The cyclic homology and 𝐾-theory of curves

Abstract

ABSTRACT. It is now possible to calculate the K"-theory of a large class of singular curves over fields of characteristic zero. Roughly speaking, the ÜT-theory of a curve is the X-theory of its (smooth) normalization plus a few shifted copies of the if-theory of the field plus a "nil part." The nil part is a vector space depending only on the analytic type of the singularities, and may be computed locally. We completely compute the nil part for seminormal curves and give a conjectural calculation in general which depends upon cyclic homology.Until recently, very little has been known about the higher algebraic Ktheory of anything but finite fields. In this note we announce the computability of the X-theory of singular curves in characteristic zero in terms of the K-theory of smooth curves and fields. If the curve is seminormal, we give a complete calculation; otherwise, the calculation depends on the validity of: CONJECTURE. Let B be a finite integral extension of a ring A, and let I be the conductor ideal. Assume A contains Q, the rational numbers. Then

Algebraic Topology: Applications and New Directions

“…We finally mention that for k a field of characteristic zero, the cyclic homology groups of A were calculated by Geller, Reid, and Weibel [12,Theorem 9.2] and that, in view of the affirmation by Cortiñas [4] of the KABI-conjecture made in [12], this gives a complete calculation of the groups K q (A, a).…”

Excision for $K$-theory of connective ring spectra

“…In a series of papers ( [11], [8], [10], [9]) Geller, Reid and Weibel explored the idea that cyclic homology should be a precise measure for the failure of excision in the algebraic K-theory of Q-algebras, and did some conjectural calculations. The problem remained open (although it IS an exercise in [13]), until Cortiñas released a preprint [3] claiming the conjecture using Suslin and Wodzicki's results on nonunital rings [18].…”

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

“…[11]. In [10] the statement of Corollary 0.2 was conjectured to hold when A and B are commutative unital Q-algebras and B is a finite integral extension of A, (KABI conjecture) and it was shown its validity permits computation of the K-theory of singular curves in terms of their cyclic homology and of the K-theory of nonsingular curves. In [12] 0.2 was conjectured for unital Q-algebras and it was shown that for * = 2 the left hand side maps surjectively onto the right hand side.…”

“…An application of 0.3 to the computation of the K-theory of particular rings -other than coordinate rings of curves-in terms of their cyclic homology was given in [13,Thm. 3.1]; see also [10,Thm. 7.3].…”

“…I learned about the existence of the KABI conjecture in 1991 from C. Weibel. His papers with Geller and Reid ( [10]) and Geller ([11], [12], [13]) demonstrated its potential applications, produced first positive results and thus made it attractive as a problem. I am thankful to all three of them for calling my attention to it through their work, as well as to J. Cuntz, D. Quillen, A. Suslin and M. Wodzicki, for their results are crucial to this paper.…”