Connective Algebraic K-theory. Issue 1 (2nd January 2014) - Dai, Shouxin; Levine, Marc
Abstract: We examine the theory of connective algebraic K -theory, , 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 to a bi-graded oriented duality theory when the base scheme is the spectrum of a field k of characteristic zero. The homology theory may be viewed as connective algebraic G -theory. We identify for X a finite type k -scheme with the image of in, where 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 of connective algebraic K -theory defined by Cai. We also show that the classifying map from algebraic cobordism identifies with the universal oriented Borel-Moore homology theory having formal group law u + υ − βuυ 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 ] CK in Ω d CK ( X ), and this fundamental class is functorial with respect to pull-back for l.c.i. morphisms.