Excision in cyclic homology theories

Transcription

1 Invent. math. 143, (2001) Digital Object Identifier (DOI) /s Excision in cyclic homology theories Michael Puschnigg Universität Münster, Mathematisches Institut, Einsteinstr. 62, Münster, Germany Oblatum 24-VI-1999 & 30-V-2000 Published online: 8 December 2000 Springer-Verlag 2000 Introduction In this paper we establish excision for Z/2Z-graded cyclic homology theories, including periodic, entire, asymptotic and local bivariant cyclic cohomology. Our work is a modification of the approach of Cuntz-Quillen [CQ] to excision in bivariant periodic cyclic homology and makes also use of the work of Wodzicki [Wo]. We study the dimension shift of periodic cyclic cohomology under the boundary map associated to an extension of algebras and obtain the possible values of this dimension shift. Excision in entire cyclic cohomology is used to calculate these groups in a number of hitherto unknown cases. Finally a bivariant and multiplicative Chern-Connes character on Kasparov s bivariant K-theory [Ka] with values in bivariant local cyclic cohomology is constructed. To put our approach into perspective we recall the previous work on excision in cyclic homology. Ever since the invention of cyclic homology by Connes and independently by Tsygan the problem of excision played a central role in the theory. It is concerned with the question of to what extent an extension 0 I A B 0 of algebras induces natural long exact sequences on cyclic homology groups. The first results on this problem were obtained by Wodzicki [Wo]. In 1987 he introduced the notion of H-unitality (homological unitality) for algebras and showed that excision in cyclic homology holds for all extensions of k-algebras (k Q a field) with k-linear section and with fixed kernel I, if and only if I is H-unital as a k-algebra. This condition is however rarely satisfied in practice. For example excision in cyclic homology does not hold for nilpotent extensions because nilpotent algebras are never H-unital (unless they are zero).

2 250 M. Puschnigg If one considers periodic cyclic homology instead, the situation becomes different. A theorem of Goodwillie says that epimorphisms with nilpotent kernels are periodic cyclic homology equivalences and therefore nilpotent extensions satisfy excision in the periodic theory. One can thus hope that stronger excision results hold in the periodic setup, and in fact Cuntz and Quillen [CQ] succeeded in 1994 in proving that arbitrary extensions of k-algebras (k Q a field) with k-linear section satisfy excision in (bivariant) periodic cyclic cohomology. Their proof uses their approach to cyclic homology via quasifree resolutions and the X-complex which they apply in the category of k-pro-algebras. It is based on the fact that in the category of pro-algebras every extension is HP-equivalent to an extension of pro-algebras with asymptotically H-unital kernel. This result immediately raises the question whether excision holds for other Z/2-graded versions of cyclic homology as the entire, asymptotic or local cyclic theories [Co], [CM], [Pu1], [Pu2]. The Cuntz-Quillen proof however does not carry over to the topological framework addressed in this paper. Before explaining our proof we want to outline the ideas of the work of Wodzicki and Cuntz-Quillen, which are relevant for us, in more detail. Let 0 ĈC (A, B) ĈC (A) ĈC (B) 0 be the extension of periodic cyclic bicomplexes associated to an extension of k-algebras. Excision is equivalent to the statement that the natural inclusion ĈC (I ) ĈC (A, B) is a continuous chain homotopy equivalence. Wodzicki considers the excision problem in Hochschild homology first and then passes to the cyclic case. He introduces an increasing Wodzicki filtration of the associated relative complex C (A, B) starting with C (I ) such that the associated graded complex is isomorphic (in positive degree) to a direct sum of (shifted) tensor powers of the bar complex Gr W (C (A, B)/C (I )) i C bar (I ) n i. He calls IH-unital if C bar (I ) is acyclic. H-unitality of I implies that C (I ) is a deformation retract of C (A, B) as well as the cyclic analogue. This establishes excision both in Hochschild- and cyclic homology for extensions with H-unital kernel. Cuntz and Quillen begin by proving excision in periodic cyclic homology for arbitrary extensions of quasifree algebras. If 0 I R B 0 is an extension of k-algebras with R quasifree [CQ1], they construct an associated extension of pro-algebras 0 lim I n R lim R/I n 0 n n

3 Excision in cyclic homology theories 251 and observe that lim I n is H-unital in the category of proalgebras. n lim R/I n is equivalent to B and lim I n is equivalent to I in periodic n n cyclic homology by a variant of Goodwillies theorem on the invariance of periodic cyclic homology under nilpotent extensions. From this they deduce excision for the considered extension of quasifree algebras by an argument analogous to that of Wodzicki. In order to apply it in a bivariant setting (i.e. on the level of chain complexes and not only on homology) Cuntz and Quillen use in a crucial manner the fact that every ideal in a quasifree algebra R is projective as an R-left module. The general case of excision is derived from this by showing that every extension is equivalent in periodic cyclic cohomology to an extension of quasifree algebras. Our approach to excision is based on the notion of a free ideal extension. An extension of k-algebras with k-linear section 0 J A B 0 is called a free ideal extension if the ideal J is free as A left-(or right) module. So this class is close to the class of quasifree extensions considered by Cuntz and Quillen. There are however quasifree extensions with projective but not free kernel and on the other hand there exist free ideal extensions of algebras which are not quasifree (for example the universal multiplicatively split and the universal invertible extensions). Similar to Wodzicki we analyze the excision problem for the Hochschild complex first and then pass to the cyclic bicomplex. We look for pairs (C (A, B), φ) consisting of an intermediate relative subcomplex C (A, B) of the relative Hochschild complex C (A, B) which contains C (J) and achainmapφ : C (A, B) C (A, B) satisfying φ is naturally chain homotopic to the identity. φ equals the identity on C (J). φ maps C (A, B) onto C (J). The identity fulfills this requirement for C (A, B) = C (J) whereas the existence of a chain map φ in the case C (A, B) = C (A, B) would be equivalent to excision in Hochschild homology. According to Wodzicki the obstruction against the existence of φ for a given intermediate relative complex C (A, B) is encoded in the bar homology of the ideal J. It turns out that the bar homology of a free ideal, although being nontrivial, is of a particularly simple form which allows to find large intermediate subcomplexes in the Hochschild complex and in the cyclic bicomplex. The homotopy invariance of periodic cyclic homology implies that large intermediate cyclic bicomplexes, and consequently also the cyclic bicomplex of the free ideal, are deformation retracts of the full relative cyclic bicomplex. Excision for free ideal extensions results. The general case follows from this as in the approach of Cuntz and Quillen.

4 252 M. Puschnigg The central part of our proof consists therefore of a detailed analysis of the bar complex C bar (J) of a free ideal J. The bar homology of an ideal (or algebra) can be described independently of the complex used to define it as the Tor-group Tor +1 (k, k). Cal- J culating these Tor groups via the standard bar resolution of k over J one recovers the bar complex of J. However, for a free ideal J, thej-module k possesses a different and smaller free resolution, depending naturally on the choice of an A-basis of J, which leads to a new complex C small (J) calculating the Tor-groups in question. The differentials of this complex are zero and the J-adic valuation of it satisfies Fil +2 J C small (J) = 0. For the original bar complex this implies that there exists a natural harmonic decomposition C bar (J) h( C 1 bar (J)) ( C +1 bar (J)) where the harmonic subspace is naturally isomorphic to the bar homology H bar (J). The canonical harmonic projection P onto the harmonic subspace is a chain map which is naturally chain homotopic to the identity. This is analogous to the Hodge decomposition in differential geometry where the operator h equals the adjoint d of the differential d. For the bar homology of a free ideal this implies that H bar (J) is different from zero (and actually infinite dimensional if B or J are so) in every degree. But, on the other hand, Fil +2 J H bar (J) = 0. So the harmonic projection of C bar (J) annihilates the subcomplex the C,bar > (J) := Fil +2 J C bar (J). All this is treated in Sect. 1 of this paper. In Sect. 2 we study the relative Hochschild- and cyclic complexes associated to a free ideal extension. The associated graded complex of the Hochschild-complex (with respect to the Wodzicki-filtration) is isomorphic (modulo a naturally contractible complex) to a direct sum of tensor powers of the bar complex C bar (J). We deduce from this the existence of a natural chain map φ : C (A, B) C (A, B) satisfying the following conditions φ is naturally chain homotopic to the identity. φ preserves Wodzicki- and J-adic filtrations. On the associated graded complex Gr W (C (A, B)) i Cbar (I ) n i the operator φ equals Gr W (φ) = i P n i where P is the harmonic projection of the bar complex of the free ideal J. If C > (A) J := Fil +1 J (Ω (A)+dΩ (A)) denotes the linear span of chains whose J-adic valuation is strictly larger than their Hochschild degree then C > (A) J is a complex containing C (J) and φ maps C > (A) J onto C (J). φ is continuous with respect to I-adic topologies associated to auxiliary ideals I J of A.

5 Excision in cyclic homology theories 253 As the basic chain map φ commutes with the cyclic differential, one obtains an intermediate cyclic bicomplex (CC > (A) J,φ)satisfying similar assertions. In Sect. 3 we use this to prove excision in periodic cyclic homology. The homotopy invariance of periodic cyclic homology ([Go]) implies that the associated graded complex Gr J (ĈC (A, B)) with respect to the J-adic filtration is naturally contractible. By iteration one obtains that the intermediate cyclic bicomplex ĈC > (A) J, and therefore also the cyclic bicomplex ĈC (J) of the ideal J, are natural deformation retracts of the relative cyclic bicomplex ĈC (A, B). This establishes excision in bivariant periodic cyclic homology for free ideal extensions, and the general case follows from this as in [CQ]. In Sect. 4 we analyze the behaviour of the boundary map in periodic cyclic (co)homology with respect to the dimension filtration, which is induced by the Hodge-filtration on the cyclic bicomplex. Note that these dimension shifts are independent of the particular construction of the boundary map: excision means that the natural inclusion ĈC (I ) ĈC (A, B) is a chain homotopy equivalence and we are asking how the induced isomorphism on (co)homology shifts dimensions. Given an extension 0 I A B 0 and a homology class α HP n (B) of dimension n, the dimension of δ(α) usually depends both on the particular extension and on the choice of α. We exhibit however a class of extensions for which the boundary map shows a remarkable rigidity. If 0 I R S 0 is an extension such that R is quasifree [CQ1] and such that δ: HP (S) HP 1 (I ) is injective, then for any α HP (B) the dimension of δ(α) only depends on the dimension of α. The precise dimension formula (an approximate formula is dim α = 3dimδ(α)) is given in Theorem 4.4. This result determines the behaviour of the bivariant Chern-Connescharacter of [Cu1] with respect to the natural filtrations on kk-theory and bivariant periodic cyclic cohomology. This was posed as a problem by Cuntz [Cu1]. The fact that the homological shift function is not one to one implies that there cannot exist a similar exact shift formula in cohomology. For arbitrary extensions with cohomological boundary map δ: HP (I ) HP +1 (B) the results of Sect. 3 establish the estimate dim(δ(α)) 3dim(α) + 3, which even holds on the level of chain complexes. We show that it is actually sharp by providing sufficiently many examples of extensions with maximal dimension shift. A typical extension where a maximal dimension shift occurs is the universal extension 0 JA n TA n A n 0 of [Cu] for A n = Σ 3n+3 C,the(3n+3)-fold algebraic suspension of C [Wo].

6 254 M. Puschnigg There exist better bounds for invertible and multiplicatively split extensions which will be presented elsewhere. A similar sharp estimate for the iterated boundary map in higher extensions is also given. The optimal bound for the dimension shift of the boundary map had been conjectured in an earlier version of this paper [Pu2]. The predicted value for the dimension shift of the iterated boundary map given there was however incorrect. In Sect. 5 we show that our proof of excision applies also to bivariant entire, bivariant asymptotic and bivariant local cyclic cohomology of Indalgebras equipped with systems of supports. We do this by verifying the continuity of the chain maps used in the previous sections with respect to the appropriate topologies. This is possible because the harmonic decomposition of the bar complex of a free ideal is given by completely explicit formulas. In Sect. 6 we present two consequences of the general excision theorems of Sect. 5. By comparing the functorial behaviour of periodic and entire cyclic cohomology we answer an old question of Connes: the entire cyclic cohomology of the algebra of smooth functions on a smooth compact manifold without boundary equals its continuous periodic cyclic cohomology and is thus isomorphic to the Z/2-graded singular homology of the manifold. Cuntz [Cu] characterized Kasparov s bivariant K-theory [Ka] as the universal stable homotopy bifunctor which satisfies excision (with respect to suitable extensions of C -algebras). By results of Sect. 5 and of [Pu1] these properties are satisfied by bivariant local cyclic cohomology which finally leads (see [Cu1]) to a Grothendieck-Riemann-Roch Theorem in Noncommutative Geometry: there exists a bivariant Chern-Connes character ch biv : KK (, ) HClc (, ) on Kasparov s bivariant KK-theory with values in bivariant local cyclic cohomology. It is multiplicative up to the period factor 2πi, extends the ordinary Chern character, and is compatible with boundary maps in six term exact sequences. A stable version of this theorem was already established in [Pu]. From there it is known that ch biv : KK (, ) Z C HClc (, ) is an isomorphism on a large class of C -algebras. The results of Sects. 2, 3 and 5 in this paper were presented in the Seminar Nichtkommutative Geometrie at Münster in the fall of This paper is the extended and revised version of the preprint [Pu2]. I want to thank Joachim Cuntz for fruitful discussions and constructive comments and criticism on an earlier version of this paper. I understand that Ralf Meyer has obtained another proof of excision in cyclic homology theory which is based on quasifree resolutions and the X-complex and thus in spirit closer to the work of Cuntz and Quillen. It is a pleasure for me to thank Hildegard Eissing for her excellent typing, lots of coffee and the friendly atmosphere at the institute. I also

7 Excision in cyclic homology theories 255 thank Nina Riedel for typing parts of the paper. Finally I want to thank my wife Wieke Benjes for her proof reading and for mastering the mysteries of the computer. Remark: We work throughout this paper in the category of nonunital algebras over a field k of characteristic zero. Contents 1 Barhomologyoffreeideals Maps of Hochschild and cyclic complexes associated to a free ideal extension Excisioninperiodiccyclichomology Dimension shifts under the boundary map in periodic cyclic homology Excision in cyclic homology theories for topological algebras Consequences Bar homology of free ideals In this section we will analyze in detail the bar homology of a certain class of ideals. These ideals are far from being H-unital but their bar homology and therefore the obstruction to excision in cyclic homology for extensions associated to such ideals is of a particularly simple form. The bar homology of ideals is known to play a crucial role in excision problems for cyclic homology. In fact for a k-algebra I Wodzicki [Wo] has established the equivalence of the following two assertions: Every extension 0 I R S 0ofk-algebras with kernel I satisfies excision in cyclic homology, i.e. induces long exact sequences of cyclic homology groups. I is H-unital, i.e. the bar homology of I vanishes. The bar homology H bar (A) of an algebra A has been introduced as the homology of the chain complex C bar n (A) := A (n+1) with n 0, n 1 n (a 0... a n ) := ( 1) i a 0... a i a i+1... a n. i=0 A more conceptual way to define it is the following. Let A be a k-algebra and let à be the algebra obtained from A by adjoining a unit. The natural extension of k-algebras 0 A à k 0 turns k and A into Ã-(left)-modules. There is a natural isomorphism H bar (A) Torà (k, A) Torà +1 (k, k).

8 256 M. Puschnigg 1.1 Free ideals We define now the class of ideals investigated in this section. Definition 1.1 An ideal J of an algebra A is called (left-) free iffitisfree as A (left-)module. This means that there exists a k-linear subspace V J such that the algebra multiplication of A defines an isomorphism à k V J of Ã-modules. Such a linear subspace V is called a basis of J. The notion of right-free ideals is defined similarly. In the sequel we will suppose that all free ideals J are free as A leftmodules and that some basis V J has been fixed. The distinguished class of extensions for which an explicit proof of excision in periodic cyclic homology will be given is the following: Definition 1.2 A free ideal extension of k-algebras is an extension 0 J A B 0 of k-algebras with k-linear section such that J is a free ideal in A. Thus a free ideal extension consists of the following data: An extension of k-algebras 0 J A f B 0 A k-linear section s : B A of f. A k-linear basis V of J as A-left-module. The k-linear subspace s(b)v which is the isomorphic image of s(b) k V under the algebra multiplication of A will be denoted by W. A map of free ideal extensions is a map of extensions preserving all additional structures. Lemma 1.3 Let 0 J A B 0 be a free ideal extension. Then a) J n is a free ideal of A with basis V n. b) J n = J k V n k J k V (n k) for 0 k nwherej 0 := Ã, V 0 := k. c) There exists a natural k-linear decomposition J V W J 2 ( J V ) W Here naturality means naturality with respect to maps of free ideal extensions. d) For every integer m 1 the ideal J possesses a natural k-linear decomposition m 2 J J m (V W)V j. j=0

9 Excision in cyclic homology theories 257 Proof: Fix k, 0 k n. Because J is free the iterated multiplication defines an isomorphism J k V (n k) J k V n k J n. On the other hand J n = J n 1 J = J n 1 (ÃV) = (J n 1 Ã)V = J n 1 V, and iteration shows that J n = J k V n k. This proves b) and a) is just the special case k = 0ofb).Thek-linear decomposition of Ã: à k s(b) J provides a decomposition à k V k k V s(b) k V J k V. The algebra multiplication of A maps this isomorphically onto J = à V V W JV = V W J 2 = ( J V ) W Part d) follows inductively from b) and c). After having established the basic notions of free ideals and free ideal extensions, we begin our analysis of the bar complex of a free ideal. 1.2 The small resolution of a free ideal We will construct a natural resolution of a free ideal J by free J-modules. Lemma 1.4 Let 0 J A B 0 be a free ideal extension of k-algebras. a) The map 0 : J k V J k W J (a v, a v ) av + a v is an epimorphism of J-modules. b) The linear decompostition of (1.3) c) defines a k-linear section s 0 of 0 with image J k V k W J k V J W. c) The natural projection J k V J k W J k WofJ-modules maps Ker 0 isomorphically onto the module J k W. We denote this isomorphism by α : Ker 0 J k W. Its inverse is given on elementary tensors by α 1 : j s(b)v j s(b)v js(b) v. Proof: Clearly 0 is a map of J-modules. Its surjectivity follows from b) which is a consequence of 1.3 c). By construction (Id s 0 0 )( J k (V W)) J k (V W) so that the projection considered in c) maps Ker 0 to J k W. Moreover J k V Ker 0 = 0 because the multiplication map

10 258 M. Puschnigg is injective on J k V. Therefore the projection defines a monomorphism α : Ker 0 J k W. Define a map of J-modules by α 1 : J k W Ker 0 j s(b)v j s(b)v js(b) v. It is well defined because s(b) k V W under the multiplication. Clearly α 1 is a right inverse to α and because α is injective α 1 is also a left inverse of α. Thus α and α 1 are isomorphisms inverse to each other. Denote the module J k (V W) by P0 small (J). It is a free J-module with basis V W. The epimorphism 0 : P0 small J 0 therefore can be considered as first step of a J-free resolution of J. By 1.4 c) the kernel of 0 is isomorphic as J-module to a direct sum of copies of J. Thus one obtains by iteration an infinite J-free resolution of J. The construction is summarized in Definition and Proposition 1.5 (Small Resolution) Let 0 J A B 0 be a free ideal extension of k-algebras. Let V J be the basis of J and let W := s(b)v. a) For n 0 define free J-modules Pn small (J) by Pn small (J) := J k (V W) k W n. A canonical basis of Pn small (J) is given by (V W) W n. b) For n > 0 define homomorphisms of J-modules n : P small n (J) P small n 1 (J) by n := ( α 1 id (n 1) W ) ( 0 id n ) W where 0 and α are defined in (1.4). c) The complex 0 0 J P small 1 n 0 (J) P small n (J) n+1 is acyclic and defines a resolution P small (J) of J by free J-modules. It is called the small resolution of J. It is natural with respect to maps of free ideal extensions. d) The small resolution is k-linearly contractible. A natural contracting homotopy is provided by the k-linear maps s n, n 0 where s 0 is defined in (1.4) b) and for n > 0 s n : P small n 1 s n := ( s 0 id n W (J) Psmall n (J) equals ) ( p id (n 1) ) W where ( p : J (V W) J W is the natural projection. e) n P small n (J) ) J Pn 1 small (J).

11 Excision in cyclic homology theories 259 Proof: Assertions a) and b) are clear. The fact that P small (J) is a complex follows from the identity 0 α 1 = 0ofLemma1.4c). It is clearly natural with respect to maps of free extensions. Its acyclicity follows from d). To check d) note that n+1 s n+1 = (α 1 p) id n W and s n n = (s 0 0 ) id n W. Thus it suffices to show that 1 s 1 +s 0 0 = Id P small. 0 Now on the subspace Ker 0 of P0 small 1 s 1 + s 0 0 = α 1 p + s 0 0 = α 1 p = id holds by Lemma 1.4 c). On the complementary subspace s 0 (J) P small 0 (J) 1 s 1 + s 0 0 = α 1 p + s 0 0 = s 0 0 = id. holds. Finally e) is clear from the definitions. Adic filtrations of the small resolution. The small resolution carries natural adic filtrations. Definition 1.6 a) Define a decreasing filtration Fil J, the J-adic filtration of P small (J) by Fil m J ( P small n (J) ) := J m n 1 Pn small (J) where the notation J l := Ãforl 0 is understood. b) If I J A is an auxiliary ideal of A, I = J, define the I-adic filtration of P small (J) by Fil m I ( P small n (J) ) := I m Pn small (J). Lemma 1.7 a) The J-adic filtration of the small resolution is preserved by the differential and the linear contracting homotopy s of P small (J). b) For I J, I = J the I-adic filtration of P small is preserved by the differential. For the contracting homotopy one has s(fil m I Psmall (J)) Fil m 1 I P small (J). We will use now the small resolution to investigate the structure of the bar complex of a free ideal. Comparison with the standard resolution. Recall that there is a standard resolution of a k-algebra J by free J-modules, the Bar-resolution P bar (J). It is defined as Pn bar (J) := J J (n+1), which is a free J-module with

12 260 M. Puschnigg canonical basis J (n+1). The differentials are given by n = b : P bar n (J) P bar n 1 (J) j 0... j n+1 n i=0 ( 1)i j 0... j i j i+1... j n+1. A k-linear contracting homotopy is provided by s n = s : P bar n 1 (J) Pbar n (J) j 0... j n 1 1 j 0... j n 1. It satisfies s n+1 s n = 0, n 0. If I J is an auxiliary ideal (here the case I = J is allowed), the I-adic filtration of P bar is defined by ( Fil m I P bar n (J) ) := I m 0 I m 1... I m n+1 m m n+1 =m m i 0 where the convention I 0 = J holds. The I-adic filtration is preserved by the differentials and the contracting linear homotopy of P bar. Let now A be a k-algebra and let M P, N Q be resolutions of A- modules M, N by based, free A-modules. Suppose that k-linear contracting homotopies of both resolutions have been fixed. Under these assumptions every A-module map f : M N possesses a canonical extension to a map f : P Q of resolutions: if f ˆ has been constructed inductively for < n define f n on the fixed basis V n of P n by f n Vn := sn Q f n 1 n P and extend it to an A-module map. Similarly if ρ : P P is a map of resolutions covering the zero map of M then ρ is chain homotopic to zero via a canonical chain homotopy. Applied to the bar resolution and the small resolution of free ideals the discussion can be summarized as follows: Definition and Proposition 1.8 Let 0 J A B 0 be a free ideal extension of k-algebras and let P bar (J) and P small (J) be the barresolution respectively the small resolution of J by free J-modules. a) There exists a canonical and natural map of resolutions ϕ : P bar (J) P small (J) covering the identity of J. b) There exists a canonical and natural map of resolutions ψ : P small (J) P bar (J) covering the identity of J. c) ϕ ψ = Id P small (J ). d) There exists a natural and canonical k-linear operator h : P bar (J) P +1 bar (J) defining a chain homotopy between ψ ϕ and the identity: Id ψ ϕ = b h + h b.

13 Excision in cyclic homology theories 261 e) The A-module maps ϕ,ψ and h preserve the J-adic filtrations of P bar (J) and P small (J). f) Let I J, I = J be an auxiliary ideal of A. The map ψ preserves I-adic filtrations. For the other maps the inclusions ϕ ( Fil m I Pbar n h ( Fil m I Pbar n (J) ) Fil m n 1 I Pn small (J) (J) ) Fil m n 1 I Pn+1 bar (J) hold. Here naturality means naturality with respect to maps of free ideal extensions. Proof: Assertions a), b), and d) follow from the preceding discussion. As assertion c) will not be needed in the sequel we leave its proof to the reader. Finally e) and f) follow from Lemma 1.7 and the explicit construction of the maps ϕ,ψ and h. 1.3 The bar homology of free ideals We are now ready to calculate the bar homology of free ideals. It turns out to be rather pathological as it consists only of the obvious classes, i.e. those cocycles which cannot be boundaries by comparison of J-adic valuations. The crucial fact however is that cycles of high J-adic filtration do not contribute to the bar homology. This will be a central point in our approach to excision as it implies that on subcomplexes of the cyclic bicomplex of high J-adic valuation free ideals behave as if they were H-unital. Lemma 1.9 Let 0 J A B 0 be a free ideal extension of k- algebras with k-linear section s. Let V be the basis of J and let W := s(b)v. Let P small (J) be the small resolution of J by free J-modules. Put C small (J) := k J Psmall (J). a) The complex C small (J) is given in degree n by Cn small (J) = (V W) W n and the differential of C small (J) vanishes. b) The J-adic filtration of P small (J) induces a J-adic filtration of C small (J) satisfying Fil m J Csmall n (J) = 0 for m > n + 1. Proof: The vanishing of the differentials of C small (J) follows from 1.5 e) because J is the kernel of the augmentation map J k. Similarly b) holds by definition of the J-adic filtration (see Definition 1.6 a)).

14 262 M. Puschnigg Note that k J Pbar (J) = C bar (J) is the bar complex of J. Theorem 1.10 Let 0 J A f B 0 be a free ideal extension of k-algebras. Let V be the A-basis of J and let s be the k-linear section of f. Let W := s(b)v. a) For all n 0 there is a natural isomorphism H bar n (J) (V W) k W n. b) The J-adic filtration of the bar complex C bar (J) induces a filtration of the bar homology H bar (J). It satisfies Fil m J Hbar n (J) = 0 for m > n + 1. Proof: As the bar resolution of a free ideal is naturally chain homotopy equivalent to the small resolution one deduces that the complexes C bar (J) and C small (J) are naturally chain homotopy equivalent. Thus Hn bar (J) = H n (C bar (J)) H n(c small (J)) = Cn small (J) (V W) W n because the differentials of C small (J) vanish. As moreover the natural chain homotopy equivalences between C bar (J) and C small (J) preserve J-adic filtrations one finds that Fil m J Hbar n (J) Fil m J H n(c small (J)) Fil m J (Csmall n (J)) = 0form > n + 1byLemma1.9b). We note the following implication for the structure of proper free ideals. Corollary 1.11 The unitalization J of a proper free ideal J A, 0 = J = A, is of infinite projective dimension as bimodule over itself. R R op Proof: The identity Torn+1 ( R, k) Hn bar (R) (consider the standard resolutions) shows that the bar homology of a k-algebra R vanishes in degrees above its projective dimension as bimodule over itself, i.e. the minimal length of a projective resolution of R in the category of R-bimodules. The conclusion is then a consequence of The harmonic decomposition of the bar complex of a free ideal The comparison of the bar complex with the small complex does not only allow to determine the bar homology of J. The fact that the differential of the small complex vanishes implies the existence of a natural Hodge-type decomposition of the bar-complex which, together with the result about the J-adic filtration, provides more profound information about the bar complex of free ideals than the bare calculation of its homology.

15 Excision in cyclic homology theories 263 Lemma 1.12 Let (C, ) be a complex in an additive category in which projectors have an image. Then the following two assertions are equivalent: a) C is chain homotopy equivalent to a complex with vanishing differential. b) C possesses a Hodge type decomposition, i.e. there exists a morphism h : C C +1 of degree one such that C Im h 1 Im +1 Ker = Im, Ker h = Im h Ker Ker h = Ker /Im The canonical projections onto the harmonic subspace and the spaces Im and Im h are given by P := Id (h + h), h and h, respectively. Proof: a) implies b): Let D be a complex with vanishing differential and let ϕ : C D, ψ : D C be chain maps homotopy inverse to each other. Then ϕ ψ is chain homotopic to Id D and equals thus the identity because the differential of D vanishes. Consequently the endomorphism P := ψ ϕ of C is idempotent and C decomposes as the direct sum C Im P Im (1 P). It clearly suffices to show that each of these summands possesses a Hodge decomposition. The differential of the complex Im P vanishes so that it has the tautological and unique Hodge decomposition Im P =. As the projector P is chain homotopic to the identity of C the complementary complex Im (1 P) is contractible. It remains to verify that contractible complexes possess a Hodge decomposition. Let C be a contractible complex and let h be a contracting chain homotopy for C. Then it is straightforward to verify that h := h h is a homotopy operator defining a Hodge decomposition as asked for in b). b) implies a): Suppose that C possesses a Hodge decomposition. Then the harmonic projection P := Id ( h + h ) defines a deformation retraction of C onto the subcomplex with differential zero. As a consequence we obtain a Hodge type decomposition of the bar complex of a free ideal. Theorem 1.13 Let 0 J A B 0 be a free ideal extension of k-algebras and let C bar (J) be the bar complex of J. Then there exists an operator h : C bar (J) Cbar +1 (J) satisfying a) The bar complex possesses the Hodge type decomposition C bar (J) = Im h 1 Im +1 where

16 264 M. Puschnigg The harmonic subspace is isomorphic to the bar homology H bar (J) of J Ker = Im Ker h = Im h The canonical projections onto, Im and Im h are given by the harmonic projection P := Id (h + h) and the projections h and h, respectively. b) The operator h is natural with respect to maps of free ideal extensions. c) Let Fil J be the J-adic filtration and let P be the harmonic projection of C bar(j). Then P ( Fil m J Cbar n (J) ) = 0 for m > n + 1. d) The operator h preserves the J-adic filtration and if I J is an auxiliary ideal of A then h ( Fil m I Cbar n (J) ) Fil m n 1 I Cn bar (J). Proof: Recall the maps of resolutions constructed in 1.8 and let ϕ : C bar (J) Csmall (J), ψ : C small (J) C bar(j) and h : C bar(j) C +1 bar (J) be the induced maps of the associated bar- respectively small complexes. Then ϕ and ψ are chain maps homotopy inverse to each other, a connecting homotopy between ψ ϕ and the identity of C bar (J) being given by h. As the differential of the small complex vanishes 1.9 the previous Lemma 1.12 applies and shows the existence and naturality of the Hodge decomposition. Part c) is a consequence of 1.10 and d) follows either from 1.8 f) after verifying that h := h h = h or from a direct inspection of the explicit formulas for h given in the next subsection. Explicit formulas for the harmonic projection. The Hodge decomposition of the bar complex of a free ideal can be calculated explicitely. Apart from formulas for the harmonic projection and the canonical homotopy operator we give a particularly simple characterization of the image of the latter one. Proposition 1.14 Let 0 J A B 0 be a free ideal extension of k-algebras with k-linear section s, let V be the A-basis of J and let W := s(b)v. Recall the k-linear decomposition J = V W J 2 of (1.3) c) and let the corresponding linear decomposition of C bar (J) into irreducible subspaces be understood. a) The harmonic projection P n of Cn bar (J) onto its harmonic subspace n vanishes on all irreducible subspaces of Cn bar (J) except those contained in (V W) 0 W 1 W n

17 Excision in cyclic homology theories 265 where it is given on simple tensors by the inductive formulas P 0 (α 0 ) = α 0 P n (α 0 s(b 1 )v 1 s(b 2 )v 2 s(b n )v n ) = α 0 P n 1 (s(b 1 )v 1 s(b 2 )v 2 s(b n )v n ) α 0 s(b 1 ) P n 1 (v 1 s(b 2 )v 2 s(b n )v n ). b) An explicit formula for the harmonic projection on simple tensors as considered in a) is given by S {1,...,n} P n (α 0 s(b 1 )v 1 s(b 2 )v 2 s(b n )v n ) = ( 1) S α 0 s(b 1 ) χ S(1) s(b 1 ) (1 χ S(1)) v 1 s(b 2 ) χ S(2)... s(b n ) (1 χ S(n)) v n where S runs over all subsets of {1,...,n} and χ S denotes the characteristic function of S {1,...,n}. c) The homotopy operator h n : Cn bar (J) Cn+1 bar (J) of the harmonic decomposition (1.13) vanishes on all irreducible subspaces of Cn bar (J) except those contained in J 0 J k 1 J 2 k W k+1 W n, 0 k n for which it is given on simple tensors by the formula h n ( j 0 j k 1 α k v k s(b k+1 )v k+1 s(b n )v n ) = = ( 1) k j 0 j k 1 α k P n k (v k s(b k+1 )v k+1... s(b n )v n ). d) The subspace Im (h n 1 ) C bar n (J) of the harmonic decomposition (1.13) equals the direct sum of the subspaces J 0 J k 1 V k W k+1 W n, 0 < k n. The bar complex of locally convex algebras. Let A be a complete, locally convex algebra. The continuous bar complex of A is then defined in degree n by the complete projective tensor product Cn bar (A) cont := A n+1 π,the differential being given by the same formula as in the algebraic case. Definition 1.15 a) A closed ideal J in a complete, locally convex algebra A is called free if there exists a closed linear subspace V J such that the multiplication in A defines a continuous isomorphism à π V J.

18 266 M. Puschnigg b) A free ideal extension of complete, locally convex algebras 0 J A f B 0 is an extension with a fixed continuous linear section s of f and a fixed closed subspace V J turning J into a free ideal of A with basis V. All statements of Sect. 1 carry over to the continuous bar complex of complete, locally convex algebras. In particular, one has Theorem 1.16 Let 0 J A B 0 be a free ideal extension of complete, locally convex algebras. Then the continuous bar complex C bar(a) cont of the free ideal J possesses a Hodge decomposition as described in (1.13). The harmonic projection and the associated homotopy operator are given by the same formulas as in (1.14). 2 Maps of Hochschild and cyclic complexes associated to a free ideal extension In this section we analyze the relative Hochschild- and cyclic complexes associated to a free ideal extension 0 J A B 0. The Wodzickifiltration of the relative complexes is used to relate them to the bar complex C bar (J). We introduce the intermediate Hochschild-complex C> (A) J of chains of high J-adic valuation in the relative Hochschild complex C (A, B) and construct a natural chain map φ : C (A, B) C (A, B) which is chain homotopic to the identity and projects the intermediate relative complex C > (A) J onto C (J). Themapφ is shown to commute with the cyclic differential and to satisfy a similar assertion with respect to the intermediate cyclic bicomplex ĈC > (A) J in the relative periodic cyclic bicomplex ĈC (A, B). We begin by recalling some basic notions about Hochschild and cyclic homology [Co]. Definition 2.1 Let A be a k-algebra. a) The A-bimodule of algebraic differential n-forms over A is defined as Ω n A := Ã A n A n+1 A n where the identifications =: C n C n a 0 da 1...da n a 0 a 1 a n da 1...da n a 1 a n are understood. The A-bimodule structure is the obvious one. b) The Hochschild complex of A is defined as (C (A), ) := (Ω A, b) b(ωda) := ( 1) ω [ω, a].

19 Excision in cyclic homology theories 267 The linear decomposition of the space of algebraic differential forms induces a linear decomposition of the Hochschild complex C (A) C (A) C (A). The spaces C (A) form a subcomplex of C (A), the quotient complex C (A)/C (A) is isomorphic to the bar complex Cbar (A). c) The Z/2-graded chain complex (CC (A), ) := ( C +2n (A), b + B ) ( Ω +2n A, b + B ) with n=0 B(a 0 da 1...da n ) := n=0 n ( 1) in da i...da n da 0...da i 1 i=0 is the (noncomplete) cyclic bicomplex of A. It is an acyclic chain complex. Connes differential B is of degree one and anticommutes with the Hochschild differential b of degree minus one. d) The Hodge-filtration of CC (A) is the decreasing filtration defined by the subcomplexes Fil m Hodge CC (A) := bω m (A) Ω k (A) The completion with respect to the Hodge-filtration is denoted by ĈC (A). This complex of adically complete vector spaces calculates the periodic cyclic homology HP (A) of A. e) Any ideal I A defines an I-adic filtration of the cyclic bicomplex CC (A). The completion of CC (A) with respect to the filtration generated by Hodge- and I-adic filtration is the complete cyclic bicomplex ĈC (Â) of the I-adic completion  of A. It calculates the periodic cyclic homology HP (Â) of Â. f) If A is a complete, locally convex algebra then the continuous Hochschild and cyclic complexes are defined similarly by using the bimodule of continuous algebraic differential forms Ω n A cont := à π A n π A n+1 π k=m A n π g) The Hodge filtration of the continuous cyclic bicomplex CC (A) cont is given by the closure of the Hodge filtration on CC (A). Let 0 J A B 0 be an extension of k-algebras with k-linear section s. Then the linear decomposition A J s(b) induces a linear decomposition of the Hochschild complex C (A) : C n (A) à A n (k J s(b)) (J s(b)) n.

20 268 M. Puschnigg In the rest of this paper it is understood that elements of C (A) are decomposed according to this splitting so that each factor of a tensor monomial α = a 0... a n is supposed either to belong to J or to s(b). The maximal strings a i... a j of factors of α, counted in cyclic order, with a i J,...,a j J are called the blocks of α. The following intermediate relative Hochschild- and cyclic complexes will play a central role in our approach to excision in cyclic homology theories. 2.1 Intermediate complexes associated to an extension Definition and Lemma 2.2 Let 0 J A B 0 be an extension of k-algebras and let C (A) be the Hochschild complex of A with its J-adic filtration Fil J.Letν : N N be a monotone increasing function. a) The spaces C >ν n (A) J := Fil ν(n)+1 J C n (A) Filν(n 1)+1 J C n (A) form a subcomplex C >ν(a) J of the Hochschild complex C (A). The quotient complex C (A)/C >ν(a) J is denoted by C ν(a) J. b) One has C >ν(a) J C >ν (A) J for ν ν. The complex C >ν(a) J associated to the choices ν(n) = n respectively ν(n) = n 1 will be denoted by C > (A) J respectively C (A) J. c) If the given extension of algebras is a free ideal extension then the extension of complexes 0 C >ν (A) J C (A) C ν (A) J 0 possesses a natural linear section. d) The space CC >ν (A) J := n=0 C >ν +2n (A) J defines a subcomplex of CC (A). Denote the quotient complex by CC ν(a). e) Suppose that 0 J A B 0 is a free ideal extension. Then the extension of cyclic bicomplexes 0 CC >ν (A) J CC (A) CC ν (A) J 0 possesses a natural k-linear section.

21 Excision in cyclic homology theories 269 f) The natural linear section of e) is continuous with respect to the Hodgefiltration and with respect to the I-adic filtration associated to any ideal I J of A. Therefore the extensions and 0 ĈC >ν (A) J ĈC (A) ĈC ν (A) J 0 0 ĈC >ν (Â) J ĈC (Â) ĈC ν (Â) J 0 possess natural, continuous, k-linear sections. g) Similar results hold for the continuous Hochschild and cyclic complexes associated to a free ideal extension of complete, locally convex algebras. Proof: All statements except c) and e) are obvious and e) is a consequence of c). For c) we proceed as follows. Fix n 0 and decompose A canonically according to 1.3 d) as A s(b) (V W) V j J ν(n)+1. ν(n) 1 j=0 View this as J-adic weight space decomposition where s(b) is of weight zero, (V W) V j is of weight j + 1andJ ν(n)+1 is of weight ν(n) + 1. It induces a weight space decomposition of tensor powers of A and we define an operator P ν : C (A) C (A) by demanding that the restrictions of P ν to C n (A) (resp. C n (A)) equal the canonical projections onto the subspaces of weight less or equal to ν(n) (resp. ν(n 1)). Then P ν is a projector with kernel equal to the subcomplex C >ν(a) J. Thus P ν identifies the quotient complex C ν(a) J with Im P ν which yields the desired splitting. Moreover it is natural with respect to maps of free ideal extensions and preserves the J-adic filtration. With respect to the I-adic filtration for an auxiliary ideal I J of A, one finds P ν (Fil m I Ωn A) = 0form >ν(n) which assures the continuity of P ν in the I-adic topology. 2.2 The Wodzicki-filtration We recall now the natural filtration of the relative Hochschild and cyclic complexes associated to an extension of algebras invented by Wodzicki [Wo]. Let 0 J A B 0 be an extension of k-algebras with k-linear section s. Definition 2.3 [Wo] There exists an increasing filtration Fil W of the Hochschild complex C (A) which is given as follows:

22 270 M. Puschnigg For every l N put Fil 4l W C n(a) := ã 0 a 1... a n at most l factors a i are in s(b) Fil 4l+1 W C n(a) := (s(b) A n ) Fil 4l+4 W C n(a) + Fil 4l W C n(a) Fil 4l+2 C n (A) := (Ã A (n 1) s(b)) Fil 4l+4 W C n(a) + Fil 4l+1 W C n(a) Fil 4l+3 C n (A) := (J A (n 1) J) Fil 4l+4 W C n(a) + (1 J A (n 2) J) Fil 4l+4 W C n(a) + Fil 4l+2 C n (A). The Wodzicki-filtration satisfies Fil 0 W C (A) = C (J) Fil l W C n(a) = C n (A) if l 4n + 4. Following Wodzicki we identify the associated graded complex of the Hochschild complex with its Wodzicki-filtration with a direct sum of tensor powers of the bar complex of J. Lemma 2.4 a) There exists a natural, filtration preserving linear isomorphism Gr W C (A) C (A). In particular the subquotients of the Wodzicki-filtration can be identified linearly with subspaces of C (A). b) The complex Fil 4l+1 W C (A)/Fil 4l W C (A) is naturally isomorphic to the direct sum of (s(b) (l+1), 0) and the complexes C (k 0,...,k j ) :=s(b) k 0 [k 0 ] C bar (J) s(b) k 1 [k 1 + 1] C bar (J) s(b) k j 1 [k j 1 + 1] C bar (J) s(b) k j [k j ] where the direct sum runs over all tuples (k 0,...,k j ) of integers satisfying k 0 > 0,...,k j 1 > 0, k j 0, k k j = l + 1. c) The complex Fil 4l+2 W C (A)/Fil 4l+1 W C (A) is naturally isomorphic to the direct sum of the complexes and C (k 1,...,k j ) C (i 0,...,i j ) := C bar (J) s(b) k 1 [k 1 + 1] s(b) k j 1 [k j 1 + 1] C bar (J) s(b) k j [k j ] := 1 s(b) i 0 [i 0 + 1] C bar (J) s(b) i j 1 [i j 1 + 1] C bar (J) s(b) i j [i j ]

23 Excision in cyclic homology theories 271 where the first resp. second direct sum runs over all tuples (k 1,...,k j ) resp. (i 0,...,i j ) of integers satisfying k k j = i i j = l + 1, k 1 > 0,...,k j > 0, i 0 0, i 1 > 0,...,i j > 0. d) The quotient complexes Fil 4l+3 W C (A)/Fil 4l+2 W C (A) are naturally contractible. e) The quotient complexes Fil 4l+4 W C (A)/Fil 4l+3 W C (A) are naturally isomorphic to the direct sum of complexes C (k 0,...,k j 1 ) := 1 s(b) k 0 [k 0 + 1] C bar (J) s(b) k j 1 [k j 1 + 1] C bar (J) where the direct sum runs over all tuples of integers satisfying k 0 > 0,...,k j 1 > 0, k k j 1 = l + 1. f) Similar results hold for the continuous Hochschild complex in the topological context. Proof: All parts except d) are evident. The isomorphism of part a) identifies Fil 4l+3 W C (A)/Fil 4l+2 W C (A) with the subspace (J A (n 1) J 1 J A n 2 J) ( Fil 4l+4 W C n(a)/fil 4l W C n(a) ) of C (A). Define a homotopy operator on this space by χ(a 0... a n ) := 1 a 0... a n.thenκ := Id (χ + χ) satisfies { κ(a 0 da 1...da n ( 1) ) = n 1 da n a 0 da 1...da n 1, a n 1 J 0, a n 1 s(b) so that H κ := χ κ i converges and defines a contracting homotopy of i=0 the considered complex. It will be necessary to refine the Wodzicki-filtration by taking into account the Hodge decomposition of the bar complex. Definition 2.5 Let 0 J A B 0 be a free ideal extension of k-algebras. Recall the harmonic decomposition (1.13) of the bar complex C bar (J) and let h be the associated homotopy operator. Define Fil r C (A) C (A), r 0, as the smallest linear subspace of the Hochschild complex C (A) such that Fil r C (A) C (A) contains all tensor monomials in C (A) with at least r+1 blocks in Im (h) and at least one factor in s(b). Fil r C (A) C (A) contains all tensor monomials in C (A) with at least rblocksinim (h) and at least one factor in s(b). We put Fil 1 C (A) := C (A).

24 272 M. Puschnigg Lemma 2.6 The Hochschild differential b maps Fil r C (A) into Fil r 1 C (A) for r = 1 and b(fil 1 C (A)) Fil 0 C (A) + C (J). Proof: This follows from the fact that Im (h) is an A-left-module and a left ideal in TJ 1.14, c) which implies that the number of blocks in Im (h) of a tensor monomial can decrease by at most one under the Hochschild differential. The second auxiliary filtration is given by Definition 2.7 Let 0 J A B 0 be a free ideal extension of k-algebras and recall the harmonic decomposition (1.13). Define C (A) as the smallest linear subspace of C (A, B) such that C (A) C (A) contains all tensor monomials in C (A) with all blocks in Ker (h) and the first or last factor in s(b). C (A) C (A) contains all tensor monomials in C (A) with all blocks in Ker (h) and the second or last factor in s(b). C (A) C (A) contains all tensor monomials in C (A) with all blocks except at most one in Ker (h) and for which the second and last factor are in J and at least one factor in s(b). 2.3 The basic chain map associated to a free ideal extension The basic chain map φ : C (A, B) C (A, B) is constructed in two steps. First we use the structure theorems about the bar complex of a free ideal of the first section to obtain Proposition 2.8 Let 0 J A B 0 be a free ideal extension of k-algebras. For every l > 0 there exists a chain map φ l : C (A) C (A) satisfying a) φ l is chain homotopic to the identity, i.e. there exists an operator χ l : C (A) C +1 (A) such that φ l = Id C (A) (b χ l + χ l b). b) The maps φ l and χ l are natural with respect to maps of free ideal extensions. c) The maps φ l and χ l preserve J-adic filtrations and Wodzicki-filtrations. d) φ l equals the identity on Fil l 1 W C (A) and χ l vanishes on Fil l 1 W C (A). e) φ l preserves the subcomplex C > (A) J := C > (A) J + C (A) J C (A) Fil0 C (A) and maps Fil l W C > (A) J into Fil l 1 W C > (A) J.

25 Excision in cyclic homology theories 273 f) If I J is an auxiliary ideal of A then ( φ l Fil k I C n (A) ) Fil k (n+1) I C n (A) ( χ l Fil k I C n (A) ) Fil k (n+1) I C n+1 (A) g) A similar statement holds for the continuous Hochschild complexes associated to free ideal extensions of complete, locally convex algebras. Proof: Identify according to 2.4 a) the Hochschild complex C (A) with the associated graded space Gr W C (A). Let for the moment χ l : Fil l W C (A)/Fil l 1 W C (A) Fil l W C +1(A)/Fil l 1 W C +1(A) be any natural linear operator preserving J-adic filtrations and extend it to all of Gr W C (A) C (A) by setting χ l = 0 in Wodzicki degrees different from l. Then the pair (χ l,φ l := Id (χ l b + bχ l )) satisfies assertions a), b), and c). We distinguish several cases. Assume first that l is of the form l = 4m +1. We decompose the complex Fil l W C (A)/Fil l 1 W C (A) according to 2.4 b). The operator χ l is constructed as a similar direct sum χ l = χ (k 0,...,k j ) where the operators χ (k 0,,k j ) are defined on s(b) k 0 C bar n 1 by the sum on the space C (k 0,...,k j ) (J) s(b) k 1... s(b) k j 1 Cn bar j (J) s(b) k j χ (k 0,,k j ) := j i=1 ( 1)κ(i) Id k 0 P Id k 1... P Id k i 1 h Id (k i +...+n j +k j ) with κ(i) := k 0 + n 1 + k k i 1. Here P denotes the harmonic projection of C bar (J) onto its harmonic subspace (1.13) and h is the associated homotopy operator (1.13). It is easily verified that on C (k 0,...,k j ) φ (k 0,...,k j ) := Id ( χ (k 0,...,k j ) + χ (k 0,...,k j ) ) = Id k 0 P Id k 1... Id k j 1 P Id k j. With this the definition in the case l = 4m + 1 is complete. The definition in the cases l = 4m+2, l = 4m is similar. In the remaining case l = 4m+3 one defines χ l on the subquotient Fil 4m+3 W C (A)/Fil 4m+2 W C (A) as the contracting homotopy operator of Lemma 2.4 d). With this the homotopy operator χ l is well defined and assertions a), b), and c) hold. The fact that C > (A) J is stable under φ l is a consequence of the definitions and is most easily proved by verifying the following claims: