K 1 of Exact Categories by Mirror Image Sequences

Abstract

We establish a presentation for K 1 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 K 1 ( P ) arises from a mirror image sequence. This provides an alternative to Nenashev's presentation in terms of “double short exact sequences.”