Cyclic homology, cdh-cohomology and negative K-theory

Abstract

We prove a blow-up formula for cyclic homology which we use to show that infinitesimal K-theory satisfies cdh-descent. Combining that result with some computations of the cdh-cohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic K-theory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero.

Localization theorems in topological Hochschild homology and topological cyclic homology

“…First, we prove the following formula for blow-ups along regular sequences, which already has been applied by Geisser and Hesselholt [17] to prove the characteristic p analogue of Weibel's conjecture. We state the theorem using the notation of [8,Section 1], and prove it in Section 8. We also prove a projective bundle theorem [43, 4.1,7.3] in Section 8.…”

Differential graded categories

“…We refer to the recent proof [28] of Weibel's conjecture [162] on the vanishing of negative K-theory for an application of the theorem. The algebraic description of topological Hochschild (co-)homology [142] would suggest that it is also preserved under topological Morita equivalence but no reference seems to exist as yet.…”

Infinitesimal cohomology and the Chern character to negative cyclic homology

“…We conclude this section with a comparison (Theorem 5.9) between infinitesimal hypercohomology H(X inf , K ) and the "infinitesimal K -theory" used in our earlier papers [12] and [13] and based upon the construction in the eponymous paper [9]. Definition 5.7 Let K inf (X ) denote the homotopy fiber of the Chern character ch :…”