Guillermo Cortiñas

The obstruction to excision in K-theory and in cyclic homology

Let f:A→B be a ring homomorphism of not necessarily unital rings and

BIVARIANT ALGEBRAIC K-THEORY

I\triangleleft{A}

The

an ideal which is mapped by f isomorphically to an ideal of B. The obstruction to excision in K-theory is the failure of the map between relative K-groups K*(A:I)→K*(B:f(I)) to be an isomorphism; it is measured by the birelative groups K*(A,B:I). Similarly the groups HN*(A,B:I) measure the obstruction to excision in negative cyclic homology. We show that the rational Jones-Goodwillie Chern character induces an isomorphism

-theory of toric varieties

ch_{*}:K_{*}(A,B:I)\otimes\mathbb{Q}\overset{\sim}{\to}HN_{*}(A\otimes\mathbb{Q},B\otimes\mathbb{Q}:I\otimes\mathbb{Q}).

The K-theory of toric varieties in positive characteristic

Abstract We show how methods from K-theory of operator algebras can be applied in a completely algebraic setting to define a bivariant, M ∞-stable, homotopy-invariant, excisive K-theory of algebras over a fixed unital ground ring H, (A, B) ↦ kk *(A, B), which is universal in the sense that it maps uniquely to any other such theory. It turns out kk is related to C. Weibels homotopy algebraic K-theory, KH. We prove that, if H is commutative and A is central as an H-bimodule, then We show further that some calculations from operator algebra KK-theory, such as the exact sequence of Pimsner-Voiculescu, carry over to algebraic kk.

Relative Chern Characters for Nilpotent Ideals

Recent advances in computational techniques for K-theory allow us to describe the K-theory of toric varieties in terms of the K-theory of fields and simple cohomological data.

Toric varieties, monoid schemes and cdh descent

We investigate when Isomorphism Conjectures, such as the ones due to Baum-Connes, Bost and Farrell-Jones, are stable under colimits of groups over directed sets (with not necessarily injective structure maps). We show in particular that both the K-theoretic Farrell-Jones Conjecture and the Bost Conjecture with coefficients hold for those groups for which Higson, Lafforgue and Skandalis have disproved the Baum-Connes Conjecture with coefficients.We present a