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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, [email protected]
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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

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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