Documenta Math. 657 A Simple Criterion for Extending Natural Transformations to Higher K-Theory

In this article we introduce a very simple an widely applicable criterion for extending natural transformations to higher K-theory. More precisely, we prove that every natural transformation defined on the Grothendieck group and with values in an additive the- ory admits a unique extension to higher K-theory. As an application, the higher trace maps and the higher Chern characters originally con- structed by Dennis and Karoubi, respectively, can be obtained in an elegant, unified, and conceptual way from our general results.


Introduction
In his foundational work, Grothendieck [10] introduced a very simple and elegant construction K 0 , the Grothendieck group, in order to formulate a farreaching generalization of the Riemann-Roch theorem.Since then, this versatile construction spawned well-beyond the realm of algebraic geometry to become one of the most important (working) tools in mathematics.Latter, through revolutionary topological techniques, Quillen [23] extended the Grothendieck group to a whole family of higher K-theory groups K n , n ≥ 0. However, in contrast with K 0 , these higher K-theory groups are rather misterious and their computation is often out of reach.In order to capture some of its flavour, Connes, Dennis, Karoubi, and others, constructed natural transformations towards simpler theories E making use of a variety of highly involved techniques; see [6,7,15].Typically, the construction of a natural transformation K 0 ⇒ E 0 is very simple, while its extension K n ⇒ E n to higher K-theory is a real "tour-de-force".For example, the trace map K 0 ⇒ HH 0 consists simply in taking the trace of an idempotent, while its extension Gonc ¸alo Tabuada K n ⇒ HH n makes use of an array of tools (Hurewicz maps, group homology, assembly maps, etc) coming from topology, algebra, representation theory, etc.These phenomena motivate the following general questions: Questions: Given a natural transformation K 0 ⇒ E 0 , is it possible to extend it to higher K-theory K n ⇒ E n ?If so, is such an extension unique ?
In this article we prove that if E verifies three very simple conditions, not only such an extension exists, but it is moreover unique.The precisely formulation of our results makes use of the language of Grothendieck derivators, a formalism which allows us to state and prove precise universal properties; see Appendix A.

Statement of results
A differential graded (=dg) category, over a fixed commutative base ring k, is a category enriched over cochain complexes of k-modules (morphisms sets are such complexes) in such a way that composition fulfills the Leibniz rule : Dg categories extend the classical notion of (dg) k-algebra and solve many of the technical problems inherent to triangulated categories; see Keller's ICM address [16].In non-commutative algebraic geometry in the sense of Bondal, Drinfeld, Kaledin, Kontsevich, Van den Bergh, and others, they are considered as differential graded enhancements of (bounded) derived categories of quasi-coherent sheaves on a hypothetic noncommutative space; see [1,8,9,14,17,18].Let E : dgcat → Spt be a functor, defined on the category of dg categories, and with values in the category of spectra [2].We say that E is an additive functor if it verifies the following three conditions: (i) filtered colimits of dg categories are mapped to filtered colimits of spectra; (ii) derived Morita equivalences (i.e.dg functors which induce an equivalence on the associated derived categories; see [16, §4.6]) are mapped to weak equivalences of spectra; (iii) split exact sequences (i.e.sequences of dg categories which become split exact after passage to the associated derived categories; see [24, §13]) are mapped to direct sums in the homotopy category of spectra.Examples of additive functors include Hochschild homology (HH), cyclic homology (HC), and algebraic K-theory (K); see [16, §5].Recall from [25] that the category dgcat carries a Quillen model structure whose weak equivalences are the derived Morita equivalences.Given an additive functor E, we obtain then an induced morphism of derivators E : HO(dgcat) → HO(Spt).Associated to E, we have also the composed functors where π s n denotes the n th stable homotopy group functor and Ab the category of abelian groups.Our answer to the questions stated in the Introduction is: Theorem 1.1.For any additive functor E, the natural map is bijective.In particular, every natural transformation φ : K 0 ⇒ E 0 admits a canonical extension φ n : K n ⇒ E n to all higher K-theory groups.
Intuitively speaking, Theorem 1.1 show us that all the information concerning a natural transformation is encoded on the Grothendieck group.Its proof relies in an essential way on the theory of non-commutative motives, a subject envisioned by Kontsevich [17,19] and whose development was initiated in [3,4,24,25,27,28].In the next section we illustrate the potential of this general result by explaining how the highly involved constructions of Dennis and Karoubi can be obtained as simple instantiations of the above theorem.Due to its generality and simplicity, we believe that Theorem 1.1 will soon be part of the toolkit of any mathematician whose research comes across the above conditions (i)-(iii).

Applications
2.1.Higher trace maps.Recall from [16, §5.3] the construction of the Hochschild homology complex HH(A) of a dg category A. This construction is functorial in A and so by promoting it to spectra we obtain a well-defined functor (2.1) As explained in loc.cit., this functor verifies conditions (i)-(iii) and hence it is additive.Now, given a k-algebra A, recall from [20,Example 8.3.6] the construction of the classical trace map Roughly, it is the map induced by sending an idempotent matrix to the image of its trace (i.e. the sum of the diagonal entries) in the quocient A/[A, A].This construction extends naturally from k-algebras to dg categories (see [26]) giving rise to a natural transformation Morally, these are the non-commutative analogues of the classical Chern character with values in even dimensional de Rham cohomology.As shown in [26] this construction extends naturally from k-algebras to dg categories giving rise to natural transformations Proposition 2.5.In Theorem 1.1 let E be the additive functor Ω 2i HC (obtained by composing (2.4) with the (2i) th -fold looping functor on Spt) and let φ be the natural transformation K 0 ⇒ (Ω 2i HC) 0 = HC 2i .Then, for every kalgebra A, the canonical extension Some explanations are in order: k denotes the dg category naturally associated to the base ring k, i.e. the dg category with only one object and with k as the dg algebra of endomorphisms (concentrated in degree zero); the symbol [k] stands for the class of k (as a module over itself) in the Grothendieck group K 0 (k) = K 0 (k).Let us now turn our attention to Nat(K, E).Recall from [24, §15] the notion of additive invariant of dg categories.Roughly speaking, it consists of a morphism of derivators defined on HO(dgcat) and with values in a triangulated derivator which verifies conditions analogous to (i)-(iii).Since the functor E is additive, the induced morphism of derivators is an additive invariant of dg categories.Following [3,Theorem 8.1] we have then a natural bijection2 Nat(K, E) A careful inspection of the proof of [3,Theorem 8.1] show us that (3.3) sends a natural transformation Φ ∈ Nat(K, E) to the element π s 0 (Φ(e)(k))([k]) of the abelian group E 0 (k).Note that this element is simply the image of [k] by the abelian group homomorphism K, E).On the one hand, we observe that the composed map (3.2) ) of the abelian group E 0 (k).On the other hand, the following equalities hold: Therefore, we have , where Φ is the unique natural transformation associated to φ under the bijection (1.2).

Proof of Proposition 2.3
The essence of the proof consists in describing the unique natural transformation Φ ∈ Nat(K, HH) which corresponds to (2.2) under the bijection (1.2).Recall from [24, §15] the construction of the universal additive invariant of dg categories Given any Quillen model category M we have an induced equivalence of categories where the left-hand side denotes the category of homotopy colimit preserving morphisms of derivators and the right-hand side the category of additive invariants of dg categories.The algebraic K-theory functor K is additive and so the induced morphism K is an additive invariant of dg categories.Thanks to equivalence (4.1), it factors then uniquely through U A .Recall from [24, Theorem 15.10] that for every dg category A we have a weak equivalence of spectra where RHom(−, −) denotes the spectral enrichment of Mot A (see [3, §A.3]).Therefore, we conclude that K can be expressed as the following composition (4.2) HO(dgcat) The Hochschild homology functor, with values in the projective Quillen model category C(k) of cochain complexes of k-modules (see [12,Theorem 2.3.11]),verifies conditions (i)-(iii).Hence, it gives rise to an additive additive invariant of dg categories which we denote by Note that, according to our notation, HH can be expressed as the following composition By construction, the morphism HH maps U A (k) to HH(k) = k.Hence, by making use of the above factorizations (4.2) and (4.3), we conclude that it induces a natural transformation Φ ∈ Nat(K, HH).We now show that the image of this natural transformation Φ by the map (1.2) is the natural transformation (2.2).By taking E = HH in bijection (3.2) we obtain: Under this bijection, the natural transformation (2.2) corresponds to the unit of the base ring k; see [26,Theorem 1.3].Hence, it suffices to show that the same holds for the natural transformation π s 0 •Φ(e)•l associated to Φ.The class [k] of k (as a module over itself) in the Grothendieck group K 0 (k) corresponds to the identity morphism in 2).This implies that Φ is in fact the unique natural transformation which corresponds to (2.2) under the bijection (1.2).Finally, let A be a k-algebra.As proved in [27,Theorem 2.8], the canonical extension φ n : K n (A) → HH n (A) of φ (i.e. the abelian group homomorphism (π s n • Φ(e) • l)(A)) agrees with the n th trace map constructed by Dennis and so the proof is finished.

Proof of Proposition 2.5
We prove first the particular case (i = 0).Let us start by describing the unique natural transformation Φ ∈ Nat(K, HC) which corresponds to φ : K 0 ⇒ HC 0 under the bijection (1.2).Observe that HC can be expressed as the following composition

3 .//
Proof of Theorem 1.1 We start by describing the natural map (1.2).As mentioned in §1, the category dgcat carries a Quillen model structure whose weak equivalences are the derived Morita equivalences; see [25, Theorem 5.3].Let us write Hmo for the associated homotopy category and l : dgcat → Hmo for the localization functor.According to our notation the map (1.2) sends a natural transformation Φ ∈ Nat(K, E) to the natural transformation π s 0 • Φ(e) • l ∈ Nat(K 0 , E 0 ).Pictorially, we have: / Ab .The functors K, E : dgcat → Spt are additive and so the following diagrams dgcat / Ho(Spt) are commutative.Moreover, the 0 th stable homotopy group functor π s 0 descends to the homotopy category Ho(Spt).These facts show us that the composed horizontal functors in the above diagram (3.1) are in fact K 0 and E 0 .We now study the set Nat(K 0 , E 0 ).Recall from [16, §5.1] the notion of additive invariant.Intutively, it consists of a functor defined on dgcat and with values Documenta Mathematica 16 (2011) 657-668 in an additive category which verifies conditions similar to (ii)-(iii).Since by hypothesis E is additive, the composed functor E 0 : dgcat l Ab is an additive invariant.Hence, as proved in [26, Proposition 4.1], we have the following natural bijection Finally, since the right-hand side in this latter equality coincides with the image of Φ by the map (3.3), we conclude that (3.3) = (3.2) • (1.2).Theorem 1.1 now follows from diagram (3.4) and the fact that both maps (3.2) and (3.3) are bijective.The canonical extension φ