Connective algebraic K-theory

TL;DR

This paper develops the theory of connective algebraic K-theory, extending it to a bi-graded duality theory, identifying its relation to algebraic G-theory and cobordism, and establishing fundamental classes for schemes.

Contribution

It introduces a bi-graded oriented duality theory for connective algebraic K-theory over fields of characteristic zero, linking it to G-theory and algebraic cobordism.

Findings

CK' theory matches the (2n,n) part of Cai's theory.

CK' is identified with the universal oriented Borel-Morel homology theory ng.

Fundamental classes in ng are functorial and relate to G-theory and Chow groups.

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 \Omega_*^{CK}:=\Omega_*\otimes_L\Z[\beta] having formal group law u+v-\beta uv with coefficient ring \Z[\beta]. As an application, we show that every pure dimension d finite type k scheme has a well-defined fundamental class [X] in \Omega_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 \beta, and to the fundamental class in CH after moding out by \beta.