Universal suspension via Non-commutative motives

arXiv:1003.4425v1 [math.KT] 23 Mar 2010 UNIVERSAL SUSPENSION VIA NON-COMMUTATIVE MOTIVES GONC¸ALO TABUADA Abstract. In this article we further the study of non-commutative motives, initiated in [5, 6, 26]. Our main result is the construction of a simple model, given in terms of infinite matrices, for the suspension in the triangulated cate- gory of non-commutative motives. As a consequence, this simple model holds in all the classical invariants such as Hochschild homology, cyclic homology and its variants (periodic, negative, . . .), algebraic K-theory, topological Hochschild homology, topological cyclic homology, . . . 1. Introduction Non-commutative motives. A differential graded (=dg) category, over a commu- tative base ring k, is a category enriched over complexes of k-modules (morphisms sets are such complexes) in such a way that composition fulfills the Leibniz rule : d(f ◦g) = (df)◦g+(−1)deg(f)f ◦(dg). Dg categories enhance and solve many of the technical problems inherent to triangulated categories; see Keller’s ICM adress [13]. In non-commutative algebraic geometry in the sense of Bondal, Drinfeld, Kaledin, Kapranov, Kontsevich, To¨en, Van den Bergh, . . . [2, 3, 7, 10, 17, 18, 31], they are considered as dg-enhancements of derived categories of (quasi-)coherent sheaves on a hypothetic non-commutative space. All the classical (functorial) invariants, such as Hochschild homology HH, cyclic homology HC, (non-connective) algebraic K-theory IK, topological Hochschild homology T HH, and topological cyclic homology T C, extend naturally from k- algebras to dg categories. In order to study all these classical invariants simultane- ously the author introduced in [26] the notion of localizing invariant. This notion, that we now recall, makes use of the language of Grothendieck derivators [9], a formalism which allows us to state and prove precise universal properties. Let L : HO(dgcat) →D be a morphism of derivators, from the derivator associated to the derived Morita model structure on dg categories (see §2.2), to a triangu- lated derivator. We say that L is a localizing invariant if it preserves filtered ho- motopy colimits as well as the terminal object, and sends exact sequences of dg categories (see §2.3) A −→B −→C 7→ L(A) −→L(B) −→L(C) −→L(A)[1] Date: August 28, 2018. 2000 Mathematics Subject Classification. 18D20, 19D35, 19D55. Key words and phrases. Non-commutative motives, Infinite matrix algebras, Algebraic K- theory, (Topological) Hochschild and cyclic homology. The author was partially supported by the Estimulo `a Investiga¸c˜ao Award 2008 - Calouste Gulbenkian Foundation. 1 2 GONC¸ALO TABUADA to distinguished triangles in the base category D(e) of D. Thanks to the work of Keller [14, 15], Thomason-Trobaugh [30], Schlichting [22], and Blumberg-Mandell [1] (see also [29]), all the mentioned invariants satisfy localization1, and so give rise to localizing invariants. In [26], the author constructed the universal localizing invariant Uloc dg : HO(dgcat) −→Motloc dg , i.e. given any triangulated derivator D, we have an induced equivalence of categories (1.1) (Uloc dg )∗: Hom!(Motloc dg , D) ∼ −→Homloc(HO(dgcat), D) , where the left-hand side denotes the category of homotopy colimit preserving mor- phisms of derivators, and the right-hand side denotes the category of localizing invariants. Because of this universality property, which is a reminiscence of mo- tives, Motloc dg is called the localizing motivator, and its base category Motloc dg (e) the category of non-commutative motives. We invite the reader to consult [5, 6, 26] for several applications of this theory of non-commutative motives. Universal suspension. The purpose of this article is to construct a simple model for the suspension in the triangulated category of non-commutative motives. Consider the k-algebra Γ of N × N-matrices A which satisfy the following two conditions : (1) the set {Ai,j | i, j ∈N} is finite; (2) there exists a natural number nA such that each row and each column has at most nA non-zero entries; see Definition 3.5. Let Σ be the quotient of Γ by the two-sided ideal consisting of those matrices with finitely many non-zero entries; see Definition 3.1. Alternatively, take the (left) localization of Γ with respect to the matrices In, n ≥0, with entries (In)i,j = 1 for i = j > n and 0 otherwise; see Proposition 3.11. The algebra Σ goes back to the work of Karoubi and Villamayor [12] on negative K-theory. Recently, it was used by Corti˜nas and Thom [4] in the construction of a bivariant algebraic K-theory. Given a dg category A, we denote by Σ(A) the tensor product of A with Σ; see §2.1. The main result of this article is the following. Theorem 1.2. For every dg category A we have a canonical isomorphism Uloc dg (Σ(A)) ∼ −→Uloc dg (A)[1] . The proof of Theorem 1.2 is based on several properties of the category of non- commutative motives (see Section 6), on an exact sequence relating A and Σ(A) (see Section 4), and on the flasquene

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