Connective algebraic K-theory

Abstract

We examine the theory of connective algebraic K-theory, CK, defined by taking the -1 connective cover of algebraic K-theory with respect to Voevodsky's slice tower in the motivic stable homotopy category. We extend CK to a bi-graded oriented duality theory (CK', CK) in case the base scheme is the spectrum of a field k of characteristic zero. The homology theory CK' may be viewed as connective algebraic G-theory. We identify CK' theory in bi-degree (2n, n) on some finite type k-scheme X with the image of K_0(M(X,n)) in K_0(M(X, n+1)), where M(X,n) is the abelian category of coherent sheaves on X with support in dimension at most n; this agrees with the (2n,n) part of the theory defined by Cai. We also show that the classifying map from algebraic cobordism identifies CK' with the universal oriented Borel-Morel homology theory Ω_*^{CK}:=Ω_*\otimes_L\Z[β] having formal group law u+v-βuv with coefficient ring \Z[β]. As an application, we show that every pure dimension d finite type k scheme has a well-defined fundamental class [X] in Ω_d^{CK}(X), and this fundamental class is functorial with respect to pull-back for lci morphisms. Finally, the fundamental class maps to the fundamental class in G-theory after inverting β, and to the fundamental class in CH after moding out by β.