Connective Algebraic K-theory. Issue 1 (2nd January 2014)

• Record Type: Journal Article
• Authors: Dai, Shouxin; Levine, Marc
• Is Part Of: Journal of K-Theory. Volume 13:Issue 1(2014)
• Page Start: 9
• Page End: 56
• Publication Date: 2014-01-02
• Subjects: K-theory -- Periodicals; 512.6605
• DOI: 10.1017/is013012007jkt249
• Languages: English
• ISSNs: 1865-2433

Abstract

<abstract abstract-type="normal"> <title>Abstract</title> <p>We examine the theory of connective algebraic <italic>K</italic>-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 <italic>K</italic>-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 <italic>k</italic> 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 <italic>G</italic>-theory. We identify <inline-graphic xlink:href="ark:/27927/pghjkpnxpf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /> for <italic>X</italic> a finite type <italic>k</italic>-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 <italic>X</italic> with support in dimension at most <italic>n</italic>; this agrees with the (2n, n) part of the theory of connective algebraic <italic>K</italic>-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 <italic>u</italic> + <italic>υ</italic> − <italic>βuυ</italic> with coefficient ring ℤ[β]. As an application, we show that every pure dimension <italic>d</italic> finite type <italic>k</italic>-scheme has a well-defined fundamental class [<italic>X</italic>]<sub><italic>CK</italic></sub> in Ω<sub><italic>d</italic></sub><sup><italic>CK</italic></sup>(<italic>X</italic>), and this fundamental class is functorial with respect to pull-back for l.c.i. morphisms.</p> </abstract>