Infinitesimal cohomology and the Chern character to negative cyclic homology

There is a Chern character from K-theory to negative cyclic homology. We show that it preserves the decomposition coming from Adams operations, at least in characteristic 0. This is done by using infinitesimal cohomology to reduce to the case of a nilpotent ideal (which had been established by Cathelineau some time ago).Comment: Included reference for identification of relative Chern and rational homotopy theory characters; some minor editing for clarit

A negative answer to a question of Bass

In this companion paper to arXiv:0802.1928 we provide an example of an isolated surface singularity $R$ over a number field such that $K_0(R) = K_0(R[t])$ but $K_0(R) \neq K_0(R[t_1,t_2])$. This answers, negatively, a question of Bass.Comment: The paper was previously part of arXiv:0802.192

Bass’ \u3ci\u3eNK\u3c/i\u3e groups and \u3ci\u3ecd h\u3c/i\u3e-fibrant Hochschild homology

The K-theory of a polynomial ring R[t ] contains the K-theory of R as a summand. For R commutative and containing Q, we describe K∗(R[t ])/K∗(R) in terms of Hochschild homology and the cohomology of Kähler differentials for the cdh topology. We use this to address Bass’ question, whether Kn(R) = Kn(R[t ]) implies Kn(R) = Kn(R[t1, t2]). The answer to this question is affirmative when R is essentially of finite type over the complex numbers, but negative in general

Bass' NKNK groups and cdhcdh-fibrant Hochschild homology

The $K$-theory of a polynomial ring $R[t]$ contains the $K$-theory of $R$ as a summand. For $R$ commutative and containing \Q, we describe $K_*(R[t])/K_*(R)$ in terms of Hochschild homology and the cohomology of K\"ahler differentials for the $cdh$ topology. We use this to address Bass' question, on whether $K_n(R)=K_n(R[t])$ implies $K_n(R)=K_n(R[t_1,t_2])$. The answer is positive over fields of infinite transcendence degree; the companion paper arXiv:1004.3829 provides a counterexample over a number field.Comment: The article was split into two parts on referee's suggestion in 4/2010. This is the first part; the second can be found at arXiv:1004.382

On the vanishing of negative K-groups

Let k be an infinite perfect field of positive characteristic p and assume that strong resolution of singularities holds over k. We prove that, if X is a d-dimensional noetherian scheme whose underlying reduced scheme is essentially of finite type over the field k, then the negative K-group K_q(X) vanishes for every q < -d. This partially affirms a conjecture of Weibel.Comment: Math. Ann. (to appear

Cohomological Hasse principle and motivic cohomology for arithmetic schemes

In 1985 Kazuya Kato formulated a fascinating framework of conjectures which generalizes the Hasse principle for the Brauer group of a global field to the so-called cohomological Hasse principle for an arithmetic scheme. In this paper we prove the prime-to-characteristic part of the cohomological Hasse principle. We also explain its implications on finiteness of motivic cohomology and special values of zeta functions.Comment: 47 pages, final versio

Bass’ \u3ci\u3eNK\u3c/i\u3e groups and \u3ci\u3ecd h\u3c/i\u3e-fibrant Hochschild homology

The K-theory of a polynomial ring R[t ] contains the K-theory of R as a summand. For R commutative and containing Q, we describe K∗(R[t ])/K∗(R) in terms of Hochschild homology and the cohomology of Kähler differentials for the cdh topology. We use this to address Bass’ question, whether Kn(R) = Kn(R[t ]) implies Kn(R) = Kn(R[t1, t2]). The answer to this question is affirmative when R is essentially of finite type over the complex numbers, but negative in general

K-regularity, cdh-fibrant hochschild homology, and a conjecture of vorst

Fil:Cortiñas, G. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina