Shellability of simplicial complexes arising in representation theory
Let \(k\) be an algebraically closed field and let \(A\) be a finite-dimensional \(k\)-algebra. Recall that an \(A\)-module \(T\) is called tilting provided: (a) the projective dimension of \(T\) is finite; (b) \(\text{Ext}^i_A(T,T)=0\) for each \(i>0\); and (c) there exists an exact sequence \(0\to A\to T_0\to T_1\to\cdots\to T_r\to 0\), with \(T_i\in\text{add }T\). Tilting theory is nowadays an important tool to understand the category of \(A\)-modules. As observed by the author, C. Ringel has noticed that the set of tilting \(A\)-modules forms a simplicial complex \(\Sigma\) and suggested its study. One of the aims of the paper under review is to give a uniform approach to this study made mainly by Riedtmann, Schofield and Unger. More than that, the author has deeepened the results and has related them with some other lines of research in representation theory.
scientific article; zbMATH DE number 1334588
scientific article