Graph representation of projective resolutions. (Q2430316)

The author generalizes the concept dimension tree and the related results for monomial algebras to a more general case -- relations algebras \(\Lambda\) by bringing Gröbner bases into play. He describes the minimal resolution graph for \(M\). The paper is organized as follows: Sections 1 and 2 contain a brief introduction to the topics to be investigated and collect some definitions and well-known facts to be used later. In Section 3, he uses the combinatorial approach to analyze the minimal projective resolution of the simple modules at the vertices. A graphical description of such resolution is given in Section 4. In Theorem 4.2, he constructs a minimal \(\Lambda\)-projective resolution for the finitely generated \(\Lambda\)-module \(M\). In Section 5, the idea of graph representation -- the so-called minimal projective resolution graph for the minimal projective resolution of a given \(\Lambda\)-module \(M\) -- leads to, among others, two immediate applications, one related to the finitistic dimensions of the named algebras \(\Lambda\) and \(\Lambda^*\) is described. Finally in the last section algorithms and examples for computing such diagraphs and applications as well are presented.