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~(*,*)) when 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_(2n,n)~' (X) for X a finite type k-scheme with the image of K_0(M_(n)(X)) in K_0(M_((n+1)) (X)), where M_((n)) (X) 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 CK_(2*,*)~' with the universal oriented Borel-Moore homology theory ?_*~(CK):=?_* ?_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]_(CK) in ?_d~(CK)(X), and this fundamental class is functorial with respect to pull-back for l.c.i. morphisms.
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