Excision in Hochschild and cyclic homology without continuous linear sections

EX CISION IN HOCHSCHILD AND CYCLIC HOMO LOGY WITHOUT CONTINUOUS LINEAR SECTIONS RALF MEYER Abstract. W e pro v e t hat con tin uous Hoch schild and cyclic homology satisfy excision for e xtensions of n uclear H-unital F r´ ec het algebras and use this to compute them f or the algebra of Whitney f unctions on an arbitrary closed subset of a smo oth manifold. Using a simi l ar excision result f or p erio dic cyclic homology , we also compute the p erio dic cyclic homology of algebras of sm ooth functions and Whitney functions on closed subsets of smooth manifolds. Contents 1. Int ro duction 1 2. Prepar ations: homologica l a lgebra and functional analysis 3 2.1. Some examples of symmetric mo no idal categories 4 2.2. Exact categor y s tructures 6 2.3. Exact chain complex e s, quasi-isomor phisms, and homology 8 2.4. Hochschild ho mology and cohomology 11 2.5. Pure conﬂations 12 3. Excision in Ho chsc hild homology 15 4. Ho chsc hild ho mology for algebra s o f smoo th functions 19 4.1. The algebra of smo oth functions with c ompact suppo rt 21 5. Application to Whitney functions 23 6. Excision in p erio dic cyclic homo logy 28 7. Conclusion and outlo ok 30 References 31 1. Introduction Ho chsc hild and cyclic homology are in v ariants of no n-commutativ e algebras that generalise diﬀerential forms and de Rham cohomolo gy for smo o th ma nifolds [4]. More precis e ly , let A : = C ∞ ( X ) for a smo oth ma nifold X be the F r´ echet algebr a of s mo oth functions o n a smo oth manifold X (w e impose no growth condition at inﬁnity). Its co ntin uous Ho chsc hild homolog y HH n ( A ) for n ∈ N is naturally isomorphic to the space of diﬀerential n -forms Ω n ( X ) on X . Its contin uous pe rio dic cyclic homology HP n ( A ) for n ∈ Z / 2 is naturally isomor phic to the de Rham 2000 Mathematics Subje ct Classiﬁc ation. 19D55. Supported by the German Research F o undation (Deutsc he F orsch ungsgemeinsc haft (DFG)) through the Institutional Strategy of the Universit y of G¨ ottingen. 1 2 RALF MEYE R cohomolog y of X made 2-p erio dic: HP n  C ∞ ( X )  ∼ = M k ∈ Z H n − 2 k dR ( X ) . And its contin uo us cyclic homology HC n ( A ) in terp olates b etw e en these tw o: HC n  C ∞ ( X )  ∼ = Ω n ( X ) d  Ω n − 1 ( X )  ⊕ ∞ M k =1 H n − 2 k dR ( X ) , where d : Ω n − 1 ( X ) → Ω n ( X ) denotes the de Rham b ounda ry map. The corre- sp onding contin uous cohomolo gy gr oups HH n ( A ), HP n ( A ) and HC n ( A ) ar e natu- rally is o morphic to the top olog ical dual spa c es of these F r´ echet spaces; in particular , HH n ( A ) is isomorphic to the s pace of de Rham n -cur rents on X . Alain Connes [4] prov es t hese co homologica l results with an explicit pro jective A -bimo dule r e s olution of A . The same method yields their homolo g ical analogues. Recently , Jean-Paul Brasselet and Markus Pﬂaum [1 ] extended these computa- tions to the alg ebra of Whitney functions on certain reg ular subsets of R n . The pro of is quite complicated b ecause the p ossible singularities of such subsets make it m uch harder to write down pro jective bimo dule res olutions. Here we use excision theorems to compute these inv ariants for the a lgebra of Whitney functions on any closed s ubset of a smo oth manifold. Our pro of is shorter and remo ves the tec hnical assumptions in [1]. Let Y be a clos ed subset of the s mo oth manifold X and let J ∞ ( X ; Y ) b e the closed ideal in C ∞ ( X ) consisting of a ll functions that hav e v a nis hing T aylor series at all p o ints of Y in so me – he nce in any – lo cal co ordinate c hart. The alg ebra o f Whitney functions o n Y is the F r´ echet algebr a E ∞ ( Y ) : = C ∞ ( X )  J ∞ ( X ; Y ) . It may depend on the embedding o f Y in to X . Since E ∞ ( Y ) is deﬁned to ﬁt into an extension of F r´ echet alge bras (1) J ∞ ( X ; Y ) ֌ C ∞ ( X ) ։ E ∞ ( Y ) , we may hope to compute its in v ariants us ing the Excision Theorem of Mariusz W o dzicki [27], whic h provides natura l long exact sequences in Ho chsc hild, cyclic, and per io dic cyclic homolo g y for suitable algebra extensions. The only is sue is whether the Excis ion Theor em a pplies to the extension (1) be- cause it need not have a contin uous linear section, and such a sectio n is assumed by previous excision statements a bo ut contin uous Hochschild (co)homolo gy of top olog- ical algebra s ; this is why Brasselet and Pﬂaum use another appr oach. Our main ta sk is, therefor e, to formulate a n Excisio n Theo rem for contin uous Ho chsc hild homo logy that does not r equire contin uous linear sections. Tha t such a theorem exists has long been known to the exp erts. Mariusz W o dzicki stated a s pec ia l case in [26, Prop osition 3] and a nno unced general results for top olo gical algebras in [27, Remark 8.5.(2)], which were, how ever, never published. The pro of of W o dzicki’s Ex cision Theor e m by Jo rge and Juan Guccione [8] works in great generality a nd, in fact, applies to the ex tension (1), but s uch gene r alisations are not formulated explicitly in [8]. The exa mple of the a lgebra o f Whitney functions has motiv a ted me to ﬁnally state and prov e such a g eneral excision theorem here. I w ork in a ra ther a bstract categoric al setup to avoid further embarrassments with insuﬃcient generality . EXCISION IN HOCHSCHILD AND CYCLIC HOM OLOGY 3 The situation in [8] is a ring extension I ֌ E ։ Q tha t is pur e , that is , remains exact after tenso ring with another Abelian group, a nd has an H- u nital kernel I . W e genera lise the notions of purit y and H-unitalit y to algebras in a n a dditive sym- metric monoidal catego r y ( C , ⊗ ) with an exact categ ory structure, that is, a class of distinguished extensions E , which w e call c onﬂations following the notation of [12, 13]. It is routine to chec k that the a rguments in [8] still work in this gener ality . Then we spe cialise to the catego ry of F r´ echet spaces with the complete pro jective top ological tensor pro duct a nd the class of all extensio ns in the usual sense. W e chec k that an extensio n of F r´ echet spaces with nuclear quotient is pur e and that the algebra J ∞ ( X ; Y ) is H-unital in the relev ant s ense, so that our Excision Theorem applies to (1). Excision in Ho chsc hild homolo gy also implies excisio n in cyclic a nd per io dic cyc lic homolo gy . Thus we compute all three homo logy theories for the algebra of Whitney functions. The case of F r´ echet a lg ebras is o ur main a pplication. W e als o discuss algebras in the catego r ies of inductive or pro jectiv e s y stems of Banach s paces, which include complete c o nv ex b orno lo gical algebr as and co mplete lo cally conv ex top olog ic al a lge- bras. F or instance, this covers the case of Whitney functions with compac t supp or t on a non-c ompact closed subset of a smo o th manifold, which is an inductiv e limit of n uclear F r´ echet alg ebras. The contin uo us Hochschild cohomology HH n ( A, A ) of a F r´ echet algebra A with co eﬃcients in A viewed a s an A -bimo dule is used in deformation quantisation theor y . F or A = C ∞ ( X ), this is naturally isomorphic to the space of smo oth n - vector ﬁelds on X , that is, the spac e of smo o th sectio ns of the vector bundle Λ n (T X ) on X . The method of Brass e let and P ﬂaum [1] also allows to co mpute this for the algebra of Whitney functions on suﬃcien tly nice subsets of R n . I hav e tried to repr ov e and generalise this us ing excisio n techniques, but did not succeed beca use purity of an extension is not enoug h for c ohomolo gic al computations. While it is lik ely that the Ho chsc hild co homology for the alg ebra of Whitney functions is alwa ys the space of Whitney n -vector ﬁelds, excis ion techniques only yield the co rresp onding res ult ab out HH n ( A, A k ), wher e A k is the Banach spac e of k -times diﬀerentiable Whitney functions, viewed a s a module ov e r the alg ebra A of Whitney functions. 2. Prep ara tion s: homological al gebra and functional anal ysis The abs tract framework to deﬁne and study algebras a nd mo dules are additive symmetric monoidal c ate gories (see [22]). W e dis cuss so me e x amples o f such ca t- egories: Ab elian gro ups with their usual tensor pro duct, F r´ echet spaces with the complete pro jective tenso r pro duct, and inductiv e or pr o jective systems of B anach spaces with the canonical extensions of the pro jective Banach spac e tensor pro duct. The additiona l s tructure o f an exact c ate gory sp eciﬁes a class o f c onﬂations to be used instead of s hort exact sequences . This allows to do homolo gical algebr a in non-Ab elian additive categories. T he original axioms by Daniel Quillen [21] are simpliﬁed slig h tly in [12]. W e nee d non-Ab elian catego ries b ecause F r´ echet spa ces or b orno logical vector spaces do no t form Ab elian catego ries. W e descr ib e so me natural exact categ ory structures for Ab elian groups, F r ´ echet spaces, and induc- tive or pr o jective systems of Banach spaces. W e also intro duce pure co nﬂations – conﬂations that remain conﬂations when they a re tensored with an ob ject. W e show that extensions of nuclear F r´ echet spaces are always pure and ar e close to being split in at lea st t wo diﬀerent wa y s : they r emain exact when we apply the 4 RALF MEYE R functors Hom( V , ) o r Hom( , V ) for a Banach space V . This is re la ted to useful exact categor y structures o n categories of inductive a nd pro jective sy s tems. 2.1. Some examples of symmetric monoidal categories. An additive sym- metric monoidal c ate gory is an additive categ ory with a bi-additive tensor pro duct op eration ⊗ , a unit ob ject 1 , a nd natural isomorphisms (2) ( A ⊗ B ) ⊗ C ∼ = A ⊗ ( B ⊗ C ) , A ⊗ B ∼ = B ⊗ A, 1 ⊗ A ∼ = A ∼ = A ⊗ 1 that satisfy several compa tibility conditions (see [22]), whic h we do not r e call here bec ause they ar e trivial to check in the e x amples we are interested in (we do not even sp ecify the natural transfor mations in the examples below b ecause they are so obvious). Roug hly sp eaking, the tensor pr o duct is asso ciative, symmetric, and monoidal up to cohere nt natural isomorphis ms . W e o mit the tensor pro duct, unit ob ject, and the na tural isomorphisms ab ov e from our notatio n a nd use the same notation for a symmetric mo noidal categor y and its underlying catego ry . The unit ob ject is determined uniquely up to iso morphism, anyw ay . The following ar e examples of additive symmetric monoidal categories : • Let Ab b e the categ ory of Abelian groups with its usual tensor pro duct ⊗ , 1 = Z , and the obvious natural isomorphisms (2). • Let F r b e the catego ry of F r´ echet spaces, that is, metrisable, complete, lo cally conv ex top ologica l v ector spaces, with contin uous linea r maps as morphisms. Let ⊗ : = ˆ ⊗ π be the complete pro jective top ologic a l tensor pro duct (see [7]). Here 1 is C (it would b e R if w e us ed r eal vector spaces). • Let Bor be the categ ory of complete, conv ex b or nological v ector spaces (see [10]) with b ounded linear maps a s mor phisms. In the follo wing, all bo rnologic al vector spa ces a re tacitly required to b e co mplete and conv ex. Let ⊗ : = ˆ ⊗ b e the co mplete pro jectiv e b or nologica l tensor pr o duct (see [9]) and let 1 = C once ag ain. • Let − − → Ban b e the categ o ry of inductive sy stems of Banach spaces . Let ⊗ be the canonical extension o f the complete pro jective top olog ical tensor pro duct for Banach spaces to − − → Ban : if ( A i ) i ∈ I and ( B j ) j ∈ J are inductive systems o f Bana ch spaces, then ( A i ) i ∈ I ⊗ ( B j ) j ∈ J : = ( A i ˆ ⊗ π B j ) i,j ∈ I × J . The unit ob ject is the co nstant inductive system C . • Let ← − − Ban b e the ca tegory of pro jective systems of Banach spaces. Let ⊗ be the canonical extension o f the complete pro jective top olog ical tensor pro duct for Banach spaces to ← − − Ban : if ( A i ) i ∈ I and ( B j ) j ∈ J are pro jective systems o f Bana ch spaces, then ( A i ) i ∈ I ⊗ ( B j ) j ∈ J : = ( A i ˆ ⊗ π B j ) i,j ∈ I × J . The unit ob ject is the co nstant pro jective system C . • Let TVS be the categor y of complete, lo ca lly co n vex, top o lo gical vector spaces with the complete pr o jective top ologica l tenso r pr o duct ⊗ : = ˆ ⊗ π and 1 = C . In e a ch ca se, the axioms of an additive symmetric monoidal categor y a re r outine to ch eck. Unlik e Fr , the ca tegories Ab , − − → Ban and ← − − Ban are close d symmetric monoida l categorie s, that is, they ha ve an internal Hom-functor (see [16]). The v ario us categories in tro duced ab ov e are related as follows. First, the precompact borno logy functor, which eq uips a F r´ e chet space with the precompact bo rnology , is a fully faithful, symmetric monoida l functor Cpt : Fr → Bor EXCISION IN HOCHSCHILD AND CYCLIC HOM OLOGY 5 from the ca tegory o f F r´ e chet spaces to the category of b or nologica l v ector spa ces (see [16, Theorems 1.29 and 1.8 7]). This means that a linear map b etw een t wo F r´ echet spaces is contin uo us if and only if it maps pr ecompact subsets again to precompact subsets and that the identit y map on the a lgebraic tenso r pro duct V ⊗ W of tw o F r´ echet spaces V and W extends to an iso morphism Cpt ( V ˆ ⊗ π W ) ∼ = Cpt( V ) ˆ ⊗ Cpt ( W ); this amoun ts to a deep theorem of Alexander Grothendieck [7] ab out precompact subsets of V ˆ ⊗ π W . Secondly , there is a fully faithful functor diss : Bor → − − → Ban , called disse ction functor , from the catego ry of b or nological vector spaces to the catego r y of inductive systems of Banach spaces. It writes a (co mplete, co n vex) b orno logical vector spa ce as a n inductive limit of an inductive sy stem of Banach spaces in a natural w ay (see [16]). It is, unfortunately , not symmetric monoidal o n all bor nologica l vector spaces. The pro blem is that dissection is not alwa ys compatible with co mpletions. But this patholog y rarely o ccurs. In par ticular, it is symmetric monoidal on the sub c ategory of F r´ echet space s by [16, Theo rem 1.166], that is, the comp osite functor diss ◦ Cpt : Fr → − − → Ban is a fully faithful and symmetric monoidal functor. The problems with co mpletions of b or nologica l vector spaces are the reaso n why lo cal cyclic cohomolog y requir es the category − − → Ban instead of Bor (see [16]). Explicitly , the functor diss ◦ Cpt : Fr → − − → Ban does the following. Let V b e a F r´ echet s pace a nd let I b e the s e t o f all compact, abso lutely conv ex , c ir cled subsets of V . Equiv alently , a subset S of V b e longs to I if there is a Bana ch s pace W and an injective, co mpact linea r map f : W → V that maps the closed unit ball of W onto S . Given S , we may ta ke W to b e the linear span of S with the g a uge norm of S . W e denote this Banach subspac e of V by V S . The set I is directed, and ( V S ) S ∈ I is an inductive system of Banach spac e s. The functor diss ◦ Cpt maps V to this inductive s ystem of Banach spac e s. The functor Cpt extends, of course , to a functor from TVS to Bor . But this functor is neither fully faithful nor symmetric monoidal, and neither is its co mpo - sition with the dissection functor. Dually , we may embed Fr into TVS – this em b edding is fully fa ithful and sym- metric monoidal by deﬁnition. W e a re going to desc rib e an analog ue of the dissec- tion functor that maps TVS to ← − − Ban (see [20]). Given a lo cally conv ex top ologica l vector space V , let I b e the set of all contin uous s emi-norms on I and let ˆ V p for p ∈ I be the Banach space co mpletion of V with resp ect to p . This deﬁnes a pro - jective system diss ∗ ( V ) o f Ba nach s paces with lim ← − diss ∗ ( V ) = V if V is co mplete. This construction deﬁnes a fully faithful functor diss ∗ : TVS → ← − − Ban . F or t wo complete, lo c a lly con vex to po logical vector spaces V a nd W , the semi-norms of the form p ˆ ⊗ π q for contin uous semi-norms p and q o n V and W gene r ate the pro jective tensor pro duct to po logy on V ⊗ W . This yields a natura l iso mo rphism diss ∗ ( V ˆ ⊗ π W ) ∼ = diss ∗ ( V ) ⊗ diss ∗ ( W ), and the functor diss ∗ is symmetric monoidal. In s o me situations, it is preferable to use the complete inductive topo logical tensor pro duct on TVS (s e e [2]). How ever, this tensor pro duct do es not furnis h another symmetric mono idal structure on TVS b ecaus e it fails to b e a sso ciative in general. It only works on sub categor ies, such as the catego ry of nuclear LF-spa c es, where it is closely rela ted to the pro jective b ornolo gic al tensor pro duct. 6 RALF MEYE R Once w e hav e a symmetric monoidal categor y , we may deﬁne algebras and uni- tal algebras ins ide this categor y , and mo dules ov er algebras a nd unitar y mo dules ov e r unital algebras (see [2 2]). Algebras in Ab are ring s. Algebras in TVS are complete locally conv ex to p o logical algebras, that is, complete lo ca lly conv ex top o - logical vector spa c es A with a join tly con tinuous a sso ciative bilinea r multiplication A × A → A ; notice that s uch algebras need not b e lo cally mult iplicatively convex. Similarly , algebr as in Bor are complete conv ex borno logical algebra s , that is, c o m- plete co nvex bor no logical vector spaces with a (jointly) b ounded asso cia tive bilinear m ultiplication. Unitality has the exp ected meaning for a lgebras in Ab , TVS , a nd Bor . If A is a n alg ebra in Ab , that is, a r ing , then left or right A -mo dules and A -bimo dules in Ab are left o r right A -mo dules and A -bimo dules in the usa l sense, and unitality for mo dules over unital a lgebras has the exp ected meaning . The same holds in the categories TVS a nd Bor . A le ft complete lo c ally conv ex topo logical mo dule ov e r a complete lo c a lly co nv ex top olog ical alg e bra A is a complete lo cally conv e x top olog ic al vector space M with a jointly con tinuous left mo dule structure A × M → M . 2.2. Exact category structures. A pair ( i , p ) of comp osable maps I i − → E p − → Q in an additiv e category is ca lled a s hort exact se qu enc e if i is a k ernel of p a nd p is a cokernel of i . W e also call the dia gram I i − → E p − → Q a n extension in this cas e. Example 2.1 . Extensions in Ab ar e group e xtensions or sho rt exact sequences in the usual sense. T he Op en Mapping Theore m s hows that a diagr a m of F r´ echet spaces I → E → Q is an extensio n in Fr if and only if it is exact a s a se q uence of v ector spaces. This b ecomes false for mo r e general extensions o f bor nological or top ological vector spa ces. An exact c at e gory is an additiv e categ ory C with a fa mily E of extensio ns , called c onﬂations ; we call the maps i a nd p in a conﬂation a n inﬂation and a deﬂation , resp ectively , following K eller [12, 1 3]. W e use the s ymbols ֌ a nd ։ to deno te inﬂations and deﬂations, and I ֌ E ։ Q to denote conﬂations. The conﬂations in an exact ca tegory must s atisfy some axioms (see [21]), which, thanks to a simpliﬁcation by B ernhard Keller in the app endix o f [12], require: • the identit y ma p on the zero ob ject is a deﬂation; • pro ducts of deﬂations ar e again deﬂations; • pull-backs of deﬂations along arbitrar y maps exis t and are ag ain deﬂatio ns; that is, in a pull ba ck diagr am A f / /   B   C g / / D , if g is a deﬂatio n, so is f ; • push-outs of inﬂations alo ng arbitrar y maps exist a nd are again inﬂations. These axioms are usually easy to verify in examples. An y exact c ategory is equiv a le n t to a full sub ca tegory o f an Ab elian catego ry closed under extensions, such that the conﬂations co rresp ond to the extensions in the ambien t Ab elian catego ry . As a consequenc e , most r esults of homologica l algebra extend easily to exact ca tegories. EXCISION IN HOCHSCHILD AND CYCLIC HOM OLOGY 7 W e now desc r ib e some examples o f exac t category str uctures on the s y mmetric monoidal categories introduced ab ov e. Example 2.2 . Let C be a ny additive categor y and let E ⊕ be the clas s of a ll split extensions; these are isomor phic to direct sum extensio ns. This is an exact ca tegory structure on C . When we do homolog ical algebra with top olog ical or b or nological alge bras, w e implicitly use this trivial exac t catego ry structure on the categ ory o f top olog ical or bo rnologic al vector spaces. The bar resolutions that we use in this context are all split exact (contractible) a nd their entries are only pr o jective w ith resp ect to mo dule ex tensions that split as extensio ns of topolo gical o r b or nological vector spaces. Of course, our purp ose her e is to s tudy alge br a extensions that are not split, so that we need mor e in teresting class es of conﬂations. The following deﬁnition is equiv alent to one b y Jean-Pie r re Schneiders [23]. Deﬁnition 2. 3. An additive category is quasi-Ab elian if a ny mo rphism in it has a k ernel a nd a co kernel and if it b eco mes an exac t ca teg ory when we ta ke all extensions as conﬂations. In the s ituation of Deﬁnition 2.3, the exa ct categor y axioms ab ov e simplify slightly (see also [19, Deﬁnition 1 .1.3]). The ﬁr st tw o axio ms b ecome automa tic and can b e omitted, and the mere ex istence of pull-ba cks a nd push-outs in the other tw o axioms is guar anteed by the existence of k ernels and cokernels. It go es without saying that Ab elian categ ories such a s Ab are quasi- Abelia n. The ca teg ory TVS is not quasi-Ab elian (see [2 0]) b ecause quotients of co mplete top ological vector spaces need not b e co mplete. But the other additive categories int ro duced ab ov e are all quasi-Ab elian: Lemma 2.4. The c ate gories Ab , Fr , Bor , − − → Ban and ← − − Ban ar e qu asi-Ab elian and henc e b e c ome exact c ate gories when we let al l exten sions b e c onﬂations. Pr o of. F or the ca tegories Ab , Fr and Bor , w e merely describ e the inﬂa tions a nd deﬂations or, equiv alently , the strict mono - and epimor phisms and leave it as a n exercise to verify the axioms. The inﬂations and deﬂations in Ab are simply the injectiv e and surjective group homomorphisms. Let f : V → W b e a contin uous linear map betw een tw o F r´ ec het spac e s. It is an inﬂation if f is a homeomorphism fro m V onto f ( V ) with the subspace top olog y; by the Closed Graph Theorem, this holds if and only if f is injectiv e and its range is clos ed. The map f is a deﬂation if a nd o nly if it is an op en s urjection; by the Closed Graph Theorem, this holds if a nd only if f is s urjective. Let f : V → W b e a b ounded linear map b etw e e n tw o b ornolo gical vector spaces. It is an inﬂation if and only if f is a bor nological isomorphism ont o f ( V ) with the subspace b or nology; equiv alently , a subset of V is b ounded if and only if its f -image is b ounded. It is a deﬂation if and o nly if f is a b ornolog ical quotient map, that is, any b ounded subset o f W is the f -imag e o f a bo unded subset of V . The ca tegory of pro jective sys tems ov er a q ua si-Ab elian ca teg ory is again q uasi- Abelia n by [19, Prop osition 7.1.5]. Since opp o site categorie s of quasi-Ab elian cate- gories ar e aga in quasi-Ab elian, the same holds for ca tegories o f inductive systems by duality . Since the category of Ba na ch spaces is quasi-Ab elian (it is a sub ca teg ory of the qua si-Ab elian categ ory Fr closed under sub ob jects, quotien ts, and conﬂa tio ns), we conclude that − − → Ban and ← − − Ban are quasi-Ab elian. 8 RALF MEYE R F urthermore, we ca n describ e the inﬂatio ns and deﬂa tions as follows (see [19, Corollar y 7 .1.4]). A morphism f : X → Y of inductiv e systems of Banach space s is an inﬂatio n (or deﬂatio n) if and only if there a re inductive systems ( X ′ i ) i ∈ I and ( Y ′ i ) i ∈ I of Ba nach spaces a nd inﬂations (or deﬂatio ns) f ′ i : X ′ i → Y ′ i for all i ∈ I that form a morphism of inductive sys tems, and isomo rphisms of inductiv e systems X ∼ = ( X ′ i ) and Y ∼ = ( Y ′ i ) that intert wine f a nd ( f ′ i ) i ∈ I . A dual statement holds for pro jective sy s tems. The same ar gument shows that a diagr a m I i − → E p − → Q in − − → Ban is a n extensio n if and only if it is the colimit of an inductive system of ex tensions of Banach spa ces.  Deﬁnition 2. 5. A functor F : C 1 → C 2 betw een tw o exact catego ries is ful ly exact if a diagr am I → E → Q is a co nﬂation in C 1 if and o nly if F ( I ) → F ( E ) → F ( Q ) is a conﬂation in C 2 . The functor s Cpt : Fr → Bor , diss : Bor → − − → Ban , and diss ∗ : F r → ← − − Ban ar e fully ex a ct. Thu s diss ◦ Cpt : Fr → − − → Ban and diss ∗ : F r → ← − − Ban are fully exact, symmetric monoidal, and fully faithful, that is, they pres erve all extra structure on our categories. The following exact category structure s a re useful in connection with nuclearit y: Deﬁnition 2.6. An extension I i − → E p − → Q in − − → Ban is ca lled lo c al ly split if the induced sequence Hom( V , I ) → Hom( V , E ) → Hom( V , Q ) is an extension (o f vector spaces ) for ea ch Banach space V (here we vie w V as a constant inductive sytem). An extension I i − → E p − → Q in ← − − Ban is called lo c al ly split if the induced seq uence Hom( I , V ) → Hom( E , V ) → Hom( Q, V ) is an extension (of vector spaces) for each Banach space V . It is r o utine to verify that ana logous deﬁnitions yield exact category str uctures for inductiv e a nd pro jective s ystems o ver any exact category . By restriction to the full sub category Fr , we a lso get new exact categor y s tructures on F r´ echet spaces. Deﬁnition 2.7. An extens io n I i − → E p − → Q of F r´ echet spaces is ind-lo c al ly split if any compact linear map V → Q for a Bana ch spa ce V lifts to a contin uous linear map V → E (then the lifting can b e chosen to b e compact as well). The extension is called pr o-lo c al ly split if any co nt inuous linear map I → V for a Ba nach spa c e V extends to a contin uo us linear map E → V . It is easy to chec k that an extensio n of F r´ echet s paces is ind-lo cally split if and only if diss ◦ Cpt maps it to a lo ca lly split extensio n in − − → Ban , and pro-lo ca lly split if and only if diss ∗ maps it to a lo cally split ex tension in ← − − Ban . 2.3. Exact c hain complexes, quasi-is omorphism s , and homo logy. All the basic to ols of homological algebra work in exact catego ries in the sa me w ay as in Ab elian categories . This includes the construction of a derived ca tegory (se e [13, 1 8]). T o keep this article easier to read, we only use a limited set of to ols , how ever. EXCISION IN HOCHSCHILD AND CYCLIC HOM OLOGY 9 Deﬁnition 2.8. A chain c omplex ( C n , d n : C n → C n − 1 ) in an e x act ca teg ory ( C , E ) is called exact (or E -exact) in deg ree n if ker d n exists and d n +1 induces a deﬂa- tion C n +1 d n +1 − − − → ker d n . E quiv alently , ker d n and ker d n +1 exist and the sequence ker d n +1 ⊆ − → C n +1 d n +1 − − − → ker d n is a conﬂation. In a ll our examples, the existence o f kernels and cokernels comes for free. In Ab , exactness just means im d n +1 = ker d n . By the Op en Mapping Theorem, the same happ ens in Fr . But exa ctness of c hain c o mplexes in Bo r , − − → Ban and ← − − Ban requires more than this set theore tic condition. Example 2 .9 . Let C b e a n additiv e categor y in which all idemp otent mor phisms hav e a r ange ob ject (this fo llows if a ll mo rphisms in C hav e kernels). Then a chain complex in C is E ⊕ -exact if and only if it is co nt ractible. Example 2.1 0 . Call a c hain complex in − − → Ban o r ← − − Ban lo cally split exac t if it is e x act with resp ect to the exact category structures deﬁned in Deﬁnition 2.6. A chain complex C • in − − → Ban is locally split exa ct if a nd o nly if Hom( V , C • ) is exact for eac h Banach spa ce V . Dually , a chain complex C • in ← − − Ban is lo cally s plit exact if a nd only if Hom( C • , V ) is exact for each B a nach space V . An y symmetric mono idal category C car ries a c anonic al for getful functor to the category of Abe lian gro ups, V 7→ [ V ] : = Hom( 1 , V ) , where 1 denotes the tensor unit. Another dual forg etful functor is deﬁned by [ V ] ∗ : = Hom( V , 1 ). Example 2.11 . The forgetful functor a cts iden tically o n Ab . On Fr a nd Bor , it yields the underlying Ab elian gr oup of a F r´ echet space or a bo rnologic al vector space. The forgetful functors on − − → Ban a nd ← − − Ban ma p inductive a nd pro jective systems of Banach spaces to their inductive and pr o jective limits, resp ectively . The for getful functor and its dual allow us to deﬁne the ho mology and cohomol- ogy of a chain complex in C . Let H ∗ ( C • ) a nd H ∗ ( C • ) for a chain complex C • be the homology of the chain complex [ C • ] and the coho mo logy of the co chain complex [ C • ] ∗ , resp ectively . This yields the usua l deﬁnition of homo lo gy and c ontinuous cohomolog y for chain complexes of F r´ echet spaces. While a chain co mplex of F r´ echet spaces is exact if and only if its ho mo logy v anishes, this fails for chain complexe s of b o r nologica l vector spaces or for chain complexes in − − → Ban and ← − − Ban ; there are even exa c t chain complexes in ← − − Ban with non-zero ho mology . F or F r´ echet s pa ces, exactness b ecomes s tronger tha n v anishing of homolo gy if we use other exact category structures like those in Deﬁnition 2.7. This is remedied by the r eﬁne d homolo gy H ∗ ( C • ) for chain co mplexes in C . Deﬁnition 2.12. Recall that any exact categ ory C can b e realised as a full, fully exact sub categ o ry of an Abelia n category (even in a canonica l wa y). W e let A b e such a n Ab elian catego r y containing C , and we let H n ( C • ) for a chain complex in C be its n th homology in the ambien t Abelian category A . This reﬁned homolog y is us e ful to extend familiar results a nd deﬁnitions from homologica l algebr a to c hain complexes in exact catego ries. By design H n ( C • ) = 0 if and only if C • is exact in degree n . 10 RALF MEYE R W e now compare the reﬁned homology with the usual ho mology for chain com- plexes of Abelia n groups and F r´ echet spa ces. F or chain complexes of Abe lia n gro ups, bo th agr ee b ecause A b is already Ab elian, so that H n ( C • ) is the usual homology of a c hain complex of Ab elian gr oups. F or a chain complex of F r´ echet spa ces, let H Fr n ( C • ) b e its r e duc e d homolo gy : the quotient o f ker d n by the closur e of d n +1 ( C n +1 ) with the quotient to po logy . This is the F r´ echet spac e that comes clo sest to the homology H n ( C • ). Ass ume that the bo undary map d • has closed range. Then it is automa tically op en as a map to im d n +1 with the subspa ce top ology fr om C n ; the map ker d n → H Fr n ( C • ) is open, anyw ay . Thus ker( d n +1 ) ֌ C n +1 ։ im d n +1 and im d n +1 ֌ ker d n ։ H Fr n ( C • ) are conﬂations of F r´ echet spaces. Since the embedding of Fr int o the a mb ient Ab elian category A is exact, these remain extensio ns in A . Hence H n ( C • ) ∼ = H Fr n ( C • ) if d • has close d range. In g eneral, H n ( C • ) is the co kernel in the Abelia n catego ry A of the map C n +1 → ker d n induced by d n +1 . Deﬁnition 2.13. A quasi-isomorphism b e t ween tw o ch ain complexes in an exact category is a chain map with an exact mapping cone. In an Ab elian category such as Ab , quasi-isomo rphisms are chain maps that induce an isomor phism on homo logy . As a conse quence, a chain map is a quasi- isomorphism if and only if it induces an isomorphism on the r eﬁne d homology . Lemma 2.14. A chain map b etwe en two chain c omplexes of F r´ echet sp ac es is a quasi-isomorphi sm with r esp e ct to the class of al l extensions if and only if it induc es an isomorphism on homolo gy. Pr o of. The mapping cone of a chain map f is again a chain co mplex of F r´ echet spaces and hence is e x act if and o nly if its homolo gy v anishes. By the P uppe long exact sequence, the latter homolog y v anishes if and only if f induces an isomorphism on homology .  Quasi-iso morphisms in Bor , − − → Ban , or ← − − Ban are more complicated to des crib e. T o prov e the excision theo rem, we mu st show that certain chain maps are quasi- isomorphisms. The arg ument s in [8] use homo logy to detect quasi- isomorphisms and, w ith our reﬁned notion of homology , carry over literally to a n y exact symmet- ric mo noidal categor y . But, in fact, w e do no t need this sophisticated notion o f homology b ecause we only need quasi-isomo rphisms of the follo wing simple type: Lemma 2. 15. L et I • i ֌ E • p ։ Q • b e a c onﬂ ation of chain c omplexes in C , that is, the maps i and p ar e chain maps and r estrict to c onﬂations I n ֌ E n ։ Q n for al l n ∈ Z . If I • is exact, t hen p is a qu asi-isomorph ism. If Q • is exact, t hen i is a quasi-isomorphi sm. Pr o of. Our conﬂatio n of chain complexes yields a long ex act homolog y seque nc e for reﬁned homology b ecause this works in Ab elian ca tegories. By exactness, the ma p i induces an isomorphism on r eﬁned homo logy if a nd only if the reﬁned homolo g y of Q • v anishes. That is, the map i is a q ua si-isomor phism if a nd only if Q • is exact. A similar ar gument shows that p is a quasi-isomo rphism if and only if I • is exact.  Besides Lemma 2.15, we also need to know that a co mpo site of t wo q uasi-iso- morphisms is again a quasi-isomo r phism – this follows b ecause reﬁned homolo gy is EXCISION IN HOCHSCHILD AND CYCLIC HOM OLOGY 11 a functor. This toge ther with L e mma 2 .15 suﬃces to verify the quas i-isomorphis ms we need. Finally , we need a suﬃcien t condition for long exa ct homology sequences. Let (3) A f − → B g − → C be chain maps betw een c hain complexes in C with g ◦ f = 0. Then we get an induced chain map fr o m the mapping co ne of f to C . W e call (3) a c oﬁbr e se quenc e if this map cone( f ) → C is a quasi-isomor phism. The Pupp e exact se quenc e provides a long exact sequence rela ting the re ﬁned homolo gies of A , B and co ne( f ). F or a coﬁbre sequence, we may iden tify the reﬁned homolo gy of C with that of cone( f ) and th us get a natural long ex act sequence · · · → H n ( A ) f ∗ − → H n ( B ) g ∗ − → H n ( C ) → H n − 1 ( A ) f ∗ − → H n − 1 ( B ) → · · · . W e get a co rresp onding long exact sequence for the unreﬁned homology pro- vided 1 is a pro jective ob ject, that is, the canonical for getful functor is exact. This is the case for Ab , Fr , Bor , and − − → Ban , but not for ← − − Ban b ecause pr o jective limits are not exa ct. Simila rly , we ge t a long coho mology exact se quence if 1 is injec- tive. This is the case for Ab , Fr , and ← − − Ban , but not for Bor and − − → Ban . There is no long exact cohomolo gy seq uence for arbitra ry co ﬁbre sequences in Bor b ecause the Hahn–Banach Theorem fails for b ornolog ical vector spaces. The dual forg etful functor on − − → Ban is not exa c t b ecause it involv es pro jectiv e limits. 2.4. Ho c hsc hild hom ology and cohomology. Let C b e a sy mmetr ic mono idal category . Let A b e an algebra in C , possibly without unit. W e ﬁrst deﬁne the Ho chsc hild homolog y and cohomolog y of A with co eﬃcient s in an A -bimo dule M . Then w e deﬁne the Ho chschild homology and cohomo logy of A without coeﬃcients. The Ho chschild homolo gy HH ∗ ( A, M ) of A with co eﬃcients M is the homolo gy of the chain complex HH ∗ ( A, M ) = ( M ⊗ A ⊗ n , b ) =  · · · → M ⊗ A ⊗ 2 b − → M ⊗ A b − → M → 0 → · · ·  in C , where b is deﬁned by categorifying the usual formula b ( x 0 ⊗ x 1 ⊗ · · · ⊗ x n ) : = n − 1 X j =0 ( − 1) j x 0 ⊗ · · · ⊗ x j x j +1 ⊗ · · · ⊗ x n + ( − 1) n x n x 0 ⊗ x 1 ⊗ · · · ⊗ x n − 1 for x 0 ∈ M , x 1 , . . . , x n ∈ A . The formula in the j th summand co rresp onds to the map Id M ⊗ A ⊗ j − 2 ⊗ m ⊗ Id A ⊗ n − j , whe r e m : A ⊗ A → A is the mu ltiplication map; the zer o th summand is the mult iplication map m M A : M ⊗ A → M tensored with Id A ⊗ n − 1 , and the last summand inv o lves the multiplication map m AM : A ⊗ M → M and the cyclic rotatio n o f tensor factors M ⊗ A ⊗ n → A ⊗ M ⊗ A ⊗ n − 1 , which exists in symmetric monoidal categor ies. The Ho chschild c ohomolo gy HH ∗ ( A, M ) o f A with co eﬃcients in M is the coho- mology of the co chain co mplex HH ∗ ( A, M ) : = (Hom( A ⊗ n , M ) , b ∗ ) =  · · · → 0 → Hom( A ⊗ 0 , M ) b ∗ − → Hom( A ⊗ 1 , M ) b ∗ − → Hom( A ⊗ 2 , M ) → · · ·  , 12 RALF MEYE R where we interpret A ⊗ 0 : = 1 and deﬁne b ∗ by ca tegorifying the usual formula b ∗ f ( a 1 , . . . , a n ) : = a 1 f ( a 2 , . . . , a n ) + n − 1 X j =1 ( − 1) j f ( a 1 , . . . , a j a j +1 , . . . , a n ) + ( − 1) n f ( a 1 , . . . , a n − 1 ) a n . Ho chsc hild homology and cohomology ar e just Abelian groups. W e may also consider the reﬁned homo logy of HH ∗ ( A, M ). Our e x cision results initially dea l with this r eﬁned Ho chsc hild homo lo gy . This ca rries over to the unr e ﬁned theo ries if 1 is pro jective. If the symmetric monoidal categ ory C is closed, we may r eplace Hom by the in ternal Hom-functor to enrich HH ∗ ( A, M ) to a co chain complex in C . This provides a reﬁned version of Ho chsc hild cohomo logy . If M is a right A -mo dule, then we may tur n it in to an A -bimodule by declaring the left multiplication map A ⊗ M → M to b e the zero map. This has the eﬀect that the last summand in the map b v anishes, so that b reduces to the ma p usua lly denoted by b ′ . Hence assertions ab out ( M ⊗ A ⊗ n , b ) for bi modules M contain assertions a bo ut ( M ⊗ A ⊗ n , b ′ ) for right mo dules M as special cases. Similarly , w e may enrich a le ft A -mo dule M to a bimo dule using the zer o map M ⊗ A → M , and our assertions sp ecialise to as sertions ab out ( A ⊗ n ⊗ M , b ′ ) for left A -mo dules M . Now w e deﬁne the Hochsc hild homo logy and cohomology of A without co ef- ﬁcient s. Le t 1 b e the tenso r unit of C . If the algebra A is unital, we simply let HH ∗ ( A ) a nd HH ∗ ( A ) b e the ho mology and cohomolo g y of the chain complex HH ∗ ( A, A ). Thu s HH ∗ ( A ) ∼ = HH ∗ ( A, A ). If C is closed, then we may form a dual ob ject A ∗ : = Hom( A, 1 ) inside C , and HH ∗ ( A ) ∼ = HH ∗ ( A, A ∗ ). F or a non-unital alge bra, the deﬁnition in volv es the unita l a lgebra genera ted by A , which is A + : = A ⊕ 1 with the mu ltiplication where the co o rdinate em- bedding 1 → A + is a unit. W e le t HH ∗ ( A ) be the kernel of the augmentation map HH ∗ ( A, A + ) → 1 induced b y the co ordina te pro jection A + → 1 . That is, HH 0 ( A ) = A and HH n ( A ) = A + ⊗ A ⊗ n for n ≥ 1, with the bo undary map b ; this is the chain complex of no n- commutativ e diﬀerential forms ov er A with the usual Ho chsc hild boundar y on non-co mm utative diﬀere n tial for ms. W e let HH ∗ ( A ) and HH ∗ ( A ) be the homology and coho mo logy of H H ∗ ( A ). It is well-known that HH ∗ ( A ) a nd HH ∗ ( A, A ) a r e quas i- isomorphic for unital A – this is a special case of Corollar y 3.5 b elow. Besides HH ∗ ( A ) and HH ∗ ( A ), w e are also interested in the Ho chschild cohomo l- ogy HH ∗ ( A, A ), which plays a n impo rtant r o le in deformation quan tisation and which, in low dimensions, sp ecialises to the centre and the space of outer deriv a- tions. Cyclic homolog y a nd p erio dic c y clic homology can b e deﬁned for algebra s in C as well by car rying ov er the usual recip es (see also [5]). Since it is well-known a n ywa y that excision in Hochsc hild (co)ho mo logy implies excision in cyc lic and p erio dic cyclic (co)homology , w e do not rep eat these deﬁnitions her e. 2.5. Pure conﬂations. Deﬁnition 2. 16. A conﬂation I ֌ E ։ Q in C is called pur e if I ⊗ V → E ⊗ V → Q ⊗ V EXCISION IN HOCHSCHILD AND CYCLIC HOM OLOGY 13 is a co nﬂation for all ob jects V of C . A chain complex C • is called pur e exact if C • ⊗ V is exact for a ll ob jects V o f C . A chain map f is called a pur e quasi- isomorphi sm if f ⊗ Id V is a quasi-isomo r phism fo r a ll ob jects V of C . Deﬁnition 2.17. A functor F : C 1 → C 2 betw een exa ct ca tegories is called r ight exact if it preser ves deﬂations . Equiv a lent ly , if I i ֌ E p ։ Q is a conﬂa tion in C 1 , then ker  F ( p )  → F ( E ) F ( p ) − − − → F ( Q ) is a conﬂation in C 2 . F or instance, a functor Fr → Fr is right exact if and only if it maps open surjections aga in to op en surjections. The tensor pro duct functors in the quasi-Ab elian categ ories A b , Fr , Bor , − − → Ban , and ← − − Ban are rig ht exact in this sense in each v ariable. If the tensor pro duct is r ight exact, then an extens ion I • i ֌ E • p ։ Q • is pure if and o nly if the natural ma p I ⊗ V → ker( p ⊗ V ) is a n isomo rphism for all V . In the categor y Ab , this map is always surjective, so tha t only its injectivity is an is s ue. In the categories Fr and Bor , this map is usually injective – counterexamples ar e r elated to counterexamples to Grothendieck’s Approximation P rop erty – and its r ange is always dense, but it is usually not surjective. It is clear that split ex tens io ns are pure. W e are going to show that any extension of F r´ echet spaces with nuclear quotient is pure. T his is alr eady known, but we take this o pp or tunit y to give tw o new pr o ofs that use lo cally split extensio ns in − − → Ban and ← − − Ban , resp ectively . Deﬁnition 2.18. An inductive system of Bana ch spa ces ( V i , ϕ ij : V i → V j ) i ∈ I is called nucle ar if for each i ∈ I there is j ∈ I ≥ i for whic h the ma p ϕ ij : V i → V j is nu clear, that is, b elong s to the pro jective top olog ical tensor pro duct V ∗ i ˆ ⊗ π V j . A pr o jective sys tem o f Banach spaces ( V i , ϕ j i : V j → V i ) i ∈ I is called nucle ar if for each i ∈ I there is j ∈ I ≥ i for which the map ϕ j i : V j → V i is nuclear. A map X → Y betw een t wo inductive or pro jectiv e sys tems o f Ba na ch spaces is called n u cle ar if it factor s as X → V → W → Y for a n uclear map b etw een Banach spaces V → W . By deﬁnition, an inductiv e system X o f Banach spaces is n uclear if a nd only if each map from a Banach space to X is n uclear, and a pro jective system X of Banach spaces is nuclear if a nd o nly if e a ch map from X to a Banach s pace is nu clear. Almost by deﬁnition, a borno logical v ector spa c e V is n uclear if and o nly if diss( V ) is n uclear in − − → Ban , and a lo ca lly co n vex top olog ical vector s pace V is nuclear if and only if diss ∗ ( V ) is nuclear in ← − − Ban (see [11]). F urthermore, a F r´ echet spa ce V is nuclear if a nd o nly if C pt( V ) is n uclear, if and only if diss Cpt( V ) is nuclear, s ee [11, Theorem (7) on page 160 ]. Prop ositio n 2.19. Extensions in − − → Ban or ← − − Ban with nucle ar quotient ar e lo c al ly split. Pr o of. Let I ֌ E ։ Q b e an extension in − − → Ban with nuclear Q . Recall that we may write it as an inductive limit of extens io ns of Ba nach space s I α ֌ E α ։ Q α . Nuclearity o f Q means that for each α there is β ≥ α for which the map Q α → Q β is nuclear. Now we recall that nuclear maps b etw een B a nach spaces may b e lifted in extensions. That is, the map Q α → Q β lifts to a bo unded linear map s α : Q α → E β 14 RALF MEYE R for so me β ≥ α . No w let V be any Ba nach spa c e. The space of morphis ms from V to Q is lim − → Hom( V , Q α ), that is, any morphism V → Q factor s thro ugh a map f : V → Q α for so me α . Then s α ◦ f : V → E β lifts f to a morphism V → E . Th us our extension in − − → Ban is lo cally split if Q is n uclear. Similarly , an extension in ← − − Ban is the limit of a pro jective system of extensio ns of Banach spaces I α ֌ E α ։ Q α . Nuclea rity of Q means that for each α there is β ≥ α for whic h the map Q β → Q α is nuclear. As ab ov e, this a llows us to lift it to a map Q β → E α . Subtracting this map from the canonical map E β → E α yields a map E β → I α that extends the cano nical map I β → I α . As above, this shows that a n y map I → V for a Ba nach space V extends to a map E → V , using that it factors through I α for some α .  Prop ositio n 2. 20. L o c al ly split extensions in − − → Ban or ← − − Ban ar e pur e lo c al ly split: if I ֌ E ։ Q is lo c al ly split, then I ⊗ V ֌ E ⊗ V ։ Q ⊗ V is a lo c al ly split extension as wel l and a for tiori an ext ension. Pr o of. First we claim that a loca lly s plit extension in − − → Ban or ← − − Ban may be wr itten as an inductive or pr o jective limit of extensio ns that a re split, but usually with incompatible se c tions, so that the limit extension do es not split. W e only wr ite this down for inductiv e systems, the cas e of pro jective sy s tems is dual. An analo g ous argument works for lo cally split extens ions of pr o jective or inductive sys tems ov er any additive category . W rite a lo cally split extension a s an inductiv e limit of extensio ns of Banach spaces I α ֌ E α ։ Q α . F or each α , the canonical map Q α → Q lifts to a map Q α → E , which is represented by a map s α : Q α → E β for some β ≥ α . F or each such pair of indices ( α, β ), w e may pull back the extension I β ֌ E β ։ Q β along the map Q α → Q β to an extension I β ֌ E ′ β ,α ։ Q α . The lifting s α induces a section Q α → E ′ β ,α for this pulled ba ck extension. The pairs ( β , α ) ab ov e form a directed set and the split extensions I β ֌ E ′ β ,α ։ Q α form an inductive system of extensions indexe d by this set; its inductive limit is the given extension I ֌ E ։ Q . Now we pr ov e the pur it y assertion. W r ite the extensio n I ֌ E ։ Q as an inductive sys tem of s plit extensions o f Banach spaces I α ֌ E α ։ Q α . Let V b e another ob ject of − − → Ban . The tenso r pro duct in − − → Ban commutes with inductive limits, so that I ⊗ V ∼ = lim − → I α ⊗ V , and so on. Since the extensions I α ֌ E α ։ Q α split, so do the ex tensions I α ⊗ V ֌ E α ⊗ V ։ Q α ⊗ V . This implies that I ⊗ V → E ⊗ V → Q ⊗ V is a lo ca lly split extension in − − → Ban .  Theorem 2.2 1 . L et I ֌ K ։ Q b e an extension of F r´ echet sp ac es. If Q is nucle ar, then I ֌ E ։ Q is b oth pur e ind-lo c al ly split and pur e pr o-lo c al ly split. Pr o of. W e only write down wh y I ⊗ V → E ⊗ V → Q ⊗ V is pro-lo cally split for any F r´ echet space V . A similar ar gument y ie lds that it is ind-lo cally s plit. If Q is a nuclear F r´ echet space, then diss ∗ ( Q ) is nuclear in ← − − Ban . Since the functor diss ∗ is fully exa ct, it maps I ֌ E ։ Q to an extension in ← − − Ban . Prop osition 2 .1 9 asserts that this ex tension is lo ca lly split. Since diss ∗ is symmetric mo noidal, it maps the diagram I ⊗ V → E ⊗ V → Q ⊗ V to diss ∗ ( I ) ⊗ diss ∗ ( V ) → diss ∗ ( I ) ⊗ diss ∗ ( V ) → diss ∗ ( I ) ⊗ diss ∗ ( V ) . This is a lo cally split ex tens io n by P rop osition 2.20. Hence the orig inal diagram was a pro-lo c a lly split extension.  EXCISION IN HOCHSCHILD AND CYCLIC HOM OLOGY 15 3. Excision in H ochschild homol ogy W e ﬁx a symmetric monoida l category C with a tensor pro duct ⊗ and an exa ct category structure E . Deﬁnition 3.1. Let A b e an alg ebra in C . W e call A homolo gic al ly u nital , brieﬂy H-unital, if the chain complex ( A ⊗ n , b ′ ) n ≥ 1 is pure exact. Let A b e an alge bra in C and let M b e a right A -mo dule. W e ca ll M homolo gic al ly unitary , brieﬂy H-unitary , if the chain complex ( M ⊗ A ⊗ n , b ′ ) n ≥ 0 is pure ex a ct. A similar deﬁnition applies to left A -mo dules. By deﬁnition, the algebra A is homologically unital if and only if it is homolog i- cally unitary when viewed as a left or right mo dule over itself. Recall that a ch ain co mplex is exact if and only if its reﬁned homo logy v anis hes. Therefore, M is homologically unitary if a nd only if H n  HH ∗ ( A, M ⊗ V )  = 0 for all V . If C is an Ab elian catego ry or the ca tegory of F r´ echet s pa ces with all extensions as conﬂa tio ns, then a chain complex is exact if a nd o nly if its homolog y v anishes. In this ca se, M is homolog ically unitar y if and only if HH ∗ ( A, M ⊗ V ) = 0 for all V . In g eneral, H-unitarity is unrelated to the v anishing of HH ∗ ( A, M ⊗ V ). R emark 3.2 . Let M be a left A -mo dule. If the chain complex ( A ⊗ n ⊗ M , b ′ ) n ≥ 0 is exact in dimensions zero and o ne, then the natural map A ⊗ A M → M induced by the mo dule structur e A ⊗ M → A is an isomorphis m. If the map A ⊗ A A → A is inv ertible, then A is called self-induc e d in [17]; if A ⊗ A M → M is an isomor phism, then the A -module M is called smo oth . As a result, H-unital algebra s ar e self- induced and H-unitary modules over self-induced algebr as are smooth in the sense of [17]. Lemma 3. 3. L et I ֌ E ։ Q b e an algebr a c onﬂation and let M b e a homolo gi- c al ly unitary I -mo dule. Then the I -m o dule structu r e on M extends uniquely to an E -mo dule stru ctur e on M . Pr o of. W e write down the proo f for right mo dules ; similar ar guments work fo r left mo dules a nd bimo dules. Since M is H-unitary , we g e t exact chain complexes ( M ⊗ I ⊗ n , b ′ ) and ( M ⊗ I ⊗ n , b ′ ) ⊗ E . The maps M ⊗ I ⊗ n ⊗ E → M ⊗ I ⊗ n induced by the multiplication map I ⊗ E → I provide a chain map b etw ee n these chain complexes above deg ree 0, that is, we get a commu ting diagra m · · · / / M ⊗ I ⊗ I ⊗ E / /   M ⊗ I ⊗ E / /   M ⊗ E / / 0 · · · / / M ⊗ I ⊗ I / / M ⊗ I / / M / / 0 . A r ight E -mo dule structure M ⊗ E → M on M ex tending the given I -mo dule structure w ould complete this comm uting diagram to a chain map. Since the rows are exact, there is a unique such completion. This deﬁnes an E -mo dule str uc tur e on M : asso ciativity follows fro m the uniqueness o f co mpleting ano ther diag ram inv olv ing maps M ⊗ I ⊗ n ⊗ E ⊗ 2 → M ⊗ I ⊗ n .  Theorem 3. 4 . L et I ֌ E ։ Q b e a pur e algebr a c onﬂation, let M b e an E , I -bimo dule. Assume t hat M is homol o gic al ly unitary as a right I -mo dule and view M as an E -bimo dule. Then t he c anonic al map HH ∗ ( I , M ) → HH ∗ ( E , M ) is a pur e quasi-isomorphism. Thus HH ∗ ( I , M ⊗ V ) ∼ = HH ∗ ( E , M ⊗ V ) for any obje ct V of C pr ovide d 1 is pr oje ctive. 16 RALF MEYE R Pr o of. This theo rem is an analo gue of [8, Theore m 2] and is prov ed b y exactly the same argument. F o r p ∈ N , let F p be the chain co mplex · · · ← 0 ← M ⊗ V b ← − M ⊗ E ⊗ V b ← − M ⊗ E ⊗ E ⊗ V ← · · · ← M ⊗ E ⊗ p ⊗ V b ← − M ⊗ I ⊗ E ⊗ p ⊗ V b ← − M ⊗ I ⊗ I ⊗ E ⊗ p ⊗ V b ← − M ⊗ I ⊗ 3 ⊗ E ⊗ p ⊗ V ← · · · with M ⊗ V in deg ree 0. Since the c o nﬂation I ֌ E ։ Q is pure, the map M ⊗ I ⊗ k ⊗ E ⊗ p ⊗ V → M ⊗ I ⊗ k − 1 ⊗ E ⊗ p +1 ⊗ V is a n inﬂation for all k , p ≥ 0. Hence the c a nonical map F p → F p +1 is an inﬂation for each p . Its co kernel is the chain complex F p +1 /F p ∼ = ( M ⊗ I ⊗ k , b ′ ) k ≥ 0 [ p + 1] ⊗ Q ⊗ E ⊗ p ⊗ V , where [ p + 1] denotes translation by p + 1. This chain complex is exa ct bec ause M is homologica lly unitary as a right I -mo dule. Since F p ֌ F p +1 ։ F p +1 /F p is a pure conﬂation o f chain complexes , Lemma 2 .15 shows that the map F p → F p +1 is a pure quasi-isomo rphism. Hence so are the embeddings F 0 → F p for all p ∈ N . F or p = 0, we get F 0 = HH ∗ ( I , M ) ⊗ V . In any ﬁxed degree n , we hav e ( F p ) n = HH n ( E , M ) ⊗ V for p ≥ n . Hence the cano nical map HH ∗ ( I , M ) → HH ∗ ( E , M ) is a pure quasi- isomorphism.  Corollary 3 . 5. L et I ֌ E ։ Q b e a pur e alge br a c onﬂation. If I is homolo gic al ly unital, then the c anonic al maps HH ∗ ( I , I ) / / HH ∗ ( E , I ) ( I ⊗ n +1 , b ) / / ( I ⊗ E ⊗ n , b ) HH ∗ ( I , I ) / / HH ∗ ( I ) ( I ⊗ n +1 , b ) / / (Ω n ( I ) , b ) ar e pu re qu asi-isomorph isms. If E is unital, then the unital ext ension of t he em- b e dding I → E induc es a pur e quasi-isomo rphism HH ∗ ( I ) → HH ∗ ( E , I ) . Thus HH ∗ ( I ) → HH ∗ ( E , I ) is invertible pr ovide d 1 is pr oje ctive. Recall that Ω n ( I ) = I + ⊗ I ⊗ n for n ≥ 1 and Ω 0 ( I ) = I . Pr o of. The pure qua si-isomor phism HH ∗ ( I , I ) ∼ HH ∗ ( E , I ) follows from Theo - rem 3.4 bec ause I is homolo gically unital if and o nly if it is homologica lly unitary as a rig h t mo dule over itself. The split extensio n I ֌ I + ։ 1 of mo dules induces a canonical split extension of chain complexes HH ∗ ( I , I ) ֌ HH ∗ ( I ) ։ ( I ⊗ n , b ′ ) n ≥ 1 [1] . Since I is homo logically unital, the chain complex ( I ⊗ n , b ′ ) is pure exact. Hence the map HH ∗ ( I , I ) ֌ HH ∗ ( I ) is a pur e quasi-iso morphism by Lemma 2.15.  Theorem 3.6. L et I ֌ E ։ Q b e a pure algebr a c onﬂation, let M b e a Q - bimo dule. Then we may view M as an E -bimo dule. If I is homolo gic al ly un ital, then the c anonic al map HH ∗ ( E , M ) → HH ∗ ( Q, M ) is a pur e quasi-isomorphism. Thus HH ∗ ( E , M ⊗ V ) ∼ = HH ∗ ( Q, M ⊗ V ) pr ovide d 1 is pr oje ctive. EXCISION IN HOCHSCHILD AND CYCLIC HOM OLOGY 17 Pr o of. This is the analogue of [8, Corollary 3 ]. Let ˜ F p for p ≥ 0 be the c hain complex · · · ← 0 ← M ⊗ V b ← − M ⊗ Q ⊗ V b ← − M ⊗ Q ⊗ Q ⊗ V ← · · · b ← − M ⊗ Q ⊗ p ⊗ V b ← − M ⊗ Q ⊗ p ⊗ E ⊗ V b ← − M ⊗ Q ⊗ p ⊗ E ⊗ E ⊗ V ← · · · with M ⊗ V in deg ree 0. One summand in b uses the obvious rig ht E -mo dule structure E ⊗ Q → Q on Q . Since the conﬂa tion I ֌ E ։ Q is pure, the ma p ˜ F p → ˜ F p +1 induced by the deﬂation E ։ Q is a deﬂa tion for each p ∈ N . Its kernel is ker( ˜ F p → ˜ F p +1 ) ∼ = M ⊗ Q ⊗ p ⊗ ( I ⊗ E ⊗ k , b ′ ) k ≥ 0 [ p + 1] ⊗ V . Since I is homologically unital, Theorem 3.4 implies that ( I ⊗ E ⊗ k , b ′ ) is pure exact. Hence the map ˜ F p → ˜ F p +1 is a pure qua si-isomor phism by Le mma 2.15. Hence so is the ma p ˜ F 0 → ˜ F p for a ny p ∈ N . This yie lds the as sertion b ecause ˜ F 0 = HH ∗ ( E , M ) ⊗ V and ( ˜ F p ) n = HH n ( Q, M ) ⊗ V in degr ee n for p ≥ n .  Theorem 3.7. L et I ֌ E ։ Q b e a pur e c onﬂation of algebr as in C and as- sume that I is homolo gic al ly unital. L et M I ֌ M E ։ M Q b e a pur e c onﬂa- tion of E - mo dules. Assume that the E -mo dule st ructur e on M Q desc ends to a Q -mo dule structur e and t hat M I is homolo gic al ly unitary as an I -mo dule. Then HH ∗ ( I , M I ) → HH ∗ ( E , M E ) → HH ∗ ( Q, M Q ) is a pur e c oﬁbr e s e quenc e. If 1 is pr oje ctive, t hen this yields a natura l long exact se quenc e · · · → HH n ( I , M I ) → HH n ( E , M E ) → HH n ( Q, M Q ) → HH n − 1 ( I , M I ) → HH n − 1 ( E , M E ) → HH n − 1 ( Q, M Q ) → · · · . Pr o of. The canonical map HH ∗ ( I , M I ) → H H ∗ ( E , M I ) is a pure quasi- isomorphism by Theore m 3.4 beca use M I is homolog ically unitar y as an I -mo dule. The canoni- cal map HH ∗ ( E , M Q ) → HH ∗ ( Q, M Q ) is a pure qua s i-isomorphis m by Theorem 3.6 bec ause I is homolo gically unital. The sequence HH ∗ ( E , M I ) ֌ HH ∗ ( E , M E ) → HH ∗ ( E , M Q ) is a pure conﬂation of c hain co mplexes a nd thus a pur e coﬁbr e se- quence b ecause the c o nﬂation M I ֌ M E ։ M Q is pure . Hence HH ∗ ( I , M I ) → HH ∗ ( E , M E ) → HH ∗ ( Q, M Q ) is a pure coﬁbr e sequence as well. If 1 is pr o jective, that is, the canonical forgetful functor ma ps conﬂations to exact sequences , then this coﬁbre sequence implies a long exa ct homology sequence.  Theorem 3. 8. L et I ֌ E ։ Q b e a pur e c onﬂation of alge br as in C . Assume that I is homolo gic al ly u nital. Then HH ∗ ( I ) → HH ∗ ( E ) → HH ∗ ( Q ) is a pur e c oﬁbr e se quenc e. If 1 is pr oje ctive in C , this yields a natur al long exact se quenc e · · · → HH n ( I ) → HH n ( E ) → HH n ( Q ) → HH n − 1 ( I ) → HH n − 1 ( E ) → · · · . If 1 is inje ctive in C , then ther e is a natur al long exact se qu enc e · · · → HH n ( I ) → HH n ( E ) → HH n ( Q ) → HH n +1 ( I ) → HH n +1 ( E ) → · · · . Pr o of. Let E + and Q + be the a lgebras obtained from E and Q by adjo ining unit elements. The algebra conﬂation I ֌ E + ։ Q + is also a co nﬂation of modules , and it is still pure b ecause it is the direct sum of the pure conﬂa tio n I ֌ E ։ Q and the split extension 0 → 1 = − → 1 . Hence Theorem 3.7 applies and yields a pure coﬁbre seque nc e HH ∗ ( I , I ) → HH ∗ ( E , E + ) → HH ∗ ( Q, Q + ). By deﬁnition, HH ∗ ( A ) ⊕ 1 = HH ∗ ( A, A + ) for A ∈ { E , Q } . Cancelling tw o co pie s of 1 , w e get 18 RALF MEYE R a pur e coﬁbre se q uence HH ∗ ( I , I ) → HH ∗ ( E ) → HH ∗ ( Q ). Finally , Corollary 3.5 yields a pure q ua si-isomor phism HH ∗ ( I , I ) → HH ∗ ( I ), so that w e get a pure coﬁbr e sequence HH ∗ ( I ) → HH ∗ ( E ) → HH ∗ ( Q ). The pro jectivit y o r injectivity of 1 ensures that we preser ve the co ﬁbre seq uence when we apply the cano nical forgetful functor or the dual space functor . Finally , this coﬁbre sequence of (co )ch ain complexes yields the asserted long exa ct sequences in Hochschild ho mo logy and cohomology .  Theorem 3.8 is o ur abstrac t E xcision Theorem for Ho chsc hild homo lo gy and cohomolog y . W e can sp ecialise it to v arious exact symmetric mono idal categories. F or the Abelia n categ ory Ab , we get W o dzicki’s origina l Excision Theorem for pure ring extensio ns with H-unital kernel. O ur notions of pur ity and H-unitality are the familiar one s in this case. The dual space functor is not ex act, s o that we do not get asser tio ns in cohomolog y . F or the Abelian ca tegory o f vector spaces o ver s ome ﬁeld instead o f A b , any extension is pure and the dual space functor is ex a ct. Hence Hochschild homolog y and coho mo logy satisfy excis ion for all extensions with homolog ically unital kernel, and the latter means simply that ( I ⊗ n , b ′ ) is exact. Now consider the quasi- Abelia n categ ory o f F r´ echet s paces (with all extensions as conﬂations ). Purity mea ns that I ˆ ⊗ π V → E ˆ ⊗ π V → Q ˆ ⊗ π V is a n extension of F r´ echet spa c es or , equiv alently , an extensio n o f vector s paces, for each F r´ echet space V . This is automatic if Q is nu clear by Theor em 2.21. F urther mo re, split extensions a re pure for tr ivial rea sons. The dual space functor is exact by the Hahn– Banach Theorem, so that w e get excision results b oth for Ho chsc hild homo logy and cohomolog y . H-unitality o f I mea ns that the c hain co mplex ( I ˆ ⊗ π n , b ′ ) n ≥ 1 ˆ ⊗ π V is exa ct for ea ch F r´ echet s pace V , and exa ctness is equiv alent to the v anis hing of homology . F urthermore, Theorem 2.21 shows that a nuclear F r´ echet algebr a I is homologica lly unital if and only if the homolo gy of the c hain complex ( I ˆ ⊗ π n , b ′ ) v anishes. Let C be an additive symmetric mo noidal ca teg ory in which all idemp otent mor - phisms split. T urn C into an exact category using only the split extensio ns E ⊕ . Then any ob ject of C is b oth pro jective and injective, and any co nﬂa tion is pure b ecause ⊗ is additive. H-unitalit y means that the chain complex ( I ⊗ n , b ′ ) is contractible. Thus the Excisio n Theorem applies to a split extensio n provided ( I ⊗ n , b ′ ) is contractible. The conclusion is that the map HH ∗ ( I ) → co ne  HH ∗ ( E ) → HH ∗ ( Q )  is a c hain homotopy equiv a lence. In the application to Whitney functions, we would like to compute HH ∗ ( Q, Q ) by homolog ical computations with E - mo dules . This is only p os sible under a n ad- ditional injectivit y as s umption: Theorem 3.9. L et I ֌ E ։ Q b e a pur e algebr a c onﬂation, let M b e a Q -bimo dule, which we also view as an E - bimo dule. Assume t hat I is homolo gic al ly unital and that M is inje ctive as an obje ct of C . Then the c anonic al map HH ∗ ( E , M ) → HH ∗ ( Q, M ) is a quasi-isomorphism, so t hat HH ∗ ( E , M ) ∼ = HH ∗ ( Q, M ) . Pr o of. Let ˜ F 0 for p ≥ 0 b e the co chain co mplex Hom( 1 , M ) b ∗ − → Hom( Q, M ) b − → Hom( Q ⊗ Q, M ) → · · · b ∗ − → Hom( Q ⊗ p , M ) b ∗ − → Hom( Q ⊗ p ⊗ E , M ) b ∗ − → Hom( Q ⊗ p ⊗ E ⊗ E , M ) b ∗ − → Hom( Q ⊗ p ⊗ E ⊗ 3 , M ) → · · · EXCISION IN HOCHSCHILD AND CYCLIC HOM OLOGY 19 where b ∗ is the Ho chsc hild cob oundar y map tha t uses the bimo dule str uc tur e on M and the obvious right E - mo dule structure Q ⊗ E → Q on Q . Since our a lgebra conﬂation is pure a nd M is injective as an ob ject of C , we g e t an exact s e q uence o f chain complexe s (4) ˜ F p ֌ ˜ F p +1 ։  Hom( Q ⊗ p ⊗ I ⊗ E ⊗ n [ p + 1] , M ) , ( b ′ ) ∗  . Theorem 3.4 implies that the chain complex V ⊗ ( I ⊗ E ⊗ k , b ′ ) is exac t for any V bec ause I is homo lo gically unital. Since M is injective, the quotient complex in (4) is exact. Hence the map ˜ F p → ˜ F p +1 is a quasi-isomo r phism b y Lemma 2.15. Then so is the map ˜ F 0 → ˜ F p for any p ∈ N . This y ie lds the ass ertion bec a use ˜ F 0 = HH ∗ ( E , M ) and ˜ F n p = HH n ( Q, M ) for p ≥ n .  Since there a re few injective F r´ echet spaces, this theorem rarely applies to the category of F r´ echet spaces with all extensions as conﬂations . One exa mple of an injectiv e n uclear F r´ echet space is Q n ∈ N C , the spa ce of Whitney functions on a discrete subset of a smoo th manifold. The Sch wartz spa ce, whic h is isomorphic to C ∞ ( X ) for a no n-discrete compa ct manifold X and to J ∞ ( X ; Y ) for a prop er closed subset of a compact manifold X , is not injectiv e. A more careful choice of the conﬂations improv es the situation. By deﬁnition, Banach spa ces are injectiv e for lo ca lly split extens io ns in ← − − Ban and he nc e for pro- lo cally split extensions of F r´ echet spac e s. This will later allow us to do some Ho chsc hild co homology computations with Banach s pace co eﬃcients for alge bras of Whitney functions. If w e restrict to split ex tens ions, then a ll ob jects of C bec ome injective, so that we get the following result: Corollary 3. 10. L et C b e an additive symmetric monoidal c ate gory, e quip it with the class of split extensions. L et I ֌ E ։ Q b e a split extension in C and let M b e a Q -bimo dule. If ( I ⊗ n , b ′ ) is s plit exact, t hen HH ∗ ( E , M ) ∼ = HH ∗ ( Q, M ) . Pr o of. Here a ny ob ject of C is injective and any conﬂation is pure. The assumption means that I is H-unital. Hence the assertion follows from Theo rem 3.6.  4. Hochschild homology for algebras of smoo th functions In this section, we work in the symmetric monoidal categor y Fr of F r´ echet spaces. Thu s ⊗ = ˆ ⊗ π is the complete pro jective top ologica l tensor pro duct of F r´ echet s paces. The resulting Ho chsc hild ho mology and co homology a re the c ontinuous Ho chsc hild homology and co homology of F r´ e chet alg e br as. W e let a ll extensions b e conﬂations unless we explicitly r equire another exact catego r y structure on Fr . Let X b e a smoo th manifold, p ossibly non-compact, and let C ∞ ( X ) b e the F r´ echet a lgebra of smoo th functions on X with the topolo g y o f lo ca lly uniform conv e r gence of a ll der iv atives. The k th con tinuous Ho chschild homology of C ∞ ( X ) is the space Ω k ( X ) o f smo oth diﬀeren tial k -forms o n X ; by deﬁnition, this is the space of s mo oth sections of the vector bundle (Λ k T X ) ∗ ∼ = Λ k (T ∗ X ). The contin u- ous Ho chsc hild cohomolog y of C ∞ ( X ) is the top ologica l dual space of contin uous linear functionals on Ω k ( X ). By deﬁnition, this is the space of distributiona l sec- tions of the vector bundle Λ k (T X ), called de Rham curr ent s of dimension k . The “c o nt inuit y” of the Ho chschild homolo gy and cohomo logy means that we work in the symmetr ic monoida l categor y Fr with the tensor pro duct ˆ ⊗ π . Thus 20 RALF MEYE R HH ∗  C ∞ ( X )  : = (C ∞ ( X ) ⊗ n , b ) in volv es the completed tensor pro ducts C ∞ ( X ) ⊗ n : = C ∞ ( X ) ˆ ⊗ π n ∼ = C ∞ ( X n ) . The co nt inuous linear functionals on C ∞ ( X ) ˆ ⊗ π n corres p o nd bijectiv ely to (join tly) contin uous n - linear functionals C ∞ ( X ) n → C b y the univ ersal pr op erty o f ˆ ⊗ π ; hence w e may descr ib e contin uo us Ho chschild cohomo logy without ˆ ⊗ π (as in [4]). The contin uous Hochschild co homology o f C ∞ ( X ) was computed by Alain Connes in [4, Section I I.6] to prepa re for the co mputation of its cyclic and p erio dic cyclic cohomolog y; his a rgument can also b e used to co mpute the cont inuous Ho chschild homology o f C ∞ ( X ). Several la ter a r gument by Jea n-Luc Br ylinski a nd Victor Nis- tor [3] a nd b y Nicolae T eleman [24] use lo calis ation near the diago nal to compute the Ho chschild homology and cohomo logy of C ∞ ( X ). This lo calisa tion appr oach is more conceptual but, as it seems , gives sligh tly less informa tion. The chain co mplex HH ∗ ( A ) = ( A ⊗ n , b ) is a chain co mplex of A -mo dules via a 0 · ( a 1 ⊗ · · · ⊗ a n ) : = ( a 0 · a 1 ) ⊗ · · · ⊗ a n for a 0 , . . . , a n ∈ A – notice that this deﬁnes a chain map, that is, a 0 · b ( ω ) = b ( a 0 · ω ) if and only if A is co mm utative. Thus HH ∗  C ∞ ( X )  inherits such a mo dule structure as well. The isomor phism HH ∗  C ∞ ( X )  ∼ = Ω ∗ ( X ) identiﬁes this mo dule structure on HH ∗  C ∞ ( X )  with the obvious mo dule structur e o n diﬀerential forms by p oint wis e m ultiplication. W e will need an even stronger result: Theorem 4.1. L et X b e a smo oth manifold. The anti-symmet risation map j : Ω k ( X ) → C ∞ ( X ) ⊗ k +1 , f 0 d f 1 ∧ · · · ∧ d f k 7→ X σ ∈ S k ( − 1) | σ | f 0 ⊗ f σ (1) ⊗ · · · ⊗ f σ ( k ) and t he m ap k : C ∞ ( X ) ⊗ k +1 → Ω k ( X ) , f 0 ⊗ f 1 ⊗ · · · ⊗ f k 7→ 1 k ! f 0 d f 1 ∧ · · · ∧ d f k ar e C ∞ ( X ) -line ar c ontinuous chai n maps b etwe en Ω ∗ ( X ) with the zer o b ou n dary map and HH ∗  C ∞ ( X )  : = (C ∞ ( X ) ⊗ n +1 , b ) that ar e inverse to e ach other up to C ∞ ( X ) -line ar c ontinu ous chai n homotopy. Mor e pr e cisely, k ◦ j = Id Ω ∗ ( X ) and j ◦ k = [ b, h ] for a C ∞ ( X ) -line ar c ontinu ous map h on HH ∗  C ∞ ( X )  of de gr e e 1 . Pr o of. The commutativit y of C ∞ ( X ) and the Leibniz rule d( f 1 f 2 ) = f 1 d f 2 + f 2 d f 1 in Ω ∗ ( X ) imply b ◦ j = 0 and k ◦ b = 0, tha t is , j a nd k a re chain maps. The equation k ◦ j = Id Ω ∗ ( X ) is obvious. The only as sertion that r equires work is to ﬁnd the C ∞ ( X )-linear c hain ho mo topy h . The existence of suc h a chain homoto py follows ea s ily from Co nnes’ ar gument in [4]. The following rema r ks provide some more details for reader s who do no t accept this one sentence as a pro of. First we r ecall how the Ho chsc hild c hain complex for a F r´ echet algebra A is related to pro jective resolutions. W e must expla in what “pro jective resolution” means. The following discussion a pplies to any algebra A in a symmetric monoidal category C . W e call an extension of A -bimo dules s emi-split if it s plits in C (but the s plitting need not b e A -linear). The semi-split extensions are the conﬂations of an exact categ ory s tructure on the catego ry of A -bimo dules. F or any ob ject V of C , we equip A ⊗ V ⊗ A with the ob vious A -bimo dule structure a nd call such A -bimo dules fr e e . F ree bimo dules are pr o jective with resp ect to semi-split bimo dule EXCISION IN HOCHSCHILD AND CYCLIC HOM OLOGY 21 extensions. The bar resolution ( A ⊗ n +2 , b ′ ) n ≥ 0 is contractible in C a nd hence a pro jective resolution of A in the exact categ ory of A - bimo dules with semi- s plit extensions as conﬂations. Hence it is chain homotopy equiv a le n t to any other pro jective A -bimo dule r esolution of A in this exac t category . The c ommutator quotient of an A -bimo dule is the cokernel of the commutator map A ⊗ M → M , a ⊗ m 7→ [ a, m ] : = a · m − m · a . Since the comm utator quotien t of a fr ee mo dule A ⊗ V ⊗ A is naturally isomorphic to A ⊗ V , the commutator quotient complex of the bar reso lution is the Ho chsc hild chain complex ( A ⊗ n +1 , b ). If the algebra A is commutativ e, then the commutator quotien t o f an A -bimo dule is still an A -mo dule in a canonical wa y . Thus the Ho chsch ild complex is a chain complex of A -mo dules in a cano nica l way . If P • is another pr o jective A -bimo dule r esolution of A , then P • is chain homotopy equiv alent to the bar resolution as a chain complex of A -bimo dules. Hence the Ho chsc hild chain complex is chain homotopy equiv alent to the commutator quotient complex P • / [ P • , A ]. If A is commutativ e , then this chain homotopy is A -linear bec ause the A -mo dule structure on co mmutator quotien ts is natural. Now we return to the F r´ echet a lg ebra C ∞ ( X ). Connes co mputes the Hochsc hild cohomolog y of C ∞ ( X ) by constructing another pro jective C ∞ ( X )-bimo dule reso- lution P • of C ∞ ( X ) for which the commutator q uotient complex P • / [ P • , C ∞ ( X )] is Ω k ( X ) with zero b oundar y map. As our disc us sion ab ov e s hows, this implies that (Ω ∗ ( X ) , 0 ) is c hain homotopy eq uiv alent to HH ∗  C ∞ ( X )  as a chain complex of C ∞ ( X )-mo dules. An insp ection of Connes ’ argument also shows that the chain maps inv olved in this homotopy equiv a lence are j and k . More precis ely , Connes’ construction only a pplies if X carries a nowhere v a n- ishing vector ﬁeld or, equiv a le ntly , if each connected component o f X is either non-compact or has v anishing Euler characteristic. The case of a gener a l smo oth manifold is reduced to this c ase b y considering X × S 1 , which do es carr y such a v ector ﬁeld, and then relating the Ho chsc hild cohomolog y of C ∞ ( X ) and C ∞ ( X × S 1 ). The functoriality of Ho chschild coho mology implies that H H ∗  C ∞ ( X )  is isomo r phic to the r a nge of the map o n HH ∗  C ∞ ( X × S 1 )  induced by the map X × S 1 → X × S 1 , ( x, z ) 7→ ( x, 1). Under the homotopy equiv alence b etw een HH ∗  C ∞ ( X × S 1 )  and (Ω ∗ ( X × S 1 ) , 0 ), this map corr esp onds to a pr o jection onto (Ω ∗ ( X ) , 0 ).  The additional statements ab out chain ho motopy equiv a lence in Theorem 4.1 seem diﬃcult to prov e with the lo calisa tion metho d b eca use the la tter inv olves con- tractible subc o mplexes that ar e either not even closed (such as the ch ain complex of functions v anishing in some neighbo ur ho o d o f the diag onal) or a re not c omple- men table (such as the c hain complex of functions that ar e ﬂat on the diagona l). 4.1. The algebra of sm o oth functions wi th compact supp ort. Now we wan t to r eplace the F r´ echet algebra C ∞ ( X ) by the dense subalgebra C ∞ c ( X ) of smo o th functions w ith compa c t supp or t. This is an LF-spa ce in a natural topolo gy: Le t ( K n ) n ∈ N be an increas ing sequence of compact subsets e xhausting X , then C ∞ c ( X ) is the strict inductive limit of the s ubspaces of C ∞ ( X ) o f smo oth functions that v anish outside K n . This is a top ologica l algebra, that is, the m ultiplication is jointly contin uous . Nev ertheless, we will view C ∞ c ( X ) as a bor nologica l algebra in the fol- lowing, that is, replace it by Cpt C ∞ c ( X ). This is preferable beca us e the pro jective bo rnologic al tensor pro duct a g rees with Grothendieck’s inductive tensor pr o duct for nuclear LF-spa c e s, s o that Cpt C ∞ c ( X ) ˆ ⊗ Cpt C ∞ c ( Y ) ∼ = Cpt C ∞ c ( X × Y ) for all 22 RALF MEYE R smo oth manifolds X a nd Y . In contrast, the pr o jective top ologica l tenso r pro duct is C ∞ c ( X × Y ) with a complicated to p o lo gy . Since we wan t tensor powers of C ∞ c ( X ) to b e C ∞ c ( X n ), w e must either deﬁne tensor pr o ducts in an ad ho c wa y as in [2 ] o r work b orno lo gically . W e turn the ca tegory o f complete b ornolo gical C ∞ c ( X )-mo dules into an exact category using the class of s plit extensions a s co nﬂations, that is, c o nﬂations a re extensions of C ∞ c ( X )-mo dules with a b ounded linear s ection. W e a lready hav e a pro jective bimo dule resolution for C ∞ ( X ) and w ant to use it to construct one for C ∞ c ( X ). Giv en a C ∞ c ( X )-mo dule M , we let M c ⊆ M be the subspace of all m ∈ M for which ther e is f ∈ C ∞ c ( X ) with m = f · m . This agrees with C ∞ c ( X ) · M beca use C ∞ ( X ) c = C ∞ c ( X ). A subset S of M c is called b ounde d if it is bo unded in M and there is a single f ∈ C ∞ c ( X ) with f · m = m for all m ∈ S . This deﬁnes a complete b ornolog y on M c . The subspace M c is still a mo dule ov er C ∞ ( X ) and, a fortiori , ov er C ∞ c ( X ). The multiplication maps C ∞ c ( X ) × M c → M c and C ∞ ( X ) × M c → M c are b oth b ounded. Prop ositio n 4.2 . The functor M 7→ M c is exact. If M is a pr oje ct ive C ∞ ( X ) - mo dule, then M c is pr oje ctive b oth as a C ∞ c ( X ) -mo dules and as a C ∞ ( X ) -mo dule. Pr o of. Exactness means that I c ֌ E c ։ Q c is a s e mi-split extension if I ֌ E ։ Q is a semi-split extension of C ∞ ( X )-mo dules. Let s : Q → E b e a b o unded linear section. Let ( ϕ n ) n ∈ N be a lo cally ﬁnite s e t of co mpactly supp orted functions with P ϕ 2 n = 1, that is , ϕ 2 n is a partition of unity . W e deﬁne s c ( f ) : = P n ∈ N ϕ n · s ( ϕ n · f ). This is still a well-deﬁned b ounded linear s e ction Q → E , but this new section restricts to a b ounded linear map Q c → E c bec ause for ea ch compact subset K we hav e ϕ n | K = 0 for almo st all n and each ϕ n has compact supp or t. As a co nsequence, the functor M 7→ M c is exact. Since a ny pr o jective C ∞ ( X )-mo dule is a direct summand of a free module C ∞ ( X ) ˆ ⊗ V , the cla im ab out pro jectivity mea ns that (C ∞ ( X ) ˆ ⊗ V ) c is pr o jective as a C ∞ ( X )-mo dule and as a C ∞ c ( X )-mo dule for a ny V . Since (C ∞ ( X ) ˆ ⊗ V ) c ∼ = C ∞ c ( X ) ˆ ⊗ V and M ˆ ⊗ V is pro jective once M is pro jective, w e m ust chec k that C ∞ c ( X ) is pro jectiv e as a C ∞ ( X )-mo dule and as a C ∞ c ( X )-mo dule. Equiv alently , the mul- tiplication maps C ∞ ( X ) ˆ ⊗ C ∞ c ( X ) → C ∞ c ( X ) a nd C ∞ c ( X ) + ˆ ⊗ C ∞ c ( X ) → C ∞ c ( X ) split by a mo dule homomorphis m. This follows fro m the following lemma, which ﬁnishes the pro of.  Lemma 4.3 . The multiplic ation map C ∞ c ( X ) ˆ ⊗ C ∞ c ( X ) → C ∞ c ( X ) has b ounde d line ar se ctions σ l and σ r that ar e a left and a right mo dule homomorphism, r esp e c- tively. Pr o of. Recall that C ∞ c ( X ) ˆ ⊗ C ∞ c ( X ) ∼ = C ∞ c ( X × X ). The m ultiplication map bec omes the map C ∞ c ( X × X ) → C ∞ c ( X ) that r estricts functions to the diag onal. It suﬃces to desc r ib e σ l , then ( σ r f )( x, y ) : = ( σ l f )( y , x ) provides σ r . W e make the Ansatz ( σ l f )( x, y ) : = f ( x ) · w ( x, y ) fo r some w ∈ C ∞ ( X × X ). W e need w ( x, x ) = 1 for all x ∈ X in or der to ge t a sectio n for the multiplication map, and we assume that the pro jection to the ﬁrst co ordinate ( x, y ) 7→ x restr icts to a prop er map o n the suppo rt of w . That is, for ea ch compact subset K ⊆ X there is a c o mpact subset L ⊆ X × X such that w ( x, y ) = 0 for all x ∈ K with ( x, y ) / ∈ L . This ensures that σ l f is supp orted in L if f is supp orted in K . It is ro utine to check that such a function w exists and that the r esulting map σ l has the required pro pe r ties.  EXCISION IN HOCHSCHILD AND CYCLIC HOM OLOGY 23 Lemma 4.3 implies that C ∞ c ( X ) is H-unita l, see the pro of of Coro llary 5.4 below. Viewing C ∞ c ( X )-bimo dules as C ∞ c ( X × X )-mo dules, the ab ov e carries over to bimo dules without change. Now the supp ort restriction functor M 7→ M c will require compa c t supp or t with respec t to b oth mo dule structures. Since C ∞ ( X ) c = C ∞ c ( X ), Prop o s ition 4.2 implies that the functor P 7→ P c maps any pro jective C ∞ ( X )-bimo dule reso lution of C ∞ ( X ) to a pro jective C ∞ c ( X )-bimo dule r esolution of C ∞ c ( X ). It is easy to c heck that the commutator quotient functor for bimo dules int er- t wines the supp ort restr ic tion functors:  M / [C ∞ ( X ) , M ]  c ∼ = M c  [C ∞ c ( X ) , M c ] . As a consequence, H H ∗  C ∞ c ( X )  is c hain homotopy equiv alent to HH ∗  C ∞ ( X )  c , where the supp ort r estriction is with resp ect to the canonical C ∞ ( X )-mo dule struc- ture on HH ∗  C ∞ ( X )  c . Theorem 4.1 now implies: Theorem 4.4. L et X b e a smo oth manifold. The anti-symmet risation map j : Ω k c ( X ) → C ∞ c ( X ) ⊗ k +1 , f 0 d f 1 ∧ · · · ∧ d f k 7→ X σ ∈ S k ( − 1) | σ | f 0 ⊗ f σ (1) ⊗ · · · ⊗ f σ ( k ) and t he m ap k : C ∞ c ( X ) ⊗ k +1 → Ω k c ( X ) , f 0 ⊗ f 1 ⊗ · · · ⊗ f k 7→ 1 k ! f 0 d f 1 ∧ · · · ∧ d f k ar e C ∞ c ( X ) -line ar b ounde d chain maps b etwe en Ω ∗ c ( X ) with the zer o b oundary map and HH ∗  C ∞ c ( X )  : = (C ∞ c ( X ) ⊗ n +1 , b ) t hat ar e inverse to e ach other up to C ∞ c ( X ) - line ar b ounde d chain homotopy. Mor e pr e cisely, k ◦ j = Id Ω ∗ c ( X ) and j ◦ k = [ b, h ] for an C ∞ c ( X ) -line ar b ounde d map h on HH ∗  C ∞ c ( X )  of de gr e e 1 . Here Ω k c ( X ) : = Ω k ( X ) c is the space of compactly supported smo oth k -forms with its canonical b ornolo gy . 5. Applica tion to Whitney functions As in the previous section, we w ork in the symmetr ic monoidal categor y Fr o f F r´ echet spaces. Let X b e a s mo o th manifold – fo r instance, an op en subse t of R n – and let Y b e a clo sed subset of X . A smo oth function on X is called ﬂat on Y if its T aylor ser ies v anishes at each p o int o f Y , that is, ( Df )( y ) = 0 for an y y ∈ Y and any diﬀerential op erator D , including o per ators of or der zero . The smo oth functions that are ﬂat on Y form a closed ideal J ∞ ( X ; Y ) in C ∞ ( X ). The quotient E ∞ ( Y ) : = C ∞ ( X )  J ∞ ( X ; Y ) is, b y deﬁnition, the F r ´ echet algebra of Whitney functions on Y . This algebra depe nds on the embedding of Y in X . Example 5.1 . If Y ⊆ X consists of a single point, then E ∞ ( Y ) is isomor phic to the F r´ echet alg e bra C [ [ x 1 , . . . , x n ] ] of forma l power series in n v ariables, where n is the dimension of X . 24 RALF MEYE R By deﬁnition, the F r´ echet algebra of Whitney functions ﬁts into an extension J ∞ ( X ; Y ) ֌ C ∞ ( X ) ։ E ∞ ( Y ) , as in (1). It is well-known that C ∞ ( X ) is nuclear, see [7]. Since nuclearit y is inherited by subspaces and quo tient s, J ∞ ( X ; Y ) and E ∞ ( Y ) are nuclear as well. Theorem 2.21 shows, therefore, that the extens io n (1) is b oth ind-lo cally split a nd pro-lo ca lly split, and this remains so if we tenso r ﬁrst with another F r´ echet spa ce V . As a consequence, (1) is a pure extension. Our next go al is to show that J ∞ ( X ; Y ) is homolog ically unital. This requires computing some complete pro jective topolo gical tensor pro ducts and ﬁnding a smo oth function with certain prop er ties. Lemma 5. 2. Ther e ar e isomorphisms C ∞ ( X ) ˆ ⊗ π C ∞ ( X ) ∼ = C ∞ ( X × X ) , J ∞ ( X ; Y ) ˆ ⊗ π J ∞ ( X ; Y ) ∼ = J ∞ ( X × X ; Y × X ∪ X × Y ) . Pr o of. It is well-known that C ∞ ( X ) ˆ ⊗ π C ∞ ( X ) ∼ = C ∞ ( X × X ), see [7]. Since all spaces inv o lved ar e n uclear, C ∞ ( X ) ˆ ⊗ π J ∞ ( X ; Y ), J ∞ ( X ; Y ) ˆ ⊗ π C ∞ ( X ), and J ∞ ( X ; Y ) ˆ ⊗ π J ∞ ( X ; Y ) are subspace s of C ∞ ( X × X ) – they ar e the clos ur es of the corre s po nding alg e braic tensor pro ducts in C ∞ ( X × X ) – and (5) J ∞ ( X ; Y ) ˆ ⊗ π J ∞ ( X ; Y ) = C ∞ ( X ) ˆ ⊗ π J ∞ ( X ; Y ) ∩ J ∞ ( X ; Y ) ˆ ⊗ π C ∞ ( X ) as subspaces of C ∞ ( X × X ). Since functions of the for m f 1 ⊗ f 2 with f 1 ∈ J ∞ ( X ; Y ), f 2 ∈ C ∞ ( X ) span a dense subspace of J ∞ ( X × X ; Y × X ), we get C ∞ ( X ) ˆ ⊗ π J ∞ ( X ; Y ) ∼ = J ∞ ( X × X ; X × Y ) . Similarly , J ∞ ( X ; Y ) ˆ ⊗ π C ∞ ( X ) ∼ = J ∞ ( X × X ; Y × X ) . Now (5) shows that a smo oth function on X × X belo ngs to J ∞ ( X ; Y ) ˆ ⊗ π J ∞ ( X ; Y ) if and only if it is ﬂat o n both Y × X and X × Y .  Prop ositio n 5.3. Ther e ar e c ontinuous line ar se ctions σ l , σ r : J ∞ ( X ; Y ) → J ∞ ( X ; Y ) ˆ ⊗ π J ∞ ( X ; Y ) for the m u ltiplic ation map µ : J ∞ ( X ; Y ) ˆ ⊗ π J ∞ ( X ; Y ) → J ∞ ( X ; Y ) , such t hat σ l is a left C ∞ ( X ) -mo dule map and σ r is a right C ∞ ( X ) -mo dule map. Pr o of. Let X 2 : = X × X and Y 2 : = Y × X ∪ X × Y ⊆ X 2 . By Lemma 5 .2, σ l and σ r are suppo sed to b e ma ps from J ∞ ( X ; Y ) to J ∞ ( X 2 ; Y 2 ). The multiplication map corres p o nds to the map µ : J ∞ ( X 2 ; Y 2 ) → J ∞ ( X ; Y ) that restricts functions to the diagonal, ( µf )( x ) : = f ( x, x ). Once we hav e found σ l , we may put σ r f ( x 1 , x 2 ) : = σ l f ( x 2 , x 1 ), so that it suﬃces to c o nstruct σ l . Let A ⊆ X 2 be the diagona l and let B : = X × Y . Then A ∩ B is the dia gonal image of Y in X 2 . W e cla im that A and B a re regula r ly s ituated (see [25]); even more, d ( x, A ∩ B ) ≤ C · ( d ( x, A ) + d ( x, B )  for some co nstant C . The distance from ( x 1 , x 2 ) to the diagonal A is d ( x 1 , x 2 ); the distance to B is d ( x 2 , Y ); and the distance to A ∩ B is at mos t inf y ∈ Y d ( x 1 , y ) + d ( x 2 , y ) ≤ inf y ∈ Y d ( x 1 , x 2 ) + d ( x 2 , y ) + d ( x 2 , y ) = d ( x 1 , x 2 ) + 2 d ( x 2 , Y ) . EXCISION IN HOCHSCHILD AND CYCLIC HOM OLOGY 25 Since A and B are r egularly situated, [25, Lemma 4.5 ] yields a m ultiplier w of J ∞ ( X 2 ; A ∩ B ) that is co nstant equal to one in a neighbourho o d of A \ ( A ∩ B ) and c onstant equa l to zero in a neigh b ourho o d of B \ ( A ∩ B ). 1 Being a m ultiplier means that w is a smo o th function on X 2 \ ( A ∩ B ) whose deriv atives g row at most po lynomially near A ∩ B , that is, for each compactly supp orted diﬀerential op erator D there is a p olynomial p D with | D w ( x ) | ≤ p D ( d ( x, A ∩ B ) − 1 ) fo r all x ∈ X 2 \ ( A ∩ B ). No w deﬁne σ l f ( x 1 , x 2 ) : = f ( x 1 ) · w ( x 1 , x 2 ) . The function σ l f is a smo oth function on X 2 \ ( A ∩ B ). Since f ⊗ 1 is ﬂa t o n Y × X ⊇ A ∩ B and w is a multiplier o f J ∞ ( X 2 ; A ∩ B ), its extension by 0 on A ∩ B , also denoted by σ l f , is a smo oth function o n X 2 that is ﬂat on Y × X . F urthermo re, the extensio n is ﬂat on ( X \ Y ) × Y b ecause w is , so that σ l f ∈ J ∞ ( X 2 ; Y 2 ). W e also hav e σ l f ( x, x ) = f ( x ) b oth for x ∈ X \ Y and x ∈ Y . Thus σ l has the requir ed prop erties.  It ca n b e chec ked tha t J ∞ ( X ; Y ) has a multiplier b ounded approximate unit as well; hence it is quasi- unital in the nota tio n of [15]. W e do not need this stro nger fact here. Corollary 5. 4. L et I : = J ∞ ( X ; Y ) , t hen the chain c omplex ( I ⊗ n , b ′ ) has a b ounde d c ontr acting homotopy, that is, I is homolo gic al ly unital for any exact c ate gory st ruc- tur e on the c ate gory of F r´ echet sp ac es. Pr o of. The maps s n : = σ r ⊗ Id I ⊗ n − 1 : I ⊗ n → I ⊗ n +1 for n ≥ 1 satisfy b ′ s 1 = Id I and b ′ s n + s n − 1 b ′ = Id I ⊗ n for n ≥ 2 beca us e σ r is a right mo dule map and a section for the mult iplication ma p. Hence ( I ⊗ n , b ′ ) is contractible. So is ( I ⊗ n , b ′ ) ⊗ V fo r a ny F r´ echet space V . Co nt ractible chain complex e s are exact.  Having chec ked that the extension (1) is pure and that its kernel is homolog ically unital, we are in a position to apply Theorem 3.8. Since 1 = C is b oth pro jective and injectiv e a s a F r´ echet space, we g e t lo ng exa ct sequences in Hochschild ho mo logy and coho mology for the algebra extensio n (1). W e have already computed the Ho chsc hild homology a nd cohomo lo gy of C ∞ ( X ) in the previous se ction. Our next task is to compute them for J ∞ ( X ; Y ), together with the maps o n Ho chsc hild homology and cohomolo g y induced by the embedding J ∞ ( X ; Y ) → C ∞ ( X ). The following computations also use the balanced tensor pro duct M ⊗ A N for a r ight A -mo dule M and a left A -mo dule N . This is deﬁned – in the abstract setting o f symmetric mo noidal categ ories with cokernels – as the cokernel of the map b ′ : M ⊗ A ⊗ N → M ⊗ N . Prop ositio n 5. 5. The Ho chschild homolo gy HH k  J ∞ ( X ; Y )  is isomorphic to the sp ac e J ∞ Ω k ( X ; Y ) of s mo oth diﬀer ent ial k -forms on X that ar e ﬂat on Y , and the map t o HH k  C ∞ ( X )  ∼ = Ω k ( X ) is e quivalent to the obvious emb e dding J ∞ Ω k ( X ; Y ) → Ω k ( X ) . The Ho chschild c ohomolo gies ar e natu r al ly isomorphi c to the top olo gic al dual s p ac es J ∞ Ω k ( X ; Y ) ∗ and Ω k ( X ) ∗ . Pr o of. Let I : = J ∞ ( X ; Y ) and E : = C ∞ ( X ). Theor em 4.1 shows that the chain complex HH ∗ ( E ) : = ( E ⊗ n , b ) n ≥ 1 is homotopy equiv alent as a chain co mplex of 1 I thank Markus Pﬂaum f or this construction of w . 26 RALF MEYE R E -mo dules to Ω ∗ ( X ) with zero as b oundary map. Hence I ⊗ E HH ∗ ( E ) ∼ = ( I ⊗ E ⊗ n − 1 , b ) n ≥ 1 = HH ∗ ( E , I ) is homotopy equiv alent (as a chain complex of I - mo dules) to I ⊗ E Ω ∗ ( X ) = J ∞ ( X ; Y ) ⊗ C ∞ ( X ) Ω ∗ ( X ) ∼ = J ∞ Ω ∗ ( X ; Y ) . The last step follows from the following more genera l co mputation. If V is a smo oth vector bundle on X and Γ ∞ ( V ) is its spa ce o f smo o th sectio ns, then J ∞ ( X ; Y ) ⊗ C ∞ ( X ) Γ ∞ ( V ) is the space of smo oth s e ctions of V that are ﬂat on Y . This is ea sy to see for trivial V , a nd Swan’s Theor em r educes the general case to this specia l case. Corollar ies 5 .4 and 3.5 show that the canonical embedding HH ∗ ( I ) → HH ∗ ( E , I ) is a pure qua si-isomor phism, so that HH ∗ ( I ) is pur ely quasi-is omorphic to I ⊗ E HH ∗ ( E ). Hence HH ∗ ( I ) ∼ = J ∞ Ω k ( X ; Y ) a s asserted. F urther mo re, our computa- tion shows that the map HH ∗ ( I ) → HH ∗ ( E ) induced by the embedding I → E is the obvious embedding J ∞ Ω k ( X ; Y ) → Ω k ( X ). The ra nge of the b oundary ma p of HH ∗ ( E ) is closed. Thus HH k ( E ) inher its a F r´ e chet top olog y – this is, of course, the standard F r´ echet space structure on Ω k ( X ). Since the dual space functor o n F r´ echet spaces is exact, there is the following universal c o eﬃcient theor em: if the b oundar y map of a chain complex of F r´ echet spaces has closed r ange, then the cohomolo g y of the top o logical dual chain co mplex is the top olog ical dual of the homo lo gy . Thus HH k ( E ) ∼ = Ω k ( X ) ∗ . Our computation shows tha t the bo undary map of HH ∗ ( I ) has clo sed ra ng e as w ell, so tha t HH k ( E ) ∼ = J ∞ Ω k ( X ; Y ) ∗ .  Since the embedding J ∞ Ω k ( X ; Y ) → Ω k ( X ) has closed range, the quotient space E ∞ Ω k ( Y ) : = Ω k ( X )  J ∞ Ω k ( X ; Y ) is a F r´ echet space in the quotient top olo gy . This is the space of Whitney diﬀer ential forms . Since Ω k ( X ) is a pro jective C ∞ ( X )-mo dule and the extension (1) is pure, the diagra m J ∞ ( X ; Y ) ⊗ C ∞ ( X ) Ω k ( X ) → C ∞ ( X ) ⊗ C ∞ ( X ) Ω k ( X ) → E ∞ ( Y ) ⊗ C ∞ ( X ) Ω k ( X ) is an extension of F r´ echet spaces . Iden tifying J ∞ ( X ; Y ) ⊗ C ∞ ( X ) Ω k ( X ) ∼ = J ∞ Ω k ( X ; Y ) , C ∞ ( X ) ⊗ C ∞ ( X ) Ω k ( X ) ∼ = Ω k ( X ) , we see that E ∞ Ω k ( Y ) ∼ = E ∞ ( Y ) ⊗ C ∞ ( X ) Ω k ( X ) . This pr ovides an alternative deﬁnition for the space of Whitney diﬀerential forms. The de Rham b oundar y map on Ω ∗ ( X ) maps the subspa ce J ∞ Ω ∗ ( X ; Y ) in to itself and hence induces a de Rham b oundar y map 0 → E ∞ Ω 0 ( Y ) d − → E ∞ Ω 1 ( Y ) d − → E ∞ Ω 2 ( Y ) d − → E ∞ Ω 3 ( Y ) → · · · on Whitney diﬀerential forms. The coho mology o f this co chain complex is the de Rha m c ohomolo gy H ∗ dR ( Y ) of Y . EXCISION IN HOCHSCHILD AND CYCLIC HOM OLOGY 27 Theorem 5.6. L et Y b e a close d su bset of a sm o oth manifold X and let C ∞ ( Y ) b e the F r´ echet algebr a of Whitney functions on Y ⊆ X . Then HH k  E ∞ ( Y )  ∼ = E ∞ Ω k ( Y ) , HC k  E ∞ ( Y )  ∼ = E ∞ Ω k ( Y )  d( E ∞ Ω k − 1 Y ) ⊕ H k − 2 dR ( Y ) ⊕ H k − 4 dR ( Y ) ⊕ · · · , HP k  E ∞ ( Y )  ∼ = M j ∈ Z H k − 2 j dR ( Y ) . Pr o of. The Excision Theorem 3.8 provides a long exact sequence · · · → HH k  J ∞ ( X ; Y )  → HH k  C ∞ ( X )  → HH k  E ∞ ( Y )  → HH k − 1  J ∞ ( X ; Y )  → HH k − 1  C ∞ ( X )  → HH k − 1  E ∞ ( Y )  → · · · bec ause J ∞ ( X ; Y ) is homologic a lly unital and the extension (1 ) is pure. W e hav e seen ab ov e that the map HH k  J ∞ ( X ; Y )  → HH k  C ∞ ( X )  is equiv alent to the map J ∞ Ω k ( X ; Y ) → Ω k ( X ). The la tter is a closed embedding and, in pa r ticular, injectiv e. Hence the long exact seq uence above yields HH k  E ∞ ( Y )  ∼ = E ∞ Ω k ( Y ) as asserted. T o go from Ho chschild homology to cyclic ho mology , w e may use the natural map B ◦ I : HH k ( A ) → HH k +1 ( A ). F or A = C ∞ ( X ), this map cor r esp onds to the de Rham b oundary map. By naturality , this remains true for E ∞ ( Y ) as w ell. The same arguments as for s mo oth manifolds now yield the form ulas for the cyclic and per io dic cyclic homology of Y .  If the de Rham b oundary map on E ∞ Ω ∗ ( Y ) has clos ed range, then HC ∗  E ∞ ( Y )  and HP ∗  E ∞ ( Y )  are F r´ echet spaces, a nd the co ho mology space s HC ∗  E ∞ ( Y )  and HP ∗  E ∞ ( Y )  are just their top ologica l duals. T ogether with Mar kus Pﬂaum, I plan to show in a fo r thcoming article that the b o undary map on E ∞ Ω ∗ ( Y ) alwa ys has closed range, and to ident ify the res ulting cohomology theory with the Alexander– Spanier cohomolo gy of Y in favourable cases. The results above car ry over with small changes to the algebra o f co mpactly suppo rted Whitney functions. There are, in fact, t wo w ays to pro ceed. Either we use Theorem 4.4 instead of 4.1 and cop y the excis ion argument above; this requires the excision theorem for ind-lo cally split extensions o f nuclear b ornolo gical algebras . Or we copy the reduction from C ∞ c ( X ) to C ∞ ( X ) for the algebra of Whitney functions. W e may deﬁne a functor M 7→ M c also for mo dules over E ∞ ( Y ), and everything said in Se c tion 4 .1 carr ies ov er to Whitney functions instea d of smo oth functions without change. This s hows that the Ho chsc hild chain co mplex for the algebra of compa c tly s uppo r ted Whitney functions is HH  E ∞ ( Y )  c . Now we can cop y the arg uments ab ov e. De Rham co homology is r e placed by compac tly suppo rted de Rham cohomolo g y , of course: Theorem 5.7. L et Y b e a close d subset of a smo oth manifold X and let E ∞ c ( Y ) b e t he b ornolo gic al algebr a of c omp actly su pp orte d Whitney fun ctions on Y ⊆ X . Then HH k  E ∞ c ( Y )  ∼ = E ∞ c Ω k ( Y ) , HC k  E ∞ c ( Y )  ∼ = E ∞ c Ω k ( Y )  d( E ∞ c Ω k − 1 Y ) ⊕ H k − 2 dR , c ( Y ) ⊕ H k − 4 dR , c ( Y ) ⊕ · · · , HP k  E ∞ c ( Y )  ∼ = M j ∈ Z H k − 2 j dR , c ( Y ) . 28 RALF MEYE R Finally , we turn to Hochschild cohomolo gy with co eﬃcients. Lemma 5.8 . L et M b e a C ∞ ( X ) -mo dule, viewe d as a symmetric C ∞ ( X ) -bimo dule. Then HH ∗ (C ∞ ( X ) , M ) is chain homotopy e quivalent to Hom C ∞ ( X ) (Ω ∗ ( X ) , M ) with zer o b oun dary map. In p articular, HH k (C ∞ ( X ) , M ) ∼ = Hom C ∞ ( X ) (Ω ∗ ( X ) , M ) . Pr o of. F or any commutativ e unital algebra A , the chain complex HH ∗ ( A, M ) is naturally isomorphic to Hom A ( HH ∗ ( A ) , M ) be cause Hom A ( A ⊗ A ⊗ n , M ) ∼ = Hom( A ⊗ n , M ) and b o n A ⊗ A ⊗ n induces the bounda ry map b ∗ on Hom( A ⊗ n , M ). Theorem 4.1 asserts that HH ∗  C ∞ ( X )  is C ∞ ( X )-linearly chain homotop y equiv alent to Ω ∗ ( X ) with zero b oundar y map. Hence we g e t an induced chain homotopy equiv alence betw een HH ∗ (C ∞ ( X ) , M ) and Hom C ∞ ( X ) (Ω ∗ ( X ) , M ) with zero bo undary map.  Let Q : = E ∞ ( Y ), E : = C ∞ ( X ), and let M b e a Q -mo dule. W e may vie w M as an E -mo dule, as a symmetric Q -bimo dule, and a s a symmetric E -bimo dule. Whenever Theo r em 3.9 applies, we get HH ∗ ( E ∞ ( Y ) , M ) ∼ = HH ∗ (C ∞ ( X ) , M ) ∼ = Hom C ∞ ( X ) (Ω ∗ ( X ) , M ) . Unfortunately , it is not clear whether this holds in the most interesting cas e M = C ∞ ( X ). But since the extension J ∞ ( X ; Y ) ֌ C ∞ ( X ) ։ E ∞ ( Y ) is pure pr o - lo cally split by Theorem 2.21, we ma y use the e x act catego ry structure of pro-lo cally split extensio ns and a pply Theorem 3 .9 whenever M is a Banach s pa ce. In partic- ular, we may use the k times diﬀer ent iable version E k ( Y ) of E ∞ ( Y ), which is the Banach spac e quotient of the Bana ch alg e br a C k ( X ) of k times con tinuously diﬀer - ent iable functions on X by the closed ideal of all k times diﬀeren tiable functions whose deriv atives of order at most k v anish on Y . Theorem 5.9. HH k  E ∞ ( Y ) , E k ( Y )  is isomorphic to the sp ac e of k times c ontin- uously diﬀer en tiable Whitney k -ve ctor ﬁelds on Y . Pr o of. This follows from the ab ov e discussion and a stra ightforw ard computatio n of Hom C ∞ ( X )  Ω ∗ ( X ) , E k ( Y )  that uses Swan’s Theorem. Of co ur se, the s pa ce of k times contin uously diﬀerentiable Whitney k -vector ﬁelds on Y is the quotient of the space of k times contin uo usly diﬀeren tiable k -vector ﬁelds on X by the subspace of v ector ﬁelds that ar e ﬂat on Y (up to order k ).  6. Excision in periodic cyclic homo l ogy Joachim Cun tz a nd Daniel Quillen established that perio dic cyclic homolo gy satisﬁes ex c ision for a ll algebra ex tens ions [6], even if the kernel is not homolo gically unital. This leads us to expect that p er io dic cyclic homolog y for a lgebras in a symmetric mono ida l catego ry should satisfy an excisio n theorem fo r all pure algebra extensions. But I do not kno w ho w to establish excision in this gener ality . Instead, I only recall t wo more sp ecial results a lready in the literature and apply them to the algebra of smo oth functions on a clo sed subset of a smo oth manifold. Let C b e a symmetric monoidal catego ry and let ← − C be the categ ory of pro jective systems in C with the induced tensor pro duct. F or instance, the category ← − − Ban of pro jective systems of Banach s pa ces is a special ca se of this whic h is closely related to top ologica l vector spaces. W e assume that C is Q -linear be cause otherwise homotopy inv ar iance and excision theorems fail. W e use lo cally split extensions as EXCISION IN HOCHSCHILD AND CYCLIC HOM OLOGY 29 conﬂations to turn ← − C into an exact ca tegory . That is , a dia gram I → E → Q in ← − C is a conﬂation if and only if 0 ← Hom( I , V ) ← Hom( E , V ) ← Hom( Q, V ) ← 0 is a short exact sequence for any V ∈ C . The tenso r unit 1 of C is also a tensor unit in ← − C . Our deﬁnition o f a lo cally split extensio n ensures that 1 is injective. In genera l, it is no t pro jective, but this problem disapp ear s if w e restrict a tten tion to suitable s ubca tegories such a s the sub c ategory of F r´ echet spaces, which w e ident ify with a full sub categ ory of ← − − Ban . F or an a lg ebra A in C or in ← − C , let HP ( A ) b e the Z / 2-g raded chain complex in ← − C that computes the p er io dic cyclic homolo gy and co homology of A (see [5, 6]). Recall that this is alwa ys a c hain complex of pro jective systems, even for A in C . The following theor em is a sp ecia l ca se of [5, Theorem 8.1] by Guillermo Cor ti˜ nas and Christian V alqui. Theorem 6.1. L et I ֌ E ։ Q b e a lo c al ly split extension in ← − C . The n the induc e d maps HP ( I ) → HP ( E ) → HP ( Q ) form a c oﬁbr e se quenc e. This alwa ys yields a cyclic six-term exact se quenc e for HP ∗ . We also get a cyclic six-term exact se quenc e for HP ∗ if HP ( I ) , HP ( E ) and H P ( Q ) ar e inje ctive in ← − C . Corollary 6.2. L et I ֌ E ։ Q b e a pr o-lo c al ly split extension of F r´ echet algebr as. Then the induc e d m aps HP ( I ) → HP ( E ) → HP ( Q ) form a c oﬁbr e se quenc e, and ther e ar e induc e d cyclic six-term exact se qu enc es for HP ∗ and HP ∗ . Pr o of. The embedding of the ca teg ory of F r´ echet spaces into ← − − Ban is fully faithful and symmetric monoidal, so that it mak es no diﬀerence in which categor y we fo r m HP . Theorem 6.1 shows that HP ( I ) → HP ( E ) → HP ( Q ) is a c oﬁbre sequence for the pro -lo cally split exa ct categor y structure and a fortiori for the usua l exact category structure on Fr . Since C is bo th injective a nd pro jective a s an ob ject of Fr , this coﬁbre sequence induces lo ng exact sequences b oth in ho mology a nd in cohomolog y .  Corollar y 6.2 applies to all extensions o f F r´ echet algebr as with nuclear quotien t by Theore m 2.21. Let Y b e a closed subset of a smo o th manifold X . Let C ∞ 0 ( Y ; X ) b e the closed ideal of all smo oth functions on X that v anis h on Y and let C ∞ ( Y ) : = C ∞ ( X )  C ∞ 0 ( Y ; X ) . W e call C ∞ ( Y ) the alg ebra o f smo oth functions on Y . By des ign, the ca no nical homomorphism from C ∞ ( Y ) to the C ∗ -algebra of contin uous functions o n Y is injectiv e, that is, f = 0 in C ∞ ( Y ) if f ( y ) = 0 for all y ∈ Y . W e get a quo tien t ma p E ∞ ( Y ) → C ∞ ( Y ) beca us e J ∞ ( X ; Y ) is co nt ained in C ∞ 0 ( Y ; X ). Theorem 6.3. The quotient map E ∞ ( Y ) → C ∞ ( Y ) induc es a pr o-lo c al ly split quasi-isomorphi sm HP  E ∞ ( Y )  → HP  C ∞ ( Y )  and henc e isomorphisms HP ∗  E ∞ ( Y )  ∼ = HP ∗  C ∞ ( Y )  , HP ∗  E ∞ ( Y )  ∼ = HP ∗  C ∞ ( Y )  . T ogether with Theorem 5.6, this yields a formula for the p erio dic cy clic homolog y of C ∞ ( Y ). 30 RALF MEYE R Pr o of. The quotient map E ∞ ( Y ) → C ∞ ( Y ) is an op en s ur jection with kernel N : = C ∞ 0 ( Y ; X )  J ∞ ( X ; Y ). The e x tension N ֌ E ∞ ( Y ) ։ C ∞ ( Y ) is pro-lo cally split bec ause C ∞ ( Y ) is n uclear. Corollary 6.2 shows that HP  E ∞ ( Y )  → HP  C ∞ ( Y )  is a quasi-isomo r phism if and only if HP ( N ) is exact. W e cla im that the alge bra N is top ologica lly nilp otent: if p is any contin uo us semi-norm on N , then there is k ∈ N such that p v a nishes o n all pro ducts o f k elements in N . Since functions in N v anish on Y , products of k functions in N v anish on Y to or der k . These are annihila ted by p for suﬃciently high k beca use any contin uo us seminorm on N only in volv es ﬁnitely man y deriv atives. Since N is top ologic ally nilp otent, the as so ciated pro-alg ebra diss ∗ ( N ) is pro- nilpo ten t in the notation of [16]. Go o dwillie’s Theorem [16, Theorem 4.3 1 ] for pro-nilp otent pr o-algebr as asser ts that HP ( N ) is contractible.  Another ca se where we can prove excision inv olves ind-alg ebras. Let C b e as ab ov e and let − → C b e the ca tegory of inductive sys tems in C with the canonica l tenso r pro duct and the lo ca lly split extensions a s conﬂations . W e equip the categ ory of pro jective s ystems over − → C with the induced ex act ca tegory str ucture wher e the conﬂations are inductive systems of conﬂations in − → C . Theorem 6.4. L et I ֌ E ։ Q b e a lo c al ly split extension in − → C . The n the induc e d maps HP ( I ) → H P ( E ) → HP ( Q ) form a c oﬁbr e se quenc e. This yields a cyclic s ix-term exact se quenc e for HP ∗ . Pr o of. The pro of of the Excis ion Theorem in [14] yields this stronger result, as ob- served in passing in [1 6, Remark 4.43]. In fact, any pro of of the excision theorem for split ex tens ions of top olog ical algebra s that I know yie lds this stronger result. The idea of the a r gument is as follows. The pro of of the excisio n theor em for split ex - tensions is p otentially constructive in the sense that one can wr ite down an explicit contracting homotopy for the cone of the ma p HP ( I ) → cone  HP ( E ) → HP ( Q )  . This explicit formula only us es the m ultiplication in E and the sec tion s : Q → E of the extension b eca us e this is a ll the da ta that there is. The computation chec king that this for m ula works use s only the as s o ciativity o f the multiplication a nd the fact that s is a section b ecaus e that is all we know. When w e ha ve a loc ally split extension, we may still write down exac tly the same formula lo cally and ch eck that it still works where it is deﬁned. These lo ca lly deﬁned cont racting homotopies ma y be incompatible beca use we may use diﬀerent sections on diﬀerent entries of our inductive sys tem. Nevertheless, they provide a lo c al contracting homo to p y and thu s establish the de s ired coﬁbre sequence.  Theorem 6.4 applies , fo r instance, to extensions o f bo rnologic a l alg ebras w ith nu clear quo tient, which b eco me lo cally split extensions in − − → Ban . In particular , this cov ers the a lgebra of smo oth functions with compa ct suppor t C ∞ c ( Y ). 7. Conclusion and outloo k W e formulated and prov ed a gener al version of W o dzicki’s E xcision Theorem for the Ho chschild homolog y of pure alg ebra ex tensions with ho mologically unital kernel in any exact symmetric monoidal ca tegory . The main issue he r e w as to ﬁnd the right setup to formulate a gener al E x cision Theo rem that applies, in particular, to extensions of F r´ echet alg ebras with n uclear quotient. EXCISION IN HOCHSCHILD AND CYCLIC HOM OLOGY 31 As an application, we c o mputed the contin uous Ho chsc hild, cyclic a nd p erio dic cyclic homology for the F r´ e chet algebr a of Whitney functions on a closed subset of a smo oth manifold. Periodic cy c lic homology could satisfy an excision theorem for all pure a lgebra extensions in Q -linear symmetric monoida l categories . But I do not know how to establish suc h a gener al result. 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Excision in Hochschild and cyclic homology without continuous linear sections

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