We study the K′-theory of a CM Henselian local ring R of finite Cohen-Macaulay type. We first describe a long exact sequence involving the groups K′i(R) and the K-groups of certain other rings, including the Auslander algebra. By examining the terms and maps in the sequence, we obtain information about K′(R).

Let X be a locally compact metrizable space. We show that the Paschke dual construction, which associates to a representation of C0(X) its commutant modulo locally compact operators, can be sheafified. We use this observation to simplify several constructions in analytic K-homology.

We discuss the analytic aspects of the geometric model for K-homology with coefficients in ℤ/kℤ constructed in [12]. In particular, using results of Rosenberg and Schochet, we construct a map from this geometric model to its analytic counterpart. Moreover, we show that this map is an isomorphism in the case of a finite CW-complex. The relationship between this map and the Freed-Melrose index theorem is also discussed. Many of these results are analogous to those of Baum and Douglas in the case of spinc manifolds, geometric K-homology, and Atiyah-Singer index theorem.

For certain product varieties, Murre's conjecture on Chow groups is investigated. More precisely, let k be an algebraically closed field, X be a smooth projective variety over k and C be a smooth projective irreducible curve over k with function field K. Then we prove that if X (resp. XK) satisfies Murre's conjectures (A) and (B) for a set of Chow-Künneth projectors {, 0 ≤ i ≤ 2dim X} of X (resp. for {()K} of XK) and if for any j, , then the product variety X × C also satisfies Murre's conjectures (A) and (B). As consequences, it is proved that if C is a curve and X is an elliptic modular threefold over k (an algebraically closed field of characteristic 0) or an abelian variety of dimension 3, then Murre's conjecture (B) is true for the fourfold X × C.

We show that the functor from curved differential graded algebras to differential graded categories, defined by the second author, sends Cartesian diagrams to homotopy Cartesian diagrams, under certain reasonable hypotheses. This is an extension to the arena of dg-categories of a construction of projective modules due to Milnor. As an example, we show that the functor satisfies descent for certain partitions of a complex manifold.

In [TVa], Bertrand Toën and Michel Vaquié defined a scheme theory for a closed monoidal category ( ⊗1). In this article, we define a notion of smoothness in this relative (and not necessarily additive) context which generalizes the notion of smoothness in the category of rings. This generalisation consists in replacing homological finiteness conditions by homotopical ones, using the Dold-Kan correspondence. To do this, we provide the category s of simplicial objects in a monoidal category and all the categories sA-mod, sA-alg (A ∈ sComm()) with compatible model structures using the work of Rezk [R]. We then give a general notion of smoothness in sComm(). We prove that this notion is a generalisation of the notion of smooth morphism in the category of rings and is stable under composition and homotopy pushouts. Finally we provide some examples of smooth morphisms, in particular in ℕ-alg and Comm(Set).

The fundamental Hochschild cohomology class of the standard Podleś quantum sphere is expressed in terms of the spectral triple of Dąabrowski and Sitarz by means of a residue formula.

We introduce a refinement of the Bloch-Wigner complex of a field F. This refinement is complex of modules over the multiplicative group of the field. Instead of computing K2(F) and Kind3(F) - as the classical Bloch-Wigner complex does - it calculates the second and third integral homology of SL2(F). On passing to F× -coinvariants we recover the classical Bloch-Wigner complex. We include the case of finite fields throughout the article.

We summarise the construction of geometric cycles and their use in describing the Kasparov K-homology of a CW-complex X. When Kasparov K-homology is twisted by a degree three element of the Čech cohomology of X then there is a corresponding construction of twisted geometric cycles for the case where X is a smooth manifold however the method that was employed does not apply in the case of CW-complexes. In this article we propose a new approach to the construction of twisted geometric cycles for CW-complexes motivated by the study of D-branes in string theory.

The most basic structure of chiral conformal field theory (CFT) is the Verlinde ring. Freed-Hopkins-Teleman have expressed the Verlinde ring for the CFT's associated to loop groups, as twisted equivariant K-theory. We build on their work to express K-theoretically the structures of full CFT. In particular, the modular invariant partition functions (which essentially parametrise the possible full CFTs) have a rich interpretation within von Neumann algebras (subfactors), which has led to the developments of structures of full CFT such as the full system (fusion ring of defect lines), nimrep (cylindrical partition function), alpha-induction etc. Modular categorical interpretations for these have followed. For the generic families of modular invariants (i.e. those associated to Dynkin diagram symmetries), we provide a K-theoretic framework for these other CFT structures, and show how they relate to D-brane charges and charge-groups. We also study conformal embeddings and the modular invariant of SU(2), as well as some families of finite group doubles. This new K-theoretic framework allows us to simplify and extend the less transparent, more ad hoc descriptions of these structures obtained previously within CFT.

We generalize several comparison results between algebraic, semi-topological and topological K-theories to the equivariant case with respect to a finite group.

Let X be an algebraic variety with an action of either the additive or multiplicative group. We calculate the additive invariants of X in terms of the additive invariants of the fixed point set, using a formula of Białynicki-Birula. The method is also generalized to calculate certain additive invariants for Chow varieties. As applications, we obtain results on the Hodge polynomial of Chow varieties in characteristic zero and the number of points for Chow varieties over finite fields. As applications, we obtain the l-adic Euler-Poincaré characteristic for the Chow varieties of certain projective varieties over a field of arbitrary characteristic. Moreover, we show that the virtual Hodge (p,0) and (0,q)-numbers of the Chow varieties and affine algebraic group varieties are zero for all p,q positive.

There is an equivalence relation on the set of smooth maps of a manifold into the stable unitary group, defined using a Chern-Simons type form, whose equivalence classes form an abelian group under ordinary block sum of matrices. This construction is functorial, and defines a differential extension of odd K-theory, fitting into natural commutative diagrams and exact sequences involving K-theory and differential forms. To prove this we obtain along the way several results concerning even and odd Chern and Chern-Simons forms.

Let be a generalized based category (see definition 1.2). In this paper, we construct a cohomology theory in the category of contravariant functors: where R is a commutative ring with identity, which generalizes Bredon cohomology involving finite, profinite or discrete groups.

We also study higher K-theory of the category of finitely generated projective objects in and the category of finitely generated objects in and obtain some finiteness and other results.

In this work, we s uggest a defnition for the category of mixed motives generated by the motive h1 (E) for E an elliptic curve without complex multiplication. We then compute the cohomology of this category. Modulo a strengthening of the Beilinson-Soulé conjecture, we show that the cohomology of our category agrees with the expected motivic cohomology groups. Finally for each pure motive (Symnh1 (E)) (–1) we construct families of nontrivial motives whose highest associated weight graded piece is (Symnh1 (E)) (–1).

We generalize the construction of the category of 1–motives with torsion in [2] as well as the construction of the category of 1–motivic sheaves Shv1 in [1] to perfect fields k (without inverting the exponential characteristic). For k transcendental over the prime field we extend a result in [1] showing that and Shv1 have equivalent bounded derived categories.

Given a number field F and a prime number p; let Fn denote the cyclotomic extension with [Fn : F] = pn; and let denote its ring of integers. We establish an analogue of the classical Iwasawa theorem for the orders of K2i (){p}.

Exact sequences in algebraic K-theory contain a lot of information. Here it is shown that by using K-theory exact sequences one can easily derive Bass’ description [1] of the SK1 of an ideal in a Dedekind domain in terms of relative reciprocities.