operads Q ! −! P ! enjoys the PBW property. This gives us a short and conceptual proof of the classical HKR theorem: the maps
Ass −! Com and
Lie −! Ass are Koszul dual, so that the classical PBW theorem implies, in this way, the HKRtheorem. With generous hindsight, this comes as no surprise: apart from using standard techniquesof localisation to reduce the proof of the HKR theorem to smooth local commutative algebras, astraightforward way to prove that the HKR theorem holds is by use of the Koszul complex of thesymmetric algebra S ( V ) and its Koszul dual coalgebra S c ( V [ − ]) .Having settled the above, we then observe there are several examples of maps of operads satisfyingthe HKR property and, in Section 4.2, we explore some of them. Of particular interest is the map Perm −! Com which factors the projection of the associative operad onto the commutative operadby passing through permutative algebras [5]. The Koszul dual map to the projection
Perm −! Com is the inclusion
Lie −! PreLie , which is known to enjoy the PBW property by [16]. In Corollary 4.3we conclude that the following HKR-type theorem holds, providing us with the computation of thecotangent homology of a smooth commutative algebra A seen as a permutative algebra. Theorem.
The permutative cotangent homology of a smooth commutative algebra A is given by amodule RT (cid:54) = ( Ω ∗ A ) which is spanned by rooted trees whose vertices are labeled by elements of theclassical space of forms Ω ∗ A and no vertex has exactly one child. Finally, we offer a technique to compute the tangent cohomology of a P algebra coming from asmooth Q -algebra under the projection f : P −! Q in case P is obtained from a filtered distributivelaw [13] between Q and R as originally defined by V. Dotsenko in [10]. The shining example ofthis phenomenon is the way in which the operad Ass is obtained from
Com and
Lie ; in this way,the reader may think of the following filtered HKR theorem as another ‘ultimate’ generalization toalgebraic operads of the classical HKR theorem for the map
Ass −! Com . Indeed, in this case, thefunctor R ¡ below is precisely V S c ( V [ − ])[ ] . Differential forms on smooth operadic algebras
Theorem.
Suppose P is obtained from Koszul operads Q and R by a filtered distributive law,so that P is isomorphic to Q ◦ R as a right R -module. Then for every smooth Q -algebra A thecotangent homology of f ∗ A is given by the endofunctorA −! R ¡ ( Ω A ) of “ R ¡ -enriched differential forms” on A. Dually to the result of homology, we were able to obtain a result for tangent cohomology. In thiscase, a choice of quasi-free resolution F gives us a functor of “poly-vector fields” A Poly ∗ ( A ) ,and we obtain the following: Theorem.
If f satisfies the HKR property then for every smooth Q -algebra A there is a quasi-isomorphism of complexes: HKR A : Def ∗ ( f ∗ A ) −! Poly ∗ ( A ) . In case of cohomology, our result on filtered distributive laws says that the tangent cohomologyof f ∗ A is given by the endofunctor R ! ( Der ( A )[ ])[ − ] . In the classical case, we recover the Liestructure on tangent homology, since Der ( A ) is a Lie algebra. Indeed, since R ! = Com , the usualdistributive law allows us to give R ! ( Der ( A )[ ])[ − ] a Lie algebra structure isomorphic to the oneon H ∗ Ass ( A ) . It is unclear, however, how one could attempt to obtain the Lie algebra structure ontangent cohomology in a more general situation. Structure.
The paper is organised as follows. In Section 2 we recall the usual HKR theoremfor smooth commutative algebras, in a way that suits our operadic approach that follows, andhoping that it will be useful for the reader to incorporate the new formalism that we then developin Section 3. Here, we recall the notions of (co)tangent (co)homology and introduce the relevantnotions of smoothness and the “full” module of differential forms. With this at hand, we introducethe HKR property and prove our main theorem. In Section 4.2, where we focus on applications,we show how to recover the classical HKR theorem from our main result and apply it to obtainnew examples: we obtain a “permutative” HKR theorem for smooth commutative algebras and a“pre-Lie” HKR theorem for smooth braces algebras. In the process of drawing some connectionsof our work to that of J. Griffin [26], we obtain an HKR theorem for operads obtained from filtereddistributive laws, and briefly outline how it recovers the HKR isomorphism at the level of Liealgebras. Finally, with the purpose of making this paper better self-contained, we collect someuseful results in an Appendix about algebraic operads, their algebras and their Kähler differentials,hoping it will be of use for a reader with some background in algebraic operads.
Notation and conventions.
For references on operads and their modules we point the readerto [18, 39], and to [4, 36, 46] for homological algebra. We allow operads to be homologicallygraded, but will make it clear when we require operads to be dg. We assume that algebras overoperads are non-dg, and we fix a closed symmetric monoidal category C like Vect over whichour algebras are defined; we always work over a field of zero characteristic. Most arguments wemake actually hold for dg algebras, taking into the account the given bigrading of the resulting . Campos and P. Tamaroff objects. For simplicity, we work with non-dg algebras. We write C . If V is a chain complex and p ∈ Z , we write V [ p ] for the chain complex for which V [ p ] n = V n − p for each n ∈ Z , and whose differential changes sign according to the parity of p .Accordingly, if Q is an operad, we write Q { p } for the operad uniquely defined by the conditionthat a Q { p } -algebra structure on V is the same as a Q -algebra structure on V [ p ] .Throughout, for two quadratic operads given by quadratic data ( V , R ) and ( V (cid:48) , R (cid:48) ) , we will only consider maps of operads induced by a map of quadratic data V −! V (cid:48) such that the induced map T ( V ) −! T ( V (cid:48) ) sends ( R ) to ( R (cid:48) ) . We remind the reader that the data ( V , R ) may contain non-binary generators and that, in this case, the weight and arity gradings in T ( V ) may not coincide;see [39, §7.1.3]. Moreover, we confine ourselves to the category of weight graded operads andtheir weight graded algebras and modules. We distinguish the weight degree from the homologicaldegree by using parentheses. Hence, while X denotes a component of homological degree 3, wewrite X ( ) for a component of weight degree 3. Acknowledgements.
We kindly thank B. Keller for explaining to us the very short proof ofLemma 3.6 which we reproduced here. We also thank V. Dotsenko, J. Bellier-Millés, N. Combeand J. Nuiten for useful conversations, comments and suggestions.
This section serves to recall the objects and results related to the classical Hochschild–Kostant–Rosenberg theorem for commutative algebras. Such objects will be presented in the way that wefind best suited for the operadic generalisation and the main results appearing in Section 3. For aclassical approach we recommend both Chapter 3 and Appendix E of [36].
Throughout, fix a non-unital commutative algebra A , and let us recall how to construct a naturalmap that relates the homology of A as a commutative algebra, its Harrison homology, and thehomology of A as an associative algebra, its Hochschild homology, through a particular functor.This is the well known Hochschild–Kostant–Rosenberg map
HKR A : C ∗ ( A , A ) −! Ω ∗ A where the left hand side is the cyclic Hochschild complex of A considered as an associative algebraand Ω ∗ A is the space of differential forms on A . We now recall the details necessary to construct thismap.For any commutative algebra A , the module of Kähler differentials Ω A of A is the symmetric A -bimodule representing the functor of derivations M Der ( A , M ) . Differential forms on smooth operadic algebras
Recall that we have a natural isomorphism of symmetric A -bimodules I / I −! Ω A , where I = ker ( µ : A ⊗ A −! A ) such that 1 ⊗ x − x ⊗ + I dx . Definition 2.1
Let be J the kernel of the multiplication of a cofibrant replacement QA of A . The cotangent complex of A with coefficients in a symmetric A -bimodule M is by definitionDef ∗ ( A , M ) = J / J ⊗ QA M . The cotangent homology of A with coefficients in M is, by definition, the homology of this com-plex, and we write it H ∗ ( A , M ) .In other words, this is the non-abelian derived functor of M Ω A ⊗ A M . Dually, we have a tangent complex of A with values in M Def ∗ ( A , M ) = hom A ( J / J , M ) and the tangent cohomology of A with values in M is, by definition, the homology of this complex,and we write it H ∗ ( A , M ) . Definition 2.2
We say A is a smooth commutative algebra if for every A -module M , H ( A , M ) = . For our convenience and that of the reader, we record now some equivalent definitions of smooth-ness, which in particular show that the cotangent homology H ∗ ( A , A ) of A is very simple in case itis smooth: it is concentrated in degree zero where it equals the module of Kähler differentials Ω A .We remind the reader from the Appendix that one can also consider relative versions of the ho-mology and cohomology theories above for a morphism of algebras. In particular, since A is acommutative algebra, we can consider the (co)homology of A relative to A ⊗ A through the multi-plication map. Proposition 2.3
Let A be a finitely generated commutative algebra over a field of characteristiczero, and let B = A ⊗ A. Then the following conditions are equivalent:(1) H ( A , M ) = for any symmetric A-bimodule M,(2) H ( A , A ) = and Ω A is a projective A-module,(3) H ( A | B , N ) = for any A-module N,(4) H ( A | B , A ) = and Ω A is a projective A-module.Proof. See [36, Appendix E].As we noted, if A is a smooth commutative algebra, the fact Ω A is projective, implies that H ∗ ( A , A ) is concentrated in degree zero and H ( A , A ) = Ω A . . Campos and P. Tamaroff This receives a map A −! Ω A , the universal derivation , and we can then form the non-unitalsymmetric algebra S A ( Ω A [ − ]) of Ω A under A . We call this the space of differential forms on A and write it Ω ∗ A . Finally, let us recall that the cyclic Hochschild complex C ∗ ( A , A ) of A is given foreach n ∈ N by C n ( A , A ) = A ⊗ A ⊗ n and we write a generic element in here by a [ a | · · · | a n ] . Thereis a map of complexes HKR A : C ∗ ( A , A ) −! Ω ∗ A such that a [ a | · · · | a n ] ada · · · da n . It will be useful to note that this map is the identity of A indegree 0, and in fact split as a map of complexes as a sum of this map and the remaining part C ∗ ( A , A ) + −! S A ( Ω A ) . where the right hand side uses the non-unital symmetric algebra functor under A . Over a fieldof characteristic zero, the HKR map is a split injection. The Hochschild–Kostant–Rosenberg the-orem [30] asserts the following stronger conclusion in case A is smooth: Theorem 2.4
For every smooth commutative algebra A of finite type over k the morphism HKR A : C ∗ ( A , A ) −! Ω ∗ A is a quasi-isomorphism. There are many proofs of Theorem 2.4 in the literature. Because it serves to illustrate the generalformalism that we will develop later, let us give a non-standard proof of this theorem: it will followonce we show that (in the dg setting), the HKR map is a quasi-isomorphism for cofibrant algebrasthat resolve smooth algebras.To see why this is enough, observe that HKR A is natural, in the sense that given a map of algebras f : B −! A we have that H ∗ ( f ) ◦ HKR B = HKR A ◦ Ω ∗ f or, what is the same, there is a commutativediagram C ∗ ( B , B ) C ∗ ( A , A ) Ω ∗ B Ω ∗ A . Since the functor H ∗ , by its very definition, preserves quasi-isomorphisms, and since the HKRmap is a quasi-isomorphism for cofibrant algebras, we deduce the following interesting lemma. Itscontent is central to develop our operadic formalism later. Lemma 2.5
The map
HKR A is a quasi-isomorphism if and only if the functor of differential formsA Ω ∗ A preserves quasi-isomorphisms Q −! A for Q a cofibrant resolution of an arbitrarysmooth algebra A.
To proceed with the proof, let us first recall that we can express C ∗ ( A , A ) as a twisted tensor product A ⊗ B A where B A = ( T c ( sA ) , δ ) is the bar construction of A , arising from the fact the associative Differential forms on smooth operadic algebras operad is Koszul self-dual. Similarly, there is a commutative-Lie bar-cobar adjunction arising fromKoszul duality between the category of conilpotent Lie coalgebras and the category of commutativealgebras C : Lie - Cog (cid:29)
Com - Alg : L . The only properties of this adjunction that we need are the following:• the counit of the adjunction CL ! id is a quasi-isomorphism and,• the commutative cobar construction of a Lie coalgebra is given by the quasi-free commutativealgebra C ( g ) = ( S ( g [ ]) , δ ) , where δ is a differential extending the cobracket of g .These two adjunctions interact in the following way: the restriction functor from conilpotent as-sociative coalgebras to conilpotent Lie coalgebras Ass - Cog −! Lie - Cog has a left adjoint U c , the universal enveloping coalgebra , which satisfies B π ∗ = U c L , where π ∗ is the forgetful functor fromcommutative algebras to associative algebras, see Lemma 3.25. Lemma 2.6
The
HKR map is a quasi-isomorphism for any free commutative dga algebra.Proof.
A derivation on a free commutative algebra A = S ( V ) is uniquely determined by specify-ing the values on generators, from where it follows that Ω A = A ⊗ V . In this way, we obtain anidentification Ω ∗ A = A ⊗ S c ( V [ − ]) . At the same time, A = C ( V [ − ]) is the commutative cobar construction of the abelian Lie coalgebra g = V [ − ] . It follows that as chain complexes C ∗ ( A , A ) = A ⊗ B C ( V [ − ])= A ⊗ U c LC ( V [ − ]) (cid:39) A ⊗ U c ( V [ − ])= A ⊗ S c ( V [ − ]) . On the third line we used that U c preserves quasi-isomorphisms, which is a consequence of thePBW theorem [40]. The resulting anti-symmetrization quasi-isomorphism ϕ A : Ω ∗ A −! C ∗ ( A , A ) gives us an inverse to HKR A when taking homology, which proves our claim.A cobar algebra is any commutative algebra C ( g ) obtained via a cobar construction of a (shifted)Lie coalgebra g . Recall that cobar algebras are triangulated and hence cofibrant and that everyalgebra admits a cofibrant replacement given by a cobar algebra, as explained in Corollary 11.3.5and Proposition B.6.6 of [39]. Proposition 2.7
The map
HKR A is a quasi-isomorphism for any commutative cobar algebra.Proof. The differential of the cobar construction A = C ( g ) splits as d g + δ . As in the previouslemma, we have that Ω A = ( A ⊗ g [ − ] , d A ⊗ + ⊗ d g + δ ) , . Campos and P. Tamaroff where we notice that the differential has an external component induced by δ , such that if we write δ x = x ( ) ⊗ x ( ) in Sweedler notation, then δ ( a ⊗ x ) = ax ( ) ⊗ x ( ) . On the other hand, the samecomputation as in the previous lemma shows that there is a morphism of A -modules, ϕ A : Ω ∗ A −! C ∗ ( A , A ) . The result now follows from taking the spectral sequence associated to the PBW filtration: theassociated morphism to ϕ A is equal, in homology, to the desired inverse of the HKR map corres-ponding to the commutative algebra S ( g ◦ ) where g ◦ is the Lie algebra g with zero bracket. Theorem 2.8
The
HKR map is a quasi-isomorphism for any smooth commutative algebra.Proof.
Let A be a smooth commutative algebra. Since A is smooth it follows in particular that H ( A , M ) = A -module M . Setting A = M we see that H ∗ ( A , A ) = Ω A , from where itfollows that for any cofibrant replacement p : Q −! A Ω p : A ⊗ Q Ω Q −! Ω A is a quasi-isomorphism: the left hand side computes H ∗ ( A , A ) and the induced map is then anisomorphism. Because the canonical map Ω Q = Q ⊗ Q Ω Q −! A ⊗ Q Ω Q is a quasi-isomorphism, it follows that Ω Q −! Ω A is a quasi-isomorphism. In view of Lemma 2.5,we see that the map HKR A is a quasi-isomorphism.Before moving on, we would like to highlight the following three points that will be revisited whenwe develop the general operadic formalism for HKR theorems:(1) For q : Q −! A a cofibrant resolution of A a smooth algebra the map q ! Ω Q −! Ω A is a quasi-isomorphism. This is intrinsic to the category of commutative algebras and thus independentof the map of operads Ass −! Com .(2) Showing that for any cofibrant algebra Q the map Def ∗ ( f ∗ Q , f ∗ Q ) −! Ω ∗ Q = S c ∗ ( Def ∗ ( Q )) is a quasi-isomorphism is independent of smoothness, and depends on the map of operads f : Ass −! Com . In particular, we can take “affine” algebras as test algebras in this step,which we called “cobar algebras” above.(3) The way the two previous points are put together is by noting that, since the functor S c preserves quasi-isomorphisms, we have that Ω ∗ q : Ω ∗ Q −! Ω ∗ A is a quasi-isomorphism. Here itis the only step where we use the universal enveloping algebra functor associated to Lie −! Ass preserves quasi-isomorphisms, by the classical PBW theorem. Differential forms on smooth operadic algebras
In this section we generalise the classical notions from the previous section to algebras over op-erads, and prove Theorem 3.22, generalising the classical Theorem 2.4 to smooth algebras formorphisms of operads satisfying a natural homological condition.
Some conventions for this section. • We fix once and for all a morphism of non-dg Koszul operads P −! Q which we assumecomes from a map of quadratic data. All (co)operads will be homologically graded with zerodifferential and by A we will always denote a Q -algebra.• For any operad O , we denote by Q : O - Alg −! O - Alg a fixed choice of cofibrant replacementfunctor for O -algebras.• When context allows, we will usually simply write Q for a cofibrant replacement of somealgebra A (cid:48) which will be clear from context.• In particular, we will sometimes need to use the composition U Q where U = U O : O - Alg −! Ass - Alg is the associative universal envelope functor, in which case we will usually to writethis
U Q when the algebra we applied it to is clear from context.
In this section we introduce the formalism of (co)tangent homology and cohomology for algebrasover an operad. We refer the reader to the article [41] of J. Millès for a thorough and comprehensivestudy of this theory, and point to the Appendix, where some useful recollections on algebra overoperads and their operadic modules is given. The reader can consult Appendix A.2 for details onderivations and associative universal envelopes of algebras over operads.
Definition 3.1
Let A be a Q -algebra. We define the tangent complex of A with values in an operadic A -module M by Def ∗ ( A , M ) = Der ( QA , M ) = hom UQA ( Ω QA , N ) . Note that we dropped the subscript Q , which will be clear from context. Observe that this is definedup to natural quasi-isomorphism, and is well-defined in the derived category of complexes.The morphism f : P −! Q induces a restriction functor f ∗ : Q - Alg −! P - Alg that assigns A to the P -algebra f ∗ A with the same underlying object as A along with the P -algebra structure given bythe composition P −! Q −! End A . Observe that we can take Q ( f ∗ A ) as a cofibrant replacement Q ( f ∗ A ) ! f ∗ ( QA ) of the P -algebra f ∗ ( QA ) . In this way, we obtain a natural mapDef ∗ ( f , M ) : Def ∗ ( A , M ) −! Def ∗ ( f ∗ A , f ∗ M ) for every operadic A -module M . Definition 3.2
The cohomology of Def ∗ ( A , M ) is, by definition, the tangent cohomology of A withvalues in M , and we will write it H ∗ Q ( A , M ) . . Campos and P. Tamaroff Remark 3.3
When A is an associative algebra, H ∗ ( A , M ) differs from the classical Hochschildcohomology groups HH ∗ ( A , N ) of A only in that we do not quotient out by inner derivations indegree zero and we discard the 0th classical Hochschild cohomology group of A with values in N so that H ∗ ( A , M ) = (cid:40) HH ∗ + ( A , M ) if ∗ (cid:62) , Der ( A , M ) if ∗ = . In a similar fashion to tangent cohomology, we define the cotangent complex.
Definition 3.4
The cotangent homology of A with coefficients in M through the cotangent complexof A which is obtained as Def ∗ ( A , M ) = Ω QA ⊗ UQA M and write it H Q ∗ ( A , M ) .Note that from Proposition A.4, in case we take f : Ass −! Com , A a commutative algebra andchoose Ω BA for the cofibrant resolution for the associative algebra A , we have thatDef ∗ ( f ∗ A , f ∗ A ) = s − C ∗ ( A , A ) + . It is useful to note there are universal coefficients for these homology theories [18]. Indeed, writingthe functors as compositions, where we write U for the associative enveloping algebra UA to lightenthe notation Def ∗ ( A , M ) = hom U ( Def ∗ ( A , U ) , M ) , Def ∗ ( A , M ) = Def ∗ ( A , U ) ⊗ U M we obtain two universal coefficient spectral sequences , E s , t = Tor Us ( − , H Q t ( A , U )) = ⇒ H Q s + t ( X , − ) , E s , t = Ext sU ( H Q t ( X , U ) , − ) = ⇒ H s + t Q ( A , − ) , that explain the relation between (co)homology theories given by Ext and Tor functors and theoperadic theories. For example, in case we do this for the associative operad, we observe that H Ass t ( A , UA ) = t (cid:62) A is commutative, the cotangent homology groups, usually known as the André–Quillenhomology groups H Com ∗ ( A , UA ) = H Com ∗ ( A , A ) are in general non-zero in higher degrees, so thereare obstructions to this comparison. In analogy with the characterization of smoothness for commutative algebras of Theorem 2.3, weintroduce the following definition. Differential forms on smooth operadic algebras
Definition 3.5
The Q -algebra A is smooth if for every operadic A -module M , H Q ( A , M ) = . or, what is the same, if H Q ( A , − ) is an exact functor in operadic A -modules.Although our focus lies on non-dg algebras, it is useful to remark that the definition is in general not invariant under quasi-isomorphisms. Indeed, suppose that q : A −! A (cid:48) is a quasi-isomorphismof dg Q -algebras and let us assume first that A (cid:48) is smooth, and let M be an operadic A -module.There is a map H Q ( A , M ) −! H Q ( A (cid:48) , ψ ! ( M )) = q ! is well-behaved (that is, flatness assumptions on q ), there is no reasonto expect this to be an isomorphism. However, if A is smooth then any cofibrant replacement of A is one of A (cid:48) , so that in this case A smooth implies A (cid:48) smooth. It is immediate that every free (i.e.affine) Q -algebra is smooth.Let us now consider a related condition: we say that A is quasi-smooth if for some —and hence,every— cofibrant replacement p : QA −! A , the induced map on Kahler differential forms Ω p : p ! Ω QA −! Ω A is a quasi-isomorphism of operadic A -modules. Before relating the notions of smoothness andquasi-smoothness, we record the following lemma: Lemma 3.6
Let q : X −! Y be a map of complexes of operadic A-modules, and suppose that forevery operadic A-module M the mapq ∗ : hom A ( Y , M ) −! hom A ( X , M ) is a quasi-isomorphism. Then q is a quasi-isomorphism.Proof. Let us take J an injective cogenerator of the category of left UA -modules. Then the p thcohomology group of hom A ( X , J ) identifies with F p ( X ) : = hom A ( H p ( X ) , J ) because J is injective. Since J is also a cogenerator, the collection of functors { F p } p ∈ Z detectsquasi-isomorphisms, which gives what we wanted. Remark.
Observe that the reverse implications is not true. Indeed, let us consider the commutativealgebra A = k [ x ] , the trivial A -module M = Y = k and the complex X : A −! A where the differentialis given by multiplication by x . Then the quotient map q : X −! Y is a quasi-isomorphism, but theinduced map hom A ( Y , M ) −! hom A ( X , M ) is not: the right hand side computes Ext ∗ A ( k , M ) , andthis may not always be concentrated in degree 0. . Campos and P. Tamaroff With this lemma at hand, we can prove the following proposition. It is interesting to compareit with Corollary 7.3.5 in [27]. While we make a statement about the behaviour of the inducedmorphism Ω p : p ! Ω B −! Ω A when f is an acyclic fibration onto a smooth algebra, that corollary makes a statement about thebehaviour of that map when f is an acyclic cofibration; in both cases the conclusion is that the mapinduced is a quasi-isomorphism. Proposition 3.7
Every smooth Q -algebra is quasi-smooth.Proof. Suppose that A is smooth, and let Q −! A be a cofibrant replacement, let us show that themap p ! Ω Q −! Ω A is a quasi-isomorphism of operadic A -modules. By the previous two lemmas, itsuffices to show that for every operadic A -module M , the induced map p ∗ : hom A ( Ω A , M ) −! hom A ( p ! Ω Q , M ) is a quasi-isomorphism. By adjunction, the codomain is naturally isomorphic tohom UQ ( Ω Q , p ∗ M ) so we obtain p ∗ identifies naturally with the map hom A ( Ω A , M ) −! hom Q ( Ω Q , M ) representingthe pullback along p p ∗ : Der ( A , M ) −! Der ( Q , p ∗ M ) . Since A is smooth and the right hand side computes H ∗ Q ( A , M ) , it follows that this map is a quasi-isomorphism: it induces the identity of H Q ( A , M ) = Der ( A , M ) . We conclude that Ω p is a quasi-isomorphism, which means that A is quasi-smooth, as we wanted. Remark 3.8
It is important to observe that the notion of quasi-smoothness may be quite weak.For example, every unital associative algebra is quasi-smooth, owing to the fact that the module ofassociative Kähler preserves quasi-isomorphisms. Indeed, in this case this functor fits into an exactsequence 0 −! Ω A −! A ⊗ A −! A −! In this section we collect some facts about (left) Koszul morphisms between weight graded operads,which we introduce. We refer the reader to the excellent monograph [42, Section 2.5] for the caseof algebras. We say a symmetric sequence X is diagonally pure if for each p ∈ N the component X p of homological degree p is concentrated in weight p . With this at hand, let us introduce the kindof left dg P -modules that interest us. Differential forms on smooth operadic algebras
Definition 3.9
A quasi-free left dg P -module ( P ◦ Y , d ) is diagonally pure if its generating sequence Y is so. A map of operads f : P −! Q is left Koszul if Q admits a diagonally pure quasi-freeresolution in the category of left P -modules. Finally, a resolution F is minimal if the differential of k ◦ P F vanishes.It is easy to see that any diagonally pure quasi-free resolution ( P ◦ Y , d ) is minimal. Indeed, theresulting complex is of the form ( Y , ¯ d ) . Since d preserves the weight degree but lowers the homo-logical degree, the differential ¯ d does too and, since Y is diagonally pure, ¯ d vanishes. Definition 3.10
We define Tor P ( k , Q ) as the homology of the dg module k ◦ P F where F is anyquasi-free resolution of Q in left dg P -modules. For each i , j ∈ P we write Tor P i ( k , Q ) ( j ) for thecomponent of Tor in homological degree i and weight degree j .It is useful to note that this is well defined, since the category of left P -modules admits a modelstructure in which the fibrations are the arity-wise surjections, the weak equivalences are the quasi-isomorphisms, and the quasi-free left modules are included in the class of cofibrant objects. It isimportant that we are working over a field of characteristic zero, so that P is Σ -cofibrant. Lemma 3.11
Let f : M −! N be a map of left P -modules and suppose that F = ( P ◦ Y , d ) is aquasi-free complex mapping onto M and that R is a resolution of N . Then there exists a map ofleft P -modules F −! R extending f , and any two such choices are homotopic. Lemma 3.12
The map f is left Koszul if and only if
Tor P ( k , Q ) is concentrated on the diagonal.Proof. Let us show we can construct minimal quasi-free resolutions F = ( P ◦ X , d ) −! Q . To dothis, let us consider an equivariant section σ of the projection Q −! k ◦ P Q , which exists since wework over a field of characteristic zero, and let X = σ ( k ◦ P Q ) , so we have an epimorphism f : P ◦ X −! Q = K − . The kernel K of this map is a left P -module, so we may repeat this and take X a minimal gener-ating set for K obtained from an equivariant section of the projection K −! k ◦ P K , along with f : P ◦ X −! K . Extend the construction above to F = P ◦ ( X ⊕ s X ) where the differential isthe unique map X ⊕ s X ! F that vanishes on X and maps s X onto K . In this way, H ( F ) is isomorphic to Q through the map f . We can now continue this process by adjoining generatorsin homological degree 2 to obtain F with H ( F ) = H ( F ) isomorphic to Q . Continuing,in the limit, we obtain the desired resolution.Since the resolution is minimal, we see that X is isomorphic to Tor P ( k , Q ) , so X must be concen-trated in the diagonal. Conversely, it is clear that if we have a diagonally pure resolution, thenTor P ( k , Q ) is concentrated on the diagonal.Let us recall the following from [16] . Definition 3.13
We say a morphism of operads f : P ! Q enjoys the PBW property if there is anendofunctor X : C −! C on the category underlying that of P -algebras so that for each P -algebra A there is a natural isomorphism f ! ( A ) −! X ( A ) . . Campos and P. Tamaroff The main result of [16], if we take monads there to be algebraic operads, is the following:
Theorem 3.14
The morphism of operads f : P ! Q satisfies the PBW property if and only if itmakes Q into a free right module over P . In this case, the functor X is a basis for Q as a right P -module. Our first main theorem shows the PBW property above is related, by Koszul duality, to the notionof left Koszul morphisms, at least when P and Q are Koszul operads. Note this is an extensionof [42, Corollary 5.9] to algebraic operads. Theorem 3.15 (Duality)
A map between Koszul operads is left Koszul if and only if its Koszul dualmap satisfies the PBW property.
To do this, we just need two technical lemmas, beginning with the following simple homologicalcriterion for freeness, which we recall from Proposition 4.1 in [16], for example. We phrase it in aslightly different way than we did there:
Lemma 3.16
A right P -module is M is free if and only if for every i , j ∈ N , the group Tor P j − i ( M , k ) ( i ) vanishes unless i = j. Note we are simply saying that the homology of M ◦ L P k is concentrated in degree 0. The secondlemma relates the derived functors k ◦ L P Q and P ! ◦ L Q ! k in case P and Q are Koszul operads. Wepoint the reader to Theorem 5.8 in [42] which proves the result for associative algebras. Lemma 3.17
Let f : P −! Q be a morphism of Koszul operads, and let f ¡ : Q ¡ −! P ¡ be its dualmorphism. For each j , i ∈ N we have a natural isomorphism: Tor Q ¡ j − i ( P ¡ , k ) ( i ) −! Tor P i ( k , Q ) ∗ ( j ) . Proof.
It suffices to note that, in the category of right P -modules, k admits a resolution P ¡ ◦ P −! k while, in the category of left Q ! -modules, k admits a resolution Q ! ◦ Q −! k . Applying the functor − ◦ P Q in the first case and the functor P ! ◦ Q ! − in the second case, we obtain two complexes P ¡ ◦ Q and P ! ◦ Q that are related by the duality described in the statement of the lemma. Proof of Theorem 3.15.
The previous three lemmas immediately imply the result.
In this section we construct, for each map f : P −! Q and each Q -algebra A , an operadic analogueof the classical HKR map . This map relates the deformation complex of the P -algebra f ∗ A to acertain space of ‘differential forms’ on A depending functorially, as in the classical setting, on Ω A . Remark 3.18
We cannot avoid making the point that, while the morphisms of operads in [16, 34]enjoying the PBW property involve a statement about the pushforward functor f ! on P -algebras,the morphisms of operads we are interested in involve a statement about the pullback functor f ∗ on Q -algebras. As mentioned in the introduction, this follows the ‘mantra’ promoted in [45]. Differential forms on smooth operadic algebras
To begin, let us take a quasi-free resolution F = ( P ◦ Y , d ) of Q in left P -modules. Let us recallfrom Lemma A.3 that if Q = ( Q ( V ) , d ) is a cofibrant resolution of A , then Ω Q ⊗ U Q is canonicallyisomorphic to V ⊗ Q ( V ) , while Ω Q , A is canonically isomorphic to V ⊗ A . We will be interested inapplying the functor k ◦ P F = Y to the space Ω Q ⊗ U Q , relative to the algebra Q , as the followingdefinition explains. Definition 3.19
We define Ω ∗ F , A , the space of differential forms on A associated to F , as the chaincomplex Y ( V ) ⊗ Q ( V ) . Its differential is the one induced from Q and F .The following proposition shows that to each resolution we may associate an ‘HKR map’. Proposition 3.20
There exists a map of chain complexes
HKR A : Def ∗ ( f ∗ A ) −! Ω ∗ F , A , for every choice of resolution F . Moreover, if Q −! A is a cofibrant resolution of A in the categoryof Q -algebras, there is a commutative diagram Def ∗ ( f ∗ Q ) Def ∗ ( f ∗ A ) Ω ∗ F , Q Ω ∗ F , A . Observe that, unlike the case where A is a commutative algebra, the source and the target of theHKR map do not admit natural operadic A -module structures. We call HKR A the Hochschild–Kostant–Rosenberg map associated to f and the algebra A . Observe, moreover, that HKR A mani-festly depends on the resolution F = ( P ◦ Y , d ) and on the cofibrant resolution Q of A . This is nota problem for us, since the map it induces on homology does not. With this at hand, we can definethe HKR property: Definition 3.21
We say that f satisfies the HKR property if the map HKR is a quasi-isomorphismfor every smooth Q -algebra. Proof Proposition 3.20.
Let A be any Q -algebra and let us pick a cofibrant resolution of A of theform Q = ( Q ( V ) , d ) . By the HKR property of f , the non-dg P -algebra ( Q ( V ) , ) admits a cofibrantresolution of the form ( P ◦ Y ( V ) , d V ) −! Q ( V ) and hence perturbing this we obtain a resolution Z = ( P ◦ Y ( V ) , d V + δ ) −! ( Q ( V ) , d ) of the P -algebra ( Q ( V ) , d ) . If we use this resolution to compute the cotangent complex of the P -algebra Q = ( Q ( V ) , d ) , we obtain a complex of the formDef ∗ ( f ∗ A ) = Ω Z ⊗ U Q = ( Y ( V ) ⊗ Q ( V ) , δ ) . Campos and P. Tamaroff and, tautologically, the cotangent complex of the Q -algebra A may be computed through the com-plex Def ∗ ( A ) = Ω Q ⊗ U Q = ( V ⊗ Q ( V ) , δ (cid:48) ) . The fact all of the constructions and isomorphisms above are natural, with the exception of theperturbation process, means that, in fact, the complex ( Y ( V ) ⊗ Q ( V ) , δ ) is obtained from thecomplex ( V ⊗ Q ( V ) , δ (cid:48) ) by the endofunctor Y relative to the algebra Q ( A ) . The commutativity ofthe diagram is then immediate, although we cannot promote Ω ∗ F , A to a bona-fide functor and henceHKR A to a natural transformation.With this at hand, our second main result for maps satisfying the HKR property is the following‘operadic HKR theorem’, which we now prove with a series of lemmas. Its immediate applicationis the corollary that follows it, which we will use heavily later on. Theorem 3.22
Every left Koszul map between Koszul operads has the HKR property.
Corollary 3.23 (PBW criterion)
Every map between Koszul operads whose dual has the PBWproperty satisfies the HKR property.Proof.
This follows immediately from Theorem 3.15.
Remark 3.24
It is natural to wonder whether a converse to this last corollary exists. The conditionthat the resolution F be diagonally pure makes a certain spectral sequence collapse and gives ourresult, modulo the computations and various lemmas that we have made use of. In a generic case,one should expect an “HKR spectral sequence” to exist coming from the resolution F and, infavourable cases, one may obtain an HKR theorem without requiring that F be diagonally pure. Itwould certainly be interesting to have an example of this behaviour.The proof of Theorem 3.22 relies on the following fundamental lemma relating the two differentconstructions arising from the twisting morphisms φ and ψ associated to the Koszul operads P and Q , respectively. Lemma 3.25
Let P ¡ and Q ¡ be the Koszul dual cooperads to P and Q . For the commutative diagramof maps of (co)operads and twisting morphisms P QP ¡ Q ¡ , f φ g ψ there is a natural isomorphism of functors B φ f ∗ = g ! B ψ : Q - Alg −! P ¡ - Cog , where f ∗ denotesthe restriction of scalars functor and g ! denotes the coinduction functor.Proof. Ignoring the additional differentials produced by the bar construction, B ψ produces thecofree conilpotent Q ¡ -coalgebra functor on the underlying chain complex and g ! is the right adjointof the corestriction of scalars functor g ∗ . Since the composition of right adjoints is a right ajoint, Differential forms on smooth operadic algebras we conclude that up to bar-differentials both B φ f ∗ and g ! B ψ correspond to the cofree conilpotent Q ¡ -coalgebra on the underlying space. The commutativity of the diagram above guarantees thatboth differentials are the same. Corollary 3.26
In the conditions of the previous lemma, if the coinduction functor g ! satisfies thePBW property, there is a quasi-isomorphism of functorsf ∗ Ω ψ (cid:39) Ω φ g ! : Q ¡ - Cog −! P - Alg . Proof.
Our hypothesis on g ! implies it preserves quasi-isomorphisms [34, Corollary 1.1]. Thisimplies that there are quasi-isomorphisms of functors Ω φ g ! ∼ −! Ω φ g ! B ψ Ω ψ = Ω φ B φ f ∗ Ω ψ ∼ −! f ∗ Ω ψ . This is what we wanted.Recall from the discussion after Proposition A.2 that the operadic A -module Ω A of Kähler differen-tials on A is the quotient of the free operadic A -module generated by symbols da for a ∈ A subjectto a generalized Leibniz rule. Lemma 3.27
The map
HKR A of Proposition 3.20 is a quasi-isomorphism for Q -algebras obtainedas a cobar construction.Proof. For a Q -algebra of the form A = Ω ψ ( V ) , with V a Q ¡ -coalgebra, let us compute the quasi-isomorphism type of Def ∗ ( f ∗ A ) . Taking the cofibrant resolution Q = Ω φ B φ f ∗ A of A in the cat-egory of P -algebras given by the bar-cobar resolution, we have that:Def ∗ ( f ∗ A ) = ( Ω Q ⊗ UQ Q , d ) by Definition 3.4 = ( B φ f ∗ A ⊗ Q , d t ) by Lemma A.3 = (cid:0) f ∗ A ⊗ g ! B ψ Ω ψ ( V ) , d t (cid:1) by Lemma 3.25 ∼ (cid:0) f ∗ A ⊗ g ! ( V ) , d t (cid:1) since g ! is PBW . Here d t denotes the only non-internal differential which is transported along the isomorphism ofgraded vector spaces provided by Lemma A.3.On the other hand, since A is quasi-free, Ω A = ( U Q ( A ) ⊗ V , d t ) , endowed with an additional trans-ferred differential. It follows that as an operadic A -module, Ω ∗ F , A = (cid:0) A ⊗ g ! ( V ) , d t (cid:1) , so in particular as chain complexes Ω ∗ F , A and Der ∗ ( f ∗ A ) are quasi-isomorphic. It remains to see thatthis quasi-isomorphism gives an quasi-inverse to HKR A . Keeping track of the quasi-isomorphismsabove, one can see that filtering Def ∗ ( f ∗ A ) by the appropriate word lengths we recover the quasi-inverse at the level of the associated graded complexes, which reduces our claim to that of free Q -algebras (with zero differential). This is what we wanted. . Campos and P. Tamaroff Proposition 3.28
Suppose that A is a smooth Q -algebra and that the functor g ! preserves quasi-isomorphisms. Then Ω ∗ F , A is a complex with homology g ! A H ( A , A ) concentrated in degree zero.Proof. This follows immediately since Def ∗ ( A , A ) = Q ⊗ U Ω Q is a complex with homology con-centrated in degree 0, where it equals H ( A , A ) = A ⊗ U Ω A , while g ! A H ( A , A ) = A ⊗ g ! ( Ω A ) , soall we need to conclude is the fact g ! preserves quasi-isomorphisms. Proof of Theorem 3.22.
Since the algebra A is smooth, we know by Proposition 3.7 that the morph-ism p ! Ω Q −! Ω A is a quasi-isomorphism, and hence so is the map Q ⊗ UQ Ω Q −! A ⊗ UA Ω A . By Proposition 3.28, the resulting map Ω ∗ F , Q ! Ω ∗ F , A is a quasi-isomorphism. We have shown inLemma 3.27 that HKR QA is a quasi-isomorphism (for QA is cofibrant) and since A is smooth, thecommutativity of the following diagramDef ∗ ( f ∗ Q , f ∗ Q ) Ω ∗ F , Q Def ∗ ( f ∗ A , f ∗ A ) Ω ∗ F , A HKR Q ∼∼ ∼ HKR A shows that HKR A is a quasi-isomorphism. This is what we wanted. As before, let us take a quasi-free resolution ( P ◦ Y , d ) of Q where Y is a diagonal endofunctor and,for the Q -algebra A , let ( Q ( V ) , d ) be a quasi-free resolution. Recall that in this case the underlyinggraded vector space to Def ∗ Q ( A ) is given by hom ( V , Q ( V )) . Definition 3.29
Let A be a Q -algebra. We define Poly ∗ ( A ) , the poly-vector fields on A relative to f ,to be the chain complex Poly ∗ ( A ) : = ( hom ( Y ( V ) , Q ( V )) , δ ) . The dual result for tangent cohomology of smooth Q -algebras is the following. The proof is quitesimilar to the case of cotangent homology, so we only sketch the details. We will make use of the‘smaller complex’ Poly ∗ ( Q , A ) : = hom ( Y ( V ) , A ) . The non-internal part of the differential makesuse of the map ( Q ( V ) , d ) ! A . Theorem 3.30
If f is left Koszul then for every smooth Q -algebra A the map HKR A : Def ∗ ( f ∗ A ) −! Poly ∗ ( A ) is a quasi-isomorphism. Lemma 3.31
The conclusion of Theorem 3.30 holds for cobar algebras. Differential forms on smooth operadic algebras
Proof.
For a Q -algebra of the form A = Ω ψ ( V ) , with V a Q ¡ -coalgebra, let us compute the quasi-isomorphism type of Der ∗ ( f ∗ A ) . Taking the cofibrant resolution Q given by the bar-cobar resolu-tion Ω φ B φ f ∗ A of f ∗ A in the category of P -algebras, we have that:Def ∗ P ( f ∗ A ) = hom Q ( Ω UQ , Q ) by Definition 3.2 = hom ( B φ f ∗ A , Q ) by Lemma A.3 = hom ( g ! B ψ Ω ψ ( V ) , Q ) by Lemma 3.25 ∼ ! hom ( g ! ( V ) , Q ) g ! is PBW . = hom ( Y ( V ) , Q ) Proceeding as in the case of homology, we obtain a quasi-inverse to the HKR map.
Proof of Theorem 3.30. If A is a smooth algebra, then for any cofibrant replacement p : Q ∼ −! A there are induced quasi-isomorphismsDer ( Q , Q ) −! Der ( Q , A ) − Der ( A , A ) since hom UQ ( Ω Q , − ) is exact. Furthermore, similarly to Proposition 3.7 one can show that Poly ∗ ( Q ) ! Poly ∗ ( Q , A ) and Poly ∗ ( A ) ! Poly ∗ ( Q , A ) are quasi-isomorphisms. The result follows from thecommutativity of the diagramDef ∗ ( f ∗ Q , f ∗ Q ) Poly ∗ ( Q ) Def ∗ ( f ∗ Q , f ∗ A ) Poly ∗ ( Q , A ) Def ∗ ( f ∗ A , f ∗ A ) Poly ∗ ( A ) HKR Q ∼∼ ∼∼ = HKR A ∼ where we now use coefficients to be able to draw the zig-zag of quasi-isomorphisms. Remark 3.32
The classical cohomological version of the HKR theorem establishes not only thatthe Hochschild cohomology and the space of poly-vector fields of a smooth commutative algebraare isomorphic as chain complexes, but that they are isomorphic Lie algebras. In the operadicsetting, tangent cohomology is also a Lie algebra via the bracket defined by the commutator ofderivations. Unless the endofunctor Y carries some extra structure, it is not clear a priori how toendow Poly ∗ ( A ) with a Lie algebra structure that makes our map an isomorphism of Lie algebras.However, we point the reader to Theorem 4.10 below where Y can be taken to be an operad itself,and where we explain how in the classical case we, do recover the Lie algebra structure on poly-vector fields. . Campos and P. Tamaroff Let us show that our formalism recovers the classical HKR theorem exactly.
Proposition 4.1
The morphism
Ass −! Com enjoys the HKR property, and the induced map
HKR A : Def ∗ ( f ∗ A ) −! Ω ∗ A coincides, up to a suspension, with the classical HKR quasi-isomorphism.Proof. We offer two points of view:(1) We can produce a resolution of the form ( Ass ◦ Lie ¡ , d ) coming from the functorial Koszulresolution on free commutative algebras g V : ( T ( S c ( V [ − ])[ ]) , d ) −! S ( V ) , which is manifestly diagonally pure, since the homological degree in S c ( V [ − ])[ ] = Lie ¡ ( V ) coincides with the weight degree.(2) The Koszul dual morphism is PBW, that is, Ass is a free right
Lie -module, so Theorem 3.15implies the result. Moreover, the lemma preceding it shows that we may take Y = Lie ¡ as inthe previous item; we need only pay attention to the shift in homological degree.We conclude, in particular, that Tor Ass ( k , Com ) (cid:39) Lie ¡ as weight graded dg Σ -modules, so we maytake V Y ( V ) = S c ( V [ − ])[ ] as the functor witnessing the classical HKR property. The HKRmap for a commutative algebra A induces an isomorphism s − HH ∗ ( A , A ) + = H ∗ ( A , A ) −! Ω ∗ A = S cA ( Ω A [ − ])[ ] . It differs from the classical HKR isomorphism map by a desuspension and in that we do not have,in degree zero, the identity map of A = HH ( A ) onto A . Otherwise, our formalism recovers theHKR map exactly. Permutative algebras.
A permutative algebra [5] is an associative algebra A such that for every x , y and z ∈ A , we have that x ( yz ) = x ( zy ) . Permutative algebras are algebras over a binary quadraticoperad, denoted Perm which is the linearisation of a set operad
Perm . The Koszul dual operadof
Perm is the operad
PreLie controlling pre-Lie algebras. Both these algebraic structures play animportant role in the study of operadic deformation theory [3, 14, 15].Clearly, every commutative algebra is a permutative algebra via the same product. Since a per-mutative product is in particular associative, there is a factorisation of the map of operads f : Ass −! Com via the permutative operad:
Ass −! Perm ψ −! Com . Differential forms on smooth operadic algebras
Proposition 4.2
The map ψ : Perm ! Com enjoys the HKR property, with generating sequence RT (cid:54) = that assigns a set I to the set of rooted trees with vertices labeled by I for which no vertexhas exactly one child.Proof. The Koszul dual map to ψ is the anti-symmetrisation map φ : Lie ! PreLie . In [16] thismap was shown to satisfy the PBW property. Moreover, in [11], it was show that the generators of
PreLie as a right
Lie -module are given by the functor RT (cid:54) = as in the statement of the proposition.The result follows from Corollary 3.23.Theorems 3.22 and 3.30 allow us to compute the permutative (co)tangent (co)homology of com-mutative algebras. Corollary 4.3
Let A be a smooth commutative algebra. • The cotangent homology H ∗ ( ψ ∗ A ) is isomorphic to the algebra of “tree-wise” differentialforms RT (cid:54) = ( Ω ∗ A ) over the classical differential forms of A. • Dually, the tangent cohomology H ∗ ( ψ ∗ A ) is isomorphic to the algebra of “tree-wise” poly-vector fields RT ∨(cid:54) = ( Poly ∗ ( A )) . Corollary 4.4
For every smooth commutative algebra A there is a quasi-isomorphism
Def ∗ Perm ( A ) −! RT ∨(cid:54) = ( Poly ∗ ( A )) . Moreover, the natural map
Def ∗ Perm ( A ) −! Def ∗ Ass ( A ) induces, in homology, the natural map RT ∨(cid:54) = ( Poly ∗ ( A )) −! Poly ∗ ( A ) given by the augmentation RT ∨(cid:54) = −! k .Enriched pre-Lie algebras of Dotsenko and Foissy. In [12] the authors define a functor that assignsto every Hopf cooperad C an operad PreLie C of C -enriched pre-Lie algebras. The example we areinterested in is the following, where this functor recovers the operad of pre-Lie algebras and theoperad of braces algebras. Such braces algebras and related structures, conceived originally in [31],appeared in the literature in several opportunities [21, 22, 35], and are relevant in deformationtheory [14], for example.(1) If C = u Com ∗ is the Hopf cooperad of unital commutative coalgebras, one obtains the operad PreLie governing pre-Lie algebras.(2) If C = u Ass ∗ is the Hopf cooperad of unital associative coalgebras, one obtains the operad Br governing brace algebras.(3) The unit map u Com ∗ −! u Ass ∗ gives the map g : PreLie −! Br constructed in [8].By Proposition 2 in that article, every connected Hopf cooperad C admits a structure of associativealgebra for the Cauchy product in the category symmetric sequences —what are usually calledtwisted associative algebras—, in such a way that every morphism of Hopf cooperads C −! C (cid:48) induces a morphism of twisted associative algebras. . Campos and P. Tamaroff Definition 4.5 (Proof of Theorem 1 in [12]) Given a species X , there is a species of enriched trees,which we write T R , where each vertex is decorated by an element of C (cid:48) with the condition thatevery vertex of maximal depth is decorated by an element of X . Similarly, T L is the species of C (cid:48) -enriched trees, with the condition that the root vertex is decorated by an element of X .The main result of [12] is as follows. Theorem.
Let ϕ : C −! C (cid:48) be a morphism of connected Hopf cooperads and let us consider C (cid:48) asa C -bimodule by viewing ϕ as a map of twisted associative algebras. Then:(1) if C (cid:48) is left C -free with generators X then the operad PreLie C (cid:48) is free as a left PreLie C -modulewith generators T L and,(2) if C (cid:48) is right C -free with generators X then the operad PreLie C (cid:48) is free as a right PreLie C -module with generators T R . An immediate consequence of this result is the following, since the map of twisted associativealgebras u Com ∗ −! u Ass ∗ is both left and right free. Corollary 4.6 (Theorem 2 in [12])
The brace operad Br is free as a left and as a right PreLie -module.
From this, we obtain the following HKR theorem for smooth brace algebras.
Theorem 4.7
The map g : PreLie −! Br satisfies the HKR property: for every smooth bracealgebra A there exists a natural quasi-isomorphism HKR A : Def ∗ ( g ∗ A ) −! Ω ∗ F , A where Ω ∗ F , A = T R ( Ω A ) where T R is the endofunctor of rooted trees with vertices of maximal depthdecorated by Lie words.Diassociative algebras. Diassociative algebras were introduced by J.-L. Loday in [37]. A diassoci-ative algebra [39, Section 13.6] consists of a vector space V along with two associative operations (cid:96) : V ⊗ V −! V and (cid:97) : V ⊗ V −! V satisfying the following set of three quadratic relations: ( x (cid:97) x ) (cid:97) x = x (cid:97) ( x (cid:96) x ) , ( x (cid:96) x ) (cid:97) x = x (cid:96) ( x (cid:97) x ) , ( x (cid:97) x ) (cid:96) x = x (cid:96) ( x (cid:96) x ) . Any permutative algebra gives rise to a diassociative algebra by defining both products to bethe permutative product, so that we have a map
Dias −! Perm , whose Koszul dual is the map
PreLie −! Dend . In [16] the authors proved that this morphism is PBW and, since it is known that
Dend is a left free
Ass -module with basis the operad of braces Br , we can use Corollary 4.6 andour main theorem to obtain the following result: Differential forms on smooth operadic algebras
Theorem 4.8
For every smooth permutative algebra A there is a quasi-isomorphism
HKR A : Def Dias ∗ ( A ) −! Ω ∗ A where the endofunctor Y is given by Ass ◦ T R and T R is the endofunctor of Theorem 4.7.The work of J. Griffin. Let us now connect our formalism with the one developed by J. Griffin.Motivated by the Hodge decomposition of Hochschild cohomology [1, 19], Griffin [26] consideredthe problem —like we do— of computing the cohomology of a pull-back algebra f ∗ A under amorphism of operads f : P −! Q . Since his motivation is slightly different from ours, there is nomention of HKR-type theorems in his paper, nor of smooth algebras.However, one can find the following result in ibidem , which relates the Quillen homology of a Q -algebra A to that of its pull-back, which can be seen as a first approximation to the problem ofcomputing the cohomology of f ∗ A , and which contains already a clear link between his and ourformalism; see Theorem 3.7-(II) in [26]. Theorem 4.9
Let f : P −! Q be a map of Koszul operads and let g : Q ¡ −! P ¡ be its Koszul dualmap. Suppose that P ¡ is a free right Q ¡ -comodule with basis X . Then for every Q -algebra A thereis an isomorphism B P ( f ∗ A ) −! X ◦ B Q ( A ) . Moreover, Griffin goes on to consider the case of maps of Koszul operads P −! Q where Q isobtained from P (cid:48) and another operad P by a filtered distributive law [13] as originally defined by V.Dotsenko in [10]; see [26, Theorems 5.15 and 5.18]. The following result offers a complementarytechnique to compute the tangent cohomology of a P -algebra coming from a smooth Q -algebraunder the projection f : P −! Q . Theorem 4.10 (Filtered HKR theorem)
Suppose P is obtained from Q and R by a filtered dis-tributive law, so that P is isomorphic to Q ◦ R as a right R -module. For every smooth Q -algebra Athe cotangent homology of f ∗ A is given by the endofunctorA −! R ¡ ( Ω A ) of “ R ¡ -enriched differential forms” on A.Proof. Since we are working over a field of characteristic zero, Theorem 5.4 in [13] guaranteesthat P ! is a free right Q ! -module with generators R ! , so the claim follows. Let us recall from [38] that we can arrange certain nine operads into a “butterfly’ diagram ofmorphisms, as in the figure above. We record those maps which we know satisfy the PBW propertyand which we know satisfy the HKR property. Most of the claims follow immediately by duality(Theorem 3.15) from the results obtained in [16], or by the following simple remark: . Campos and P. Tamaroff Ass Perm ComZinbDendDiasPreLieLieLeib P B W b y [ ] H K R b y d u a l i t y HK R bydu a lit y H K R b y d u a l i t y P B W b y [ ] P B W by [ ] N o t P B W b y [ ] Figure 1: The operadic butterfly.
Remark 4.11
Note that if f : P −! Q is PBW, then we must have dim k P ( n ) (cid:54) dim k Q ( n ) for each n ∈ N . In particular, since dim k Dias ( ) > dim k Ass ( ) , dim k Leib ( ) > dim k Lie ( ) , and sincedim k PreLie ( ) > dim k Ass ( ) , the respective maps in Figure 1 are not PBW.It would be interesting to determine if the remaining arrows enjoy the HKR or the PBW propertyor if, perhaps, they enjoy none of the two. Remark 4.12
It is well known [6] that the map of operads
Lie −! PreLie makes its codomaina free left module. However, the generators exhibiting
PreLie as a left free
Lie -module are notconcentrated in weight zero, so that, as expected, the map from
Perm onto
Com is not PBW. Infact, in general, the extra weight degree we have considered means a map f : P −! Q that is leftfree will not be left Koszul unless it is the identity, which shows that it is crucial to replace the ‘leftfree’ condition to a left Koszul condition. A Recollections on operads
A.1 Operads and their algebras and modules
Let us fix a reduced symmetric operad P and write P - Alg for the category of dg P -algebras. Theoperad P , viewed as a monad, gives the left adjoint P : Σ dgMod −! P - Alg to the forgetful functor P - Alg −! Σ dgMod . Differential forms on smooth operadic algebras
Fix a dg P -algebra A as before. An operadic A-module is a dg Σ -module M along with an action γ M : P ◦ ( A , M ) −! M so that γ M ( ◦ ( γ A , γ M )) = γ M ( γ ◦ ( , )) . Here P ◦ ( A , M ) is the submodule of P ( A ⊕ M ) which is linear in M .It is useful to note that if P = As and if A is an P -algebra or, what is the same, an associativealgebra, then an operadic A -module is the same as an A -bimodule and not a left (or right) A -module. Similarly, the operadic modules for commutative algebras are the symmetric bimodules.In fact, there is a functor U P : P - Alg −! Ass - Alg , the last being the category of dga algebras, so that the category of operadic A -modules is isomorphicto the category of left U P ( A ) -modules of the associative algebra U P ( A ) . Definition A.1
We call U P ( A ) the associative enveloping algebra of A .Concretely, U P ( A ) is spanned by trees with one leaf pointed by the only element in k under therelation that identifies the corolla with root µ ◦ i ν with the corolla with root µ and ν acting on theleaves i , i + , . . . , and we will write a generic element by u ( a , . . . , a i − , − , a i + , . . . , a n ) where u is an operation of P and the empty slot corresponds to the leaf marked by k . The algebrastructure is defined by concatenation through the pointed leaf and the root through the partialcomposition ◦ i of P . We refer the reader to [33] for a useful reinterpretation of U P through thelanguage of 2-colored operads.As useful examples, we note that in the case of associative and Lie algebras, we recover the usualnotion of enveloping algebra: for an associative algebra A we have that U A s ( A ) = A ⊗ A op , for a Liealgebra L we have that U Lie ( L ) = U ( L ) ; note in both cases we are considering non-unital algebrasand non-unital actions.Given a map of P -algebras f : B −! A , we obtain two maps f ∗ : A Mod −! B Mod and f ! : B Mod −! A Mod corresponding respectively to the restriction and extension of scalars, and a map U P ( f ) : U P ( B ) −! U P ( A ) . Then the previous two adjoint functors are simply the usual functors of restriction and extensionfor U P ( f ) . We can also describe the free modules as follows. If X is a dg Σ -module, we have acoequalizer diagram P ( P ( A ) , X ) P ( A , X ) A ◦ P X . Campos and P. Tamaroff where the arrows are γ ( , ) and 1 ( γ A , ) and A ◦ P X is the free operadic A -module on X , so that A ◦ P − : Σ dgMod −! A Mod is left adjoint to the forgetful functor A Mod −! Σ dgMod . Graphically, generators of A ◦ P X correspond to corollas with their root labeled by an operation of P , all whose leaves are labeled by elements of A except for one, which is labeled by an element of X , and we impose the relations for each i , l , n ∈ N , each pair of operations µ , ν ∈ P with µ of arity n and each n -tuple ( a , . . . , a n ) of elements of A , µ ( a , . . . , a i , ν ( a i + , . . . , a i + l ) , a i + l + , . . . , a n , x ) = ( µ ◦ i ν )( a , . . . , a n , x ) . In case A or P are graded, signs will appear owing to the Koszul sign rule. A.2 Derivations and Kähler differentials
As before, let us fix an operad P , and let us also fix a P -algebra A . If M is an operadic A -modulethen a P -derivation of M is a linear map d : A −! M such that γ M ( ◦ (cid:48) d ) = d γ A . For a fixed choice f : B −! A of a map P -algebras, we say d is B -linear whenever it vanishes on the image of f .Following [27], we write Der B ( A , M ) for the complex of such derivations, which defines a functorDer B ( A , − ) : A Mod −! k Ch . In particular, if u : A −! U is a map of P -algebras, then U is an operadic A -module and we canconsider Der B ( A , U ) the complex of B -linear derivations A −! U . We refer the reader to [39,§12.3.19] for a proof of the following: Proposition A.2
The functor
Der B ( A , − ) is representable. We call the representing module the module of relative Kähler differentials and write it Ω A | B .Explicitly, Ω A is the coequalizer of the diagram A ◦ P d P ( A ) A ◦ P dA Ω A so that Ω A is the free operadic A -module on a copy dA of A where we additionally impose therelations that, for each i , l , n ∈ N , each pair of operations µ , ν ∈ P with µ of arity n and each n -tuple ( a , . . . , a n ) of elements of A , where a (cid:48) = d ν ( a i + , . . . , a i + l ) : µ ( a , . . . , a i , a (cid:48) , a i + l + , . . . , a n ) = l ∑ t = ( µ ◦ i ν )( a , . . . , da i + t , . . . , a n ) Differential forms on smooth operadic algebras
The arrows are as follows: the uppermost arrow is induced from the map1 ( , γ A ) : P ( A , d P ( A )) −! P ( A , dA ) , while the lowermost arrow is induced from the following three maps:(1) the arrow P ( A , d P ( A )) −! P ( A , P ( A , dA )) induced from the infinitesimal composite 1 ◦ (cid:48) d : d P ( A ) −! P ( A , dA ) obtained from the isomorphism d : A −! dA ,(2) the arrow P ( A , P ( A , dA )) −! ( P ◦ ( ) P )( A , dA ) which is an inclusion and(3) the arrow γ ( ) ( , ) .The module of relative Kähler differentials Ω A | B is defined similarly, with the extra relation that dB =
0. It is functorial in both arguments in the following way. If we have a pair of morphisms B f −−! A g −−! C of P -algebras we can consider any A -linear derivation of A as a B -linear derivation,so we get a morphism Ω C | g : Ω C | B −! Ω C | A representing the restriction. Similarly, any B -linear derivation d : C −! M defines a B -linear de-rivation g ∗ d : A −! f ∗ M so we obtain a morphism Ω g | B : g ! Ω A | B −! Ω C | B . The following lemma describes Kähler differentials and derivations of free algebras. In particular,it follows the corresponding complexes of derivations and of differentials of P -algebras of theform ( P ( V ) , d ) are simple, and correspond to “nc-vector fields” X : V −! P ( V ) determined on thecoordinates v ∈ V by some vector field X : ∂ v X ( ∂ v ) , and to “nc-differential forms” f ( v ) dv where f ( v ) ∈ Y is a function on the coordinates. Lemma A.3
Let A = P ( V ) be the free P -algebra on V . Write i : V −! P ( V ) for the canonicalinclusion. Then Ω A is canonically isomorphic to the free operadic X -module generated by V , andwe have isomorphisms of complexesi ∗ : Der ( A ) −! hom ( V , A ) , i ∗ : A ⊗ V −! A ⊗ UA Ω A that assign a derivation f : A −! A to its restriction f i and x ⊗ v to the class of xdv.Proof. Since X is free, any derivation f : X −! X is determined by its restriction to V , and i ∗ is abijection. From this and the Yoneda lemma it follows that Ω X is the free left U X -module generatedby V , and hence that the canonical map X ⊗ V −! X ⊗ UX Ω X is an isomorphism.In particular, if we consider a commutative algebra A and the bar-cobar resolution Y = Ω BA , weget the following: Lemma A.4
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