This article gives a complete classification of generically split projective homogeneous varieties. This project was begun in our previous article [PS10], but here we remove all restrictions on the characteristic of the base field, give a new uniform proof that works in all cases and in particular includes the case PGO2n+ which was missing in [PS10].

In [TV], Bertrand Toën and Michel Vaquié define a scheme theory for a closed monoidal category (,⊗, 1) One of the key ingredients of this theory is the definition of a Zariski topology on the category of commutative monoidal objects in . The purpose of this article is to prove that under some hypotheses, Zariski open subobjects of affine schemes can be classified almost as in the usual case of rings (ℤ-mod,⊗,ℤ). The main result states that for any commutative monoidal object A in , the locale of Zariski open subobjects of the affine scheme Spec(A) is associated to a topological space whose points are prime ideals of A and whose open subsets are defined by the same formula as in rings. As a consequence, we can compare the notions of scheme over in [D] and in [TV].

We describe derived moduli functors for a range of problems involving schemes and quasi-coherent sheaves, and give cohomological conditions for them to be representable by derived geometric n-stacks. Examples of problems represented by derived geometric 1-stacks are derived moduli of polarised projective varieties, derived moduli of vector bundles, and derived moduli of abelian varieties.

Let A be a ring with anti-involution and F a nice functor (tensor or symmetric power, for example) from finitely-generated projective A-modules to abelian groups. We show that the homology of the hyperbolic unitary groups Un,n(A) with coefficients in F(A2n) can be expressed stably (i.e. after taking the colimit over n) by the homology of these groups with untwisted coefficients and functor homology groups that we can compute in suitable cases (for example, when A is a field of characteristic 0 or a ring without ℤ-torsion and F a tensor power). This extends the result where A is a finite field, which was dealt with previously by C. Vespa and the author (Ann. Sci. ENS, 2010).

The proof begins by relating, without any assumption on F, our homology groups to the homology of a category of hermitian spaces with coefficients twisted by F. Then, when F is polynomial, we establish — following a method due to Scorichenko — an isomorphism between this homology and the homology of another category of (possibly degenerate) hermitian spaces, which is computable (in good cases) by standard methods of homological algebra in functor categories (using adjunctions, Künneth formula…). We give some examples.

Finally, we deal with the analogous problem for non-hyperbolic unitary groups in some special cases, for example euclidean orthogonal groups On (A) (the ring A being here commutative). The isomorphism between functor homology and group homology with twisted coefficients does not hold in full generality; nevertheless we succeed to get it when A is a field or, for example, a subring of ℚ containing ℤ[1/2]. The method, which is similar to that in the previous case, uses a general result of symmetrisation in functor homology proved at the beginning of the article.

The main result of this paper is a computation of the motivic cohomology of varieties of n × m-matrices of of rank m, including both the ring structure and the action of the reduced power operations. The argument proceeds by a comparison of the general linear group-scheme with a Tate suspension of a space which is -equivalent to projective n — 1-space with a disjoint basepoint.

In this paper we extend the computation of the the typical curves of algebraic K-theory done by Lars Hesselholt and Ib Madsen to general tensor algebras. The models used allow us to determine the stages of the Taylor tower of algebraic K-theory as a functor of augmented algebras, as defined by Tom Goodwillie, when evaluated on derived tensor algebras.

For R a discrete ring, and M a simplicial R-bimodule, we let R(M) denote the (derived) tensor algebra of M over R, and πR denote the ring of formal (derived) power series in M over R. We define a natural transformation of functors of simplicial R-bimodules Φ: which is closely related to Waldhausen's equivalence We show that Φ induces an equivalence on any finite stage of Goodwillie's Taylor towers of the functors at any simplicial bimodule. This is used to show that there is an equivalence of functors , and for connected bimodules, also an equivalence

We prove that uniform Roe C*-algebras C*uX associated to some expander graphs X coming from discrete groups with property (τ) are not K-exact. In particular, we show that this is the case for the expander obtained as Cayley graphs of a sequence of alternating groups (with appropriately chosen generating sets).

We study the K-theory of ring C*-algebras associated to rings of integers in global function fields with only a single infinite place. First, we compute the torsion-free part of the K-groups of these ring C*-algebras. Secondly, we show that, under a certain primeness condition, the torsion part of K-theory determines the inertia degrees at infinity of our function fields.