The Obstruction to Excision in K-theory and in Cyclic Homology

Let f : A → B be a ring homomorphism of not necessarily unital rings and I ⊳ A an ideal which is mapped by f isomorphically to an ideal of B. The obstruction to excision in K-theory is the failure of the map between relative K-groups K∗(A : I) → K∗(B : f(I)) to be an isomorphism; it is measured by the birelative groups K∗(A,B : I). Similarly the groups HN∗(A,B : I) measure the obstruction to excision in negative cyclic homology. We show that the rational Jones-Goodwillie Chern character induces an isomorphism ch∗ : K∗(A,B : I)⊗ Q ∼ → HN∗(A⊗ Q, B ⊗ Q : I ⊗ Q). 0. Introduction Algebraic K-theory does not satisfy excision. This means that if f : A → B is a ring homomorphism and I ⊳ A is an ideal carried isomorphically to an ideal of B, then the map of relative K-groups K∗(A : I) → K∗(B : I) := K∗(B : f(I)) is not an isomorphism in general. The obstruction is measured by birelative groups K∗(A,B : I) which are defined so as to fit in a long exact sequence Kn+1(B : I) → Kn(A,B : I) → Kn(A : I) → Kn(B : I) Similarly the obstruction to excision in negative cyclic homology is measured by birelative groups HN∗(A,B : I). K-theory and negative cyclic homology are related by a character KnA → HNnA, the Jones-Goodwillie Chern character ([17, Ch.II]; see also [21, §8.4]). Tensoring with Q and composing with the natural map HNn(A) ⊗Q → HNn(A⊗Q) we obtain a rational Chern character (1) chn : K Q n (A) := Kn(A)⊗Q → HNn(A⊗Q) =: HN Q n (A) The main theorem of this paper is the following. Main theorem 0.1. Let f : A → B be a homomorphism of not necessarily unital rings and I ⊳ A an ideal which is carried by f isomorphically onto an ideal of B. Then (1) induces an isomorphism ch∗ : K Q ∗ (A,B : I) ∼ → HN ∗ (A,B : I). If moreover A and B are Q-algebras, then K∗(A,B : I) is a Q-vectorspace, and the exponent Q is not needed. (*) Partially supported by CONICET, the ICTP’s Associateship and the Ramón y Cajal fellowship and by grants UBACyT X066, ANPCyT PICT 03-12330 and MTM00958. 1 2 GUILLERMO CORTIÑAS The theorem above can also be stated in terms of cyclic homology (denoted HC), as we shall see presently. We recall that, unlike HN , HC commutes with ⊗Q, so that HC ∗ (A) := HC∗(A⊗Q) = HC∗(A)⊗Q. Corollary 0.2. There is a natural isomorphism ν∗ : K Q ∗ (A,B : I) ∼= HC Q ∗−1(A,B : I). Proof. Write HP for periodic cyclic homology. There is a long exact sequence ([21, 5.1.5]) (2) HCn−1(A,B : I) → HNn(A,B : I) → HPn(A,B : I) → HCn−2(A,B : I). Cuntz-Quillen’s excision theorem [9] establishes that (3) HP ∗ (A,B : I) = 0. Here HP ∗ ( ) := HP∗( ⊗Q). From (3) and from (2) applied to A⊗Q, B⊗Q and I ⊗Q, it follows that HC ∗−1(A,B : I) = HN Q ∗ (A,B : I). Next we shall review some related results in the literature, so as to put ours in perspective. Bass ([2, Thm. XII.8.3]; see also [19, pp. 295-298]) proved K-theory satisfies excision in nonpositive degrees; in our setting this means Kn(A,B : I) = 0 for n ≤ 0. The analogous result for the negative degrees of cyclic homology is also true. In fact, by definition ([21, 2.1.15]) if R is a unital ring and J ⊳ R an ideal then HCn(R : J) = 0 for n < 0 and HC0(R, J) = J/[R, J ], the quotient by the subgroup generated by the commutators [r, j] = rj − jr. Hence in the situation of 0.1 HC−1(A,B : I) = coker(I/[A, I] → I/[B, I]) = 0. Thus 0.2 is true for ∗ nonpositive, as both birelative groups vanish. The particular case of 0.2 when ∗ = 1 was proved in [11]. In [10] the statement of Corollary 0.2 was conjectured to hold when A and B are commutative unital Q-algebras and B is a finite integral extension of A, (KABI conjecture) and it was shown its validity permits computation of the K-theory of singular curves in terms of their cyclic homology and of the K-theory of nonsingular curves. In [12] 0.2 was conjectured for unital Q-algebras and it was shown that for ∗ = 2 the left hand side maps surjectively onto the right hand side. A special case of the main theorem concerns the birelative groups K∗(A : I, J) associated to any pair of ideals I, J ⊳A; they are defined so as to fit in a long exact sequence Kn+1(A/J : I + J/J) → Kn(A : I, J) → Kn(A : I) → Kn(A/J : I + J/J). Note that if I∩J = 0 then A → A/J maps I isomorphically onto I+J/J = I/I∩J , whence 0.2 applies, and we have an a rational isomorphism (4) K ∗ (A : I, J) ∼ → HC ∗−1(A : I, J). For ∗ ≤ 1 it is well-known that both birelative Kand cyclic homology groups vanish, whence (4) follows. The case ∗ = 2 was proved independently in [18] and [20]. One can further generalize this to the case when I ∩ J is nilpotent, using a theorem of Goodwillie’s ([17, Main Thm.] see also [21, §11.3]), which says that if I ⊳ R is a nilpotent ideal of a ring R, then there is a natural isomorphism (5) K ∗ (R : I) ∼= HC Q ∗−1(R : I). THE OBSTRUCTION TO EXCISION IN K-THEORY AND IN CYCLIC HOMOLOGY 3 We point out that, even if Goodwillie states his theorem for R unital, the nonunital case follows from the unital ([5, §4.2]; see also Lemma 1.1 below). As in the case of 0.2, (5) can also be stated in terms of negative cyclic homology. Actually Goodwillie shows (see [17, 0.3]) that ch∗ induces an isomorphism (6) ch∗ : K Q ∗ (R : I) ∼ → HN ∗ (R : I) and then uses the singly relative version of (2) in combination with another theorem of his, ([16, II.5.1]; see also [9, 3.5]), which says that HP ∗ (R, I) = 0 if I is nilpotent. Now we use (5) to generalize (4). Corollary 0.3. Let A be a ring and I, J ⊳ A ideals such that I ∩ J is nilpotent. Then there is a rational isomorphism of birelative groups K ∗ (A : I, J) ∼ → HC ∗−1(A : I, J). Proof. The case I ∩ J = 0 is explained above. To prove the general case, consider the intermediate groups K n fitting in the long exact sequence (7) HN n+1(R) // K n (R) // K n (R) chn // HN n (R) (This notation will be justified below, see (11)). By 0.1, K satisfies excision; by (6), it is invariant under nilpotent extensions, or nilinvariant. But a diagram chase shows that any homology theory of rings H satisfying both excision and nilinvariance verifies H∗(A : I, J) = 0 if I ∩ J is nilpotent. Applying this to both K and HP, we obtain K ∗ (A : I, J) ∼= HN ∗ (A : I, J) ∼= HC Q ∗−1(A : I, J) The result of the corollary above for A unital was announced in [23]; however the proof in loc.cit. turned out to have a gap (see [13, pp. 591, line 1]). An application of 0.3 to the computation of the K-theory of particular rings –other than coordinate rings of curves– in terms of their cyclic homology was given in [13, Thm. 3.1]; see also [10, Thm. 7.3]. As another precedent of the main theorem of this paper, we must cite the work of Suslin and Wodzicki. To state their theorems we introduce some notation. We say that a ring I is excisive for a homology theory H if H∗(A,B; I) = 0 for every homomorphism A → B as in 0.1. In [28], M. Wodzicki characterized those rings which are excisive for cyclic homology as those whose bar homology vanishes (8) A is HC − excisive ⇐⇒ H ∗ A⊗Q = 0. In fact (8) is a particular case of [28, (3)]. He also showed that if A is excisive for rational K-theory then it is excisive for rational cyclic homology ([28, (4)]) and conjectured that the converse also holds. The latter was proved by Suslin and Wodzicki; they showed ([24, Thm. A]) (9) H ∗ (A)⊗Q = 0 ⇒ A is excisive for K -theory. They proved further that if A is a Q-algebra then (9) still holds even if we do not tensor with Q ([24, Thm. B]). Note that (9) is a formal consequence of 0.2 and (8). Actually our proof of 0.1 involves proving a version of Suslin-Wodzicki’s theorem for a certain type of pro-rings (Theorem 3.16). 4 GUILLERMO CORTIÑAS Sketch of Proof of 0.1. The assertion concerning Q-algebras follows (see 4.1 below) from Weibel’s result [26] (see also [24, 1.9]) that for all n ≥ 2, K-theory with Z/n coefficients satisfies excision for such algebras. The rest of the proof has four parts: a) An abstraction of arguments of Cuntz and Quillen and its combination with the tautological characters of [5]. (This is done in Section 1 below). We consider functors from the category Ass1 = Ass1(k) of unital rings over a commutative ring k to fibrant spectra which preserve products up to homotopy. If X is such a functor and I ⊳ A ∈ Ass1 an ideal, we put X(A : I) := hofiber(X(A) → X(A/I)), X̂(A/I) := holim n X(A/I), X̂(A : I) := holim n X(A : I). If f : A → B ∈ Ass1 is as in 0.1, we set X(A,B : I) = hofiber(X(A : I) → X(B : I)), X̂(A,B : I) = hofiber(X̂(A : I) → X̂(B : I)). We call I ∞-excisive if X̂(A,B : I) is weakly contractible for every homomorphism A → B as above. We say that X is nilinvariant if X(A : I) is contractible for every nilpotent ideal I ⊳ A. We show that if X is nilinvariant and every ideal I of every free unital algebra is ∞-excisive then X is excisive (Proposition 1.6). This gives a criterion for proving excision which generalizes that used by Cuntz and Quillen in the particular case of HP in [9]. Next, given any not necessarily excisive or nilinvariant functor X as above, we consider its noncommutative infinitesimal hypercohomology ([5, §5]) H(Ainf , X). This is a nilinvariant functor and is equipped with a natural map H(Ainf , X) → X(A). Write τX(A) for the delooping of the fiber of the latter map. We have a homotopy fibration (10) H(Ainf , X) → X(A) c −→ τX(A). We call c the tautological character. Applying the Cuntz-Quillen criterion to H(Ainf , X), we obtain (Theorem 1.8) that if every ideal of every free unital algebra F ∈ Ass1 is ∞-excisive for both X and τX then the map X(A,B : I) → τX(A,B : I) is a weak equivalence for every algebra homomorphism as in 0.1. b)Agreement of the tautological and rational Jones-Goodwillie characters. (Section 2). We apply (10) to X = K, the nonconnective rational K-theory spectrum. We show that there is a natural isomorphism HN n (A) ∼= τK n (A) under which the tautological character cn is identified with chn (Theorem 2.1). In particular, the infintesimal K-theory groups (11) K n A := πnH(Ainf ,K ) (n ∈ Z). are the “intermediate groups” of (7). Theorem 2.1 is of independent interest as it shows that HN can be functorially derived from K, and extends to all n ∈ Z previous results of the author ([4, 6.3]; [5, 6.2]; [6, 5.1]). c)Suslin-Wodzicki’s theorem for A. (Section 3). We show that if A is a ring such that for all r ≥ 0 the pro-vectorspace H r (A )⊗Q := {H r (A )⊗Q}n is zero, then A is∞-excisive forK (Theorem 3.16). Note that in the particular case when A = A, the latter assertion and (9) coincide, since A is just the constant THE OBSTRUCTION TO EXCISION IN K-THEORY AND IN CYCLIC HOMOLOGY 5 pro-ring A in this case. Our proof follows the strategy of Suslin and Wodzicki’s proof of (9), (see the summary at the beginning of Section 3) and adapts it to the pro-setting. Some technical results on pro-spaces needed in this section are proved in the Appendix. d) Application of known results on bar and cyclic pro-homology. (Section 4). By parts (a), (b) and (c), to finish the proof it is enough to show that if I is an ideal of a free unital ring, then both (i) and (ii) below hold. (i) H r (I )⊗Q = 0 (r ≥ 0). (ii) I is ∞-excisive for HN. By (2) and (3) the latter property is equivalent to (ii)’ I is ∞-excisive for HC. As ⊗Q commutes with both H and HC and sends free unital rings to free unital Q-algebras, it suffices to verify (i) and (ii)’ for ideals of free unital Q-algebras. Both of these are well-known and are straightforward from results in the literature; see 4.2 for details. The rest of this paper is organized as follows. In Section 1 we carry out part (a) of the sketch above. We generalize Cuntz-Quillen’s excision principle (Proposition 1.6), recall the construction of the tautological character (1.7) and obtain a criterion for proving that the latter computes the obstruction to excision (Theorem 1.8). Part (b) corresponds to Section 2, where we show that the tautological character for rational K-theory is the rational Jones-Goodwillie character (Theorem 2.1). Section 3 is devoted to part (c); we prove a version of (9) for pro-rings of the form A (Theorem 3.16). The proof of 0.1 is completed in Section 4, where we carry out part (d) of the sketch and show that if in the situation of the Main Theorem, A and B are Q-algebras, then the groups K∗(A,B : I) are Q-vectorspaces (Lemma 4.1). Notation for pro-spaces as well as some technical results on them which are used in Section 3 are the subject of the Appendix. Note on notation. In this paper space=simplicial set. Acknowledgement. I learned about the existence of the KABI conjecture in 1991 from C. Weibel. His papers with Geller and Reid ([10]) and Geller ([11], [12],[13]) demonstrated its potential applications, produced first positive results and thus made it attractive as a problem. I am thankful to all three of them for calling my attention to it through their work, as well as to J. Cuntz, D. Quillen, A. Suslin and M. Wodzicki, for their results are crucial to this paper. Special thanks to the editor and the referees for the work and dedication spent in improving my paper. 1. Cuntz-Quillen excision principle Summary. In this section we formulate in an abstract setting the method used by Cuntz and Quillen to prove excision for periodic cyclic homology of algebras over a field of characteristic zero. In their proof they first show that HP satisfies excision for pro-ideals of the form I where I is an ideal of a free algebra and then combine this with the invariance of HP under nilpotent extensions to prove excision holds for all ideals. Our setting is that of functors X from the category of algebras over a commutative ring k to that of fibrant spectra. We assume that X preserves products up to homotopy. We show that if X is invariant (up to weak equivalence) under nilpotent extensions and satisfies excision for pro-rings of the 6 GUILLERMO CORTIÑAS form I whenever I is an ideal of a free unital algebra, then X satisfies excision for all algebras (Prop. 1.6). Then we apply this to give a criterion for proving that the tautological character c : X → τX of [5] induces an equivalence at the level of the groups which measure the obstruction to excision (Theorem 1.8). We consider associative, not necessarily unital algebras over a fixed ground ring k. We write Ass := Ass(k) for the category of algebras, Ass1 for the subcategory of unital algebras and unit preserving maps, and (12) A 7→ Ã for the left adjoint functor to the inclusion Ass1 ⊂ Ass. By definition, Ã = A⊕ k, (a, λ)(b, μ) := (ab+ μa+ λb, λμ). Throughout this section X will be a fixed functor from Ass1 to fibrant spectra (terminology for spectra is as in [25]). We shall assume that X preserves finite products up to homotopy; this means that if A,B ∈ Ass1 then the canonical map is a weak equivalence: (13) X(A×B) ∼ → X(A)×X(B). Let A ∈ Ass1, I ⊳ A an ideal, π : A → A/I the projection. Write X(A : I) for the homotopy fiber of X(π). We say that A ∈ Ass is excisive (with respect to our fixed functor X) if for every commutative diagram (14) A Oo ~~ ~~ ~~ ~ o