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Connective Algebraic K-theory
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- Journal:
- Journal of K-Theory / Volume 13 / Issue 1 / February 2014
- Published online by Cambridge University Press:
- 02 January 2014, pp. 9-56
- Print publication:
- February 2014
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- Article
- Export citation
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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 ΩdCK(X), and this fundamental class is functorial with respect to pull-back for l.c.i. morphisms.
, defined by taking the −1 connective cover of algebraic 
when the base scheme is the spectrum of a field 
may be viewed as connective algebraic 
for 
in 
, where 
is the abelian category of coherent sheaves on 
with the universal oriented Borel-Moore homology theory 
having formal group law 