Connective Algebraic K-theory. Issue 1 (2nd January 2014) - Dai, Shouxin; Levine, Marc
Abstract We examine the theory of connective algebraic K-theory, <inline-graphic xlink:href="ark:/27927/pghjkpnx9r" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />, 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 <inline-graphic xlink:href="ark:/27927/pghjkpnx9r" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> to a bi-graded oriented duality theory <inline-graphic xlink:href="ark:/27927/pghjkpnxmd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> when the base scheme is the spectrum of a field k of characteristic zero. The homology theory <inline-graphic xlink:href="ark:/27927/pghjkpnxkw" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> may be viewed as connective algebraic G-theory. We identify <inline-graphic xlink:href="ark:/27927/pghjkpnxpf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> for X a finite type k-scheme with the image of <inline-graphic xlink:href="ark:/27927/pghjkpnxcs" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> in <inline-graphic xlink:href="ark:/27927/pghjkpnxb8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />, where <inline-graphic xlink:href="ark:/27927/pghjkpnxjc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> 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 <inline-graphic xlink:href="ark:/27927/pghjkpnxhv" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> with the universal oriented Borel-Moore homology theory <inline-graphic xlink:href="ark:/27927/pghjkpnxnx" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> 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.