Connective algebraic Ktheory

General Introduction
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[arXiv:1212.0228] Connective algebraic Ktheory from arXiv Front: math.AT by Shouxin Dai, Marc Levine We examine the theory of connective algebraic Ktheory, CK, defined by taking the 1 connective cover of algebraic Ktheory with respect to Voevodsky's slice tower in the motivic stable homotopy category. We extend CK to a bigraded 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 Gtheory. We identify CK' theory in bidegree (2n, n) on some finite type kscheme X with the image of K0(M(X,n)) in K0(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 BorelMorel homology theory \Omega*^{CK}:=\Omega*\otimesL\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 welldefined fundamental class [X] in \Omegad^{CK}(X), and this fundamental class is functorial with respect to pullback for lci morphisms. Finally, the fundamental class maps to the fundamental class in Gtheory after inverting \beta, and to the fundamental class in CH after moding out by \beta.
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Paper References
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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
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Computations and Examples
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History and Applications
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Some Research Articles
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Other Information
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