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Marco Robalo I3M - Montpellier June 24, 2013 Noncommutative Motives and K-theory Marco Robalo I3M - Montpellier June 24, 2013 Contents 1 Noncommutative Algebraic Geometry 1 2 Motives 4 3 Noncommutative Motives 6 4 K-Theory and Noncommutative Motives 8 5 Relation with the work of Cisinski-Tabuada 11 6 Future Works 12 In these notes we explain a new approach to the theory of noncommutative motives suggested by B.Toen, M. Vaquie and G. Vezzosi, which I have been studying in my ongoing doctoral thesis with B. Toen (see [30, 31]). The idea of a motivic theory for noncommutative spaces was originally proposed by M. Kont- sevich [20, 21] and in the later years G.Tabuada and D-C. Cisinski [38, 12, 11, 39] developped a formal setting that makes this idea precise. The new approach we want to explain in this talk is independent of the approach by Cisinski-Tabuada (as all the proofs) and the motivation for it is simple: to have a natural theory of noncommutative motives that is naturally comparable to the motivic stable A1-homotopy theory of Morel-Voevodsky. At the end of this talk we dedicate a special section to explain how our theory relates to methods of Cisinski-Tabuada. We assume the reader to be familiar with the theory of ( , 1)-categories as developed by J. Lurie in [22, 23]. 1 Throughout these notes we fix k a commutative base ring. 1 Noncommutative Algebraic Geometry The original guiding principle of noncommutative geometry is that we can replace the study of a given space X by the study of its ring of functions. This principle appears in the very foundations of algebraic geometry. Grothendieck took it to a new level by understanding that the abelian category Qcoh(X) of quasi-coherent 1 sheaves of an algebraic varitiety X contains most of the information encoded by X. For instance, everything that at that time was considered as a “cohomology theory” can be obtained by means of the two steps 1 Qcoh( ) cohomology Schemes − / Abelian Categories / Abelian Groups Grothendieck was also the first one to realize that the second factorization (aka, homological algebra) factors through a more fundamental step cohomology Abelian Categories / Abelian Groups 4 homotopy groups ✏ (k-linear) homotopy theories where the vertical arrow sends a k-abelian category A to the k-linear homotopy theory of complexes of objects in A studied up quasi-isomorphisms. The problem was that at that time homotopy theories were studied by means of the strict formal (categorical) inversion of those morphisms we wish to consider as isomorphism. The resulting object (the classical derived category) is very poorly behaved and many of the expected information of the homotopy theory is lost in the process. In the 90’s, dg-categories2 appeared as a technological enhancement of this classical derived category (see [7, 9, 10] and [19, 5] for an introduction) and Kontsevich envisioned them as the perfect objects for noncommutative geometry (the reason will become clear below). Nowadays, Kontsevich vision adds to the understanding that: “homotopy theories” = ( , 1)-categories 1 “k-linear homotopy theories” = k-linear ( , 1)-categories =3 k-dg-categories 1 Under this vision, the assignement X L (X) realizes the dotted map first forseen by Grothendieck. 7! qcoh Here Lqcoh(X)isthebigk-dg-category obtained from the dg-enhanced category of complexes of sheaves with quasi-coherent cohomology by inverting the quasi-isomorphisms, this time not in the world of categories but within dg-categories. This big dg-category is generated by its full sub dg-category Lpe(X) spanned by the (homotopy) compact objects - so-called perfect complexes. We will return to this below. We should also emphasize that dg-categories are important not only for algebraic geometry. They appear out of many di↵erent mathematical contexts Algebraic Geometry Symplectic Geometry Lqcoh Fukaya cat. ) u dg categories −5 O i DQ modules − complex geometry Matrix Factorizations dg algebras − 1In fact, the replacement X Qcoh(X)issostrongthatinsomecasesitallowustoreconstructthewholescheme(seethe thesis of P. Gabriel for the notion! of spectrum of an abelian category [14]). 2Don’t worry! We will not repeat here the definition of a dg-category. 3This equality is known to be true when k is the sphere spectrum (in this case a k-dg-category is just a spectral category by definition - see [?]). When k is a classical commutative ring this equality is still just a philosophical principle. People are working to prove this. 2 and can be used as a unifying language. The map from dg-algebras to dg-categories is particularly impor- tant. It sends a dg-algebra A to the dg-category with one object and A as a complex of endomorphisms with the composition given by the associative product on A. The theory of modules makes sense for any dg-category. If T is a small dg-category, we define its big dg-category of dg-modules as the dg-category of dg-enriched functors from T to the category of complexes of k-modules with its natural dg-enrichement. This carries a natural k-linear homotopy theory and we will write T for the dg-category that codifies it. If T the dg-category associated to a dg-algebra A, T is a dg-enhancement of the classical derived category of A. The fact that Lpe(X) generates the whole Lqcoh(X) can now be madeb precise by means of an equivalence L\(X) L (X). b pe ' qcoh Let us now review some standard invariants that we can extract from a small dg-category: The Hoschschild Homology of a dg-category T , HH(T ) (see [6] for a precise construction of this • invariant with values in spectra). The cyclic homology and the periodic cyclic homology of a dg- category are defined by an appropriate base-change of HH(T ). By the Hochschild-Konstant-Rosenberg theorem, the periodic cyclic homology of the small dg-category Lpe(X) is isomorphic to the de Rham cohomology of X. More recentely, Kaledin [?] introduced a noncommutative version of the cristalline cohomology. • Connective K-theory, Non-connective K-theory and A1-invariant K-theory. We will come back to this • later in the talk (Section 4). At this point we merely want to emphazise that all these K-theory flavours can be defined directly at the level of dg-categories and that their values at Lpe(X) recover the values associated to X; More recently, A. Blanc in his thesis [6], introduce a noncommutative version of topological K-theory. • top If X is a scheme defined over C, K (Lpe(X)) recovers the topological K-theory of the underlying space of complex points in X; Comparison results like the ones above enhance the understanding of dg-categories as good bodies for noncommutative spaces. An important feature of all these theories is the invariance along morphisms T T 0 such that the map ! induced between the big dg-categories of modules T T 0 is an equivalence. Such morphims are called Morita equivalences. The study of dg-categories up to! Morita equivalence admits a (combinatorial) model structure (see [37]) and throughout this notes we willb let cg(k)idem denote its underlying ( , 1)-category. It admits a natural monoidal structure given by the derivedD tensor product of dg-categories and1 by the results of [41] it admits an internal-hom (which can be explicitely described). We now introduce a bit of notation. Let A be a dg-algebra and A its dg-category of modules. We let Ape denote the full sub dg-category of A spanned by the (homotopy) compact objects. Of course, we have b Ape A. The following is a crucial result in noncommutative algebraic geometry: b ' b Theoremdb b 1.1. (Bondal-Van den Bergh [8]) Let X be a scheme over k. Then, if X is quasi-compact and quasi-separated there is a dg-algebra AX such that Lpe(X) is Morita equivalent to the small dg-category AX pe. dIn other words, every quasi-compact and quasi-separated schemes becomes affine in the noncommutative world. Definition 1.2. (Toen-Vaquie [42]) A dg-category T g(k)idem is said to be of finite type if it is a compact object in the ( , 1)-category g(k)idem.Wesaythat2TDis smooth and proper if it is a dualizable object. In particular, smooth1 and properD dg-categories are of finite type. 3 We will denote g(k)ft the full subcategory of g(k)idem spanned by the dg-categories of finite type. We can easily see thatD it is closed under the derived tensorD product of dg-categories. Moreover, we can prove that a small dg-category T is of finite type if and only if it is Morita equivalent to a small dg-category of the form Ape for a dg-algebra A which is (homotopy) compact in the homotopy theory of dg-algebras (see [42]). This description, together with the result of Bondal- Van den Bergh motivates the following definition: b Definition 1.3. The ( , 1)-category of smooth non-commutative spaces cS(k) is the opposite of g(k)ft 1 N D and we have Proposition 1.4. (Toen-Vaquie [42, 3.27]) Let X be a smooth and proper variety over k. Then Lpe(X) is a smooth and proper dg-category. In particular, it is of finite type. The assignement X Lpe(X) is known not to be fully faithful (see [44] for an explicit example and [33] for a more complete discussion! about the behavior of this assignement). In his program [20, 21] Kontsevich envisioned also that similarly to schemes, noncommutative spaces should also admit a motivic theory and one of the interesting questions is how better Lpe behaves at the motivic level.
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