In this article we further the study of non-commutative motives, initiated in [12, 43]. Our main result is the construction of a symmetric monoidal structure on the localizing motivator Motlocdg of dg categories. As an application, we obtain : (1) a computation of the spectra of morphisms in Motlocdg in terms of non-connective algebraic K-theory; (2) a fully-faithful embedding of Kontsevich's category KMMk of non-commutative mixed motives into the base category Motlocdg(e) of the localizing motivator; (3) a simple construction of the Chern character maps from non-connective algebraic K-theory to negative and periodic cyclic homology; (4) a precise connection between Toën's secondary K-theory and the Grothendieck ring of KMMk; (5) a description of the Euler characteristic in KMMk in terms of Hochschild homology.

We prove that every functor defined on dg categories, which is derived Morita invariant, localizing, and 1-homotopy invariant, satisfies the fundamental theorem. As an application, we recover in a unified and conceptual way, Weibel and Kassel's fundamental theorems in homotopy algebraic K-theory, and periodic cyclic homology, respectively.

We construct new invariants of quadratic forms over commutative rings, using ideas from Topology. More precisely, we define a hermitian analog of the Bott class with target algebraic K-theory, based on the classification of Clifford modules. These invariants of quadratic forms go beyond the classical invariants defined via the Clifford algebra. An appendix by J.-P. Serre, of independent interest, describes the “square root” of the Bott class in the general framework of lambda rings.

The basic properties of locally finite triangulated categories are discussed. The focus is on Auslander–Reiten theory and the lattice of thick subcategories.

We extend McClure's results regarding restriction maps in equivariant K-theory to bivariant K-theory:

Let G be a compact Lie group and A and B be G-C*-algebras. Suppose that KKHn(A, B) is a finitely generated R(G)-module for every H ≤ G closed and n ∈ ℤ. Then, if KKF*(A, B) = 0 for all F ≤ Gfinite cyclic, then KKG*(A, B) = 0.

We give a new construction of a weak form of Steenrod operations for Chow groups modulo a prime number p for a certain class of varieties. This class contains projective homogeneous varieties which are either split or considered over a field admitting some form of resolution of singularities, for example any field of characteristic not p. These reduced Steenrod operations are sufficient for some applications to the theory of quadratic forms.

Let C be a pseudo-abelian symmetric monoidal category, and X a Schur-finite object of C. We study the problem of rationality of the motivic zeta function ζx(t) of X. Since the coefficient ring is not a field, there are several variants of rationality — uniform, global, determinantal and pointwise rationality. We show that ζx(t) is determinantally rational, and we give an example of C and X for which the motivic zeta function is not uniformly rational.

We show localization, excision and descent theorems for K-theory of Deligne-Mumford stacks. Our approach employs the Nisnevich site which is a complete, regular and bounded cd-structure on the category of such stacks and restricts to the usual Nisnevich site on schemes. By combining excision with a refinement of localization sequences due to Krishna and Töen, we show that K-theory of perfect complexes on tame Deligne-Mumford stacks satisfies Nisnevich descent.

Recently, Bondarko constructed a DG category of motives, whose homotopy category is equivalent to Voevodsky's category of effective geometric motives, assuming resolution of singularities. Soon after, Levine extended this idea to construct a DG category whose homotopy category is equivalent to the full subcategory of motives over a base-scheme S generated by the motives of smooth projective S-schemes, assuming that S is itself smooth over a perfect field. In both constructions, the tensor structure requires ℚ-coefficients. In this article, it is shown how to provide a tensor structure on the homotopy category mentioned above, when S is semi-local and essentially smooth over a field of characteristic zero. This is done by defining a pseudo-tensor structure on the DG category of motives constructed by Levine, which induces a tensor structure on its homotopy category.

In this paper we define complex equivariant K-theory for actions of Lie groupoids using finite-dimensional vector bundles. For a Bredon-compatible Lie groupoid , this defines a periodic cohomology theory on the category of finite -CW-complexes. We also establish an analogue of the completion theorem of Atiyah and Segal. Some examples are discussed.

In this paper we show that a conjecture of Voevodsky about the slices of the motivic cobordism spectrum implies a statement about the slices of motivic Landweber spectra. Over perfect fields these slices are given by the coefficients of the corresponding topological Landweber spectrum and the motivic Eilenberg MacLane spectrum. We also prove a cohomological version of Landweber exactness which applies to the compact objects of the stable motivic homotopy category.

Using triality, we define a relative Arason invariant for orthogonal involutions on a -possibly division- central simple algebra of degree 8. This invariant detects hyperbolicity, but it does not detect isomorphism. We produce explicit examples, in index 4 and 8, of non isomorphic involutions with trivial relative Arason invariant.

We prove that a leafwise homotopy equivalence between compact foliated manifolds induces a well defined bounded operator between all Sobolov spaces of leafwise (for the natural foliations of the graphs of the original foliations) differential forms with coefficients in a leafwise flat bundle. We further prove that the associated map on the leafwise reduced L2 cohomology is an isomorphism which only depends on the leafwise homotopy class of the homotopy equivalence.

We prove a formula for the cup product on the ℓ-adic cohomology of the complement of a linear subspace arrangement.

Let p be an odd prime and L/F a p-extension of number fields with Galois group G. The aim of this paper is to provide answers to a question of Kahn concerning lower bounds for the order of the kernel and cokernel of the functorial map K2F → K2LG. To this end, we first determine a norm index formula for generalized Tate kernels and then express our lower bounds in terms of the ramification in L/F.

We show that for the homotopy category K(Ab) of complexes of abelian groups, both Brown representability and Brown representability for the dual fail. We also provide an example of a localizing subcategory of K(Ab) for which the inclusion into K(Ab) does not have a right adjoint.

Let C*(E) be the graph C*-algebra associated to a graph E and let J be a gauge-invariant ideal in C*(E). We compute the cyclic six-term exact sequence in K-theory associated to the extension

in terms of the adjacency matrix associated to E. The ordered six-term exact sequence is a complete stable isomorphism invariant for several classes of graph C*-algebras, for instance those containing a unique proper nontrivial ideal. Further, in many other cases, finite collections of such sequences constitute complete invariants.

Our results allow for explicit computation of the invariant, giving an exact sequence in terms of kernels and cokernels of matrices determined by the vertex matrix of E.

Let Γ be an Ã2 subgroup of PGL3(), where is a local field with residue field of order q. The module of coinvariants C(,ℤ)Γ is shown to be finite, where is the projective plane over . If the group Γ is of Tits type and if q ≢ 1 (mod 3) then the exact value of the order of the class [1]K0 in the K-theory of the (full) crossed product C*-algebra C(Ω) ⋊ Γ is determined, where Ω is the Furstenberg boundary of PGL3(). For groups of Tits type, this verifies a conjecture of G. Robertson and T. Steger.

We construct a Baum-Douglas type model for K-homology with coefficients in ℤ/kℤ. The basic geometric object in a cycle is a spinc ℤ/kℤ-manifold. The relationship between these cycles and the topological side of the Freed-Melrose index theorem is discussed in detail. Finally, using inductive limits, we construct geometric models for K-homology with coefficients in any countable abelian group.