On \(K_ 1\) of an abelian category
Let \({\mathcal C}\) be an exact category. After Quillen its higher \(K\)-groups are defined as \[ K_ i({\mathcal C})\colon = \pi_{i+1} (BQ({\mathcal C}),0) \] where \(Q({\mathcal C})\) is a certain category naturally associated to \({\mathcal C}\), and \(BQ({\mathcal C})\) is its classifying space. Due to Gillet and Grayson the \(H\)-space \(G({\mathcal C})\colon = \Omega BQ({\mathcal C})\) can be described explicitly as the geometric realization of a simplicial monoid \(g({\mathcal C})\) with \(q\)-simplices \[ \Delta_ q\colon = \{(K_ 0 \hookrightarrow \cdots \hookrightarrow K_ q, L_ 0 \hookrightarrow \cdots\hookrightarrow L_ q,\;\Phi_{ij}\colon\;K_ j/K_ i \simeq L_ j/L_ i)\} \] consisting of pairs of strings of admissible monomorphisms in \({\mathcal C}\) together with compatible systems of isomorphisms identifying the corresponding subquotients. The simplicial structure is the obvious one, addition is induced by taking direct sums and the inversion by interchanging the two strings. From this it is clear that elements in \[ K_ 1({\mathcal C})= \pi_ 1(G({\mathcal C}),0)= \pi_ 1(| g({\mathcal C})|, 0) \] can be represented by diagrams of pairs of objects in \({\mathcal C}\) which correspond to closed loops in \(| g({\mathcal C})|\). The minimal size of the diagram representing a class in \(K_ 1\) cannot be predicted a priori in general. It can be bounded however if \({\mathcal C}\) happens to be an abelian category as the author shows in this note. Explicitly, his result is the following: Let \({\mathcal C}\) be an abelian category. Let \[ 0\to A\to X\to C\to 0 \qquad \text{and} \qquad 0\to B\to Y\to D\to 0 \] be a pair of exact sequences in \({\mathcal C}\), and let \(\Theta\colon A\oplus C\oplus Y\to B\oplus D\oplus X\) be an isomorphism. Then the loop \[ (0,0)\to (A,A)\to (A\oplus C\oplus Y, X\oplus Y)\to (B\oplus D\oplus X, Y\oplus X)\leftarrow (B,B)\leftarrow (0,0) \] in \(| g({\mathcal C})|\) passing through the indicated vertices and following the edges given by inclusion of the pair of objects of the source into the pair of objects of the target of an arrow defines an element of \(K_ 1({\mathcal C})\), and every element of this group is obtained in the described way.
scientific article; zbMATH DE number 554221
scientific article