Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T16:51:17.106Z Has data issue: false hasContentIssue false
  • English
  • Français

K1 of Exact Categories by Mirror Image Sequences

Published online by Cambridge University Press:  19 April 2012

Clayton Sherman
Clayton Sherman*
Affiliation:
Department of Mathematics, Missouri State University, Springfield, MO 65897, USAcsherman@missouristate.edu
Get access

Abstract

We establish a presentation for K1 of any small exact category P, based on the notion of “mirror image sequence,” originally introduced by Grayson in 1979; as part of the proof, we show that every element of K1(P) arises from a mirror image sequence. This provides an alternative to Nenashev's presentation in terms of “double short exact sequences.”

Keywords

Algebraic K-theoryK1double short exact sequencemirror image sequencePrimary 19D99
Type
Research Article
Information
Journal of K-Theory , Volume 11 , Issue 1 , February 2013 , pp. 155 - 181
Copyright
Copyright © ISOPP 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Gillet, H. and Grayson, D., On the Loop Space of the Q-construction, Ill. J. of Math. 31 (1987), 574–597.Google Scholar
2.Grayson, D., Localization for Flat Modules in Algebraic K-Theory, J. of Algebra 61 (1979), 463–496.Google Scholar
3.Nenashev, A., Double short exact sequences produce all elements of Quillen's K 1, in Banaszak, , et al, (ed), Algebraic K-Theory, Contemp. Math. 199 (1996), 151–160.Google Scholar
4.Nenashev, A., Double short exact sequences and K 1 of an exact category, K-Theory 14 (1998), 23–41.Google Scholar
5.Nenashev, A., K 1 by generators and relations, J. Pure and Applied Algebra 131 (1998), 195–212.CrossRefGoogle Scholar
6.Sherman, C., On K 1 of an Abelian Category, J. of Algebra 163 (1994), 568–582.Google Scholar
7.Sherman, C., On K 1 of an Exact Category, K-Theory 14 (1998), 1–22.CrossRefGoogle Scholar
8.Sherman, C., Connecting homomorphisms in localization sequences: II, K-Theory 32 (2004), 365–389.Google Scholar
9.Sherman, C., G 1 of Certain Commutative Noetherian Rings, in progress.Google Scholar