\(K_1\) by generators and relations
The author obtains a presentation for the algebraic \(K\)-theory group \(K_1({\mathcal A})\), where \({\mathcal A}\) is an exact category. The generators are double short exact sequences; that is, they are pairs of short exact sequences with the same objects but possibly with different morphisms. An automorphism \(\alpha \colon A \to A\) yields a double short exact sequence \[ 0 \to 0 \to A \to A \to 0, \quad 0 \to 0 \to A \to A \to 0, \] where the endomorphisms of \(A\) are the identity and \(\alpha\), and a double short exact sequence can be regarded as a generalised automorphism. The members of \(K_1({\mathcal A})\) can be regarded as equivalence classes of double short exact sequences, because every member of \(K_1({\mathcal A})\) is the image of a generator.
scientific article; zbMATH DE number 1308230