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Using hypercohomology, we can extend cyclic homology from algebras to all schemes over a ring k. By ‘extend’ we mean that the usual cyclic homology of any commutative algebra agrees with the cyclic homology of its corresponding affine scheme. The purpose of this paper is to show that there is a cyclic homology theory HC∗ of schemes over a commutative ring… (More)

Given a ring R, it is known that the topological space BGl(R) is an infinite loop space. One way to construct an infinite loop structure is to consider the category = of free R-modules, or rather its classifying space BF=, as food for suitable infinite loop space machines. These machines produce connective spectra whose zeroth space is (BF=) + = Z×BGl(R).… (More)

We relate the negative K-theory of a normal surface to a resolution of singularities. The only nonzero K-groups are K −2, which counts loops in the exceptional fiber, and K −1, which is related to the divisor class groups of the complete local rings at the singularities. We also verify two conjectures of Srinivas about K0-regularity and K−1 of a surface.… (More)

Let KR be a nonconnective spectrum whose homotopy groups give the algebraic K-theory of the ring R. We give a description of the associated homology theory KR∗(X) associated to KR. We also show that the various constructions of KR in the literature are homotopy equivalent, and so give the same homology theory

- Charles Weibel
- 1994

We relate the “Hodge filtration” of the cohomology of a complex algebraic variety X to the “Hodge decomposition” of its cyclic homology. If X is smooth and projective, HC (i) n (X) is the quotient of the Betti cohomology H 2i−n(X(C);C) by the (i + 1) piece of the Hodge filtration.

Using recent results of Voevodsky, Suslin-Voevodsky and BlochLichtenbaum, we completely determine the 2-torsion subgroups of the K-theory of the integers Z. The result is periodic of order 8, and there are no 2-torsion elements except those which have been known for over 20 years. There is no 2-torsion except for the Z/2 summands in degrees 8n + 1 and 8n +… (More)

- Charles Weibel
- 2005

This survey describes the algebraic K-groups of local and global fields, and the K-groups of rings of integers in these fields. We have used the result of Rost and Voevodsky to determine the odd torsion in these groups.

- Paolo Salmon, Claudio Pedrini, Charles Weibel

In this paper we study several algebraic invariants of a real curve X: the Picard group Pic(X), the Brauer group Br(X), the K-theory matrix invariant SK1(X), the étale cohomology groups H(X,Z/2), and the sheaves H associated to these cohomology groups. We relate these to two topological invariants of the space X(R) of real points of X: the number c of… (More)

- CLAUDIO PEDRINI, CHARLES WEIBEL
- 2000

If X is a smooth curve defined over the real numbers R, we show that Kn(X) is the sum of a divisible group and a finite elementary abelian 2-group when n ≥ 2. We determine the torsion subgroup of Kn(X), which is a finite sum of copies of Q/Z and Z/2, only depending on the topological invariants of X(R) and X(C), and show that (for n ≥ 2) these torsion… (More)