Koszul cycles and Eliahou-Kervaire type resolutions
The aim of this paper is to compute a minimal free resolution of a finitely generated \(R\) module \(M\), in terms of a basis of \(H(M)\), where \((R, {\mathfrak M})\) is a regular local ring containing its residue class field, \(H(M)\) is the homology of the Koszul complex \(K (M)\) of \(M\) with respect to a minimal system \((x_1, \dots, x_n)\) of generators of \({\mathfrak M}\). Such a problem was already considered in previous papers by the same authors, but here it is treated in a more general situation. A difficult point in the procedure is the computation of a basis for \(H(M)\). So the authors consider the special case of stable ideals, in which it is easy to compute the Koszul cycles, finding again the Eliahou-Kervaire resolution. A relation between stability of \(I\) and proper sequences suggests to consider modules \(M\) for which \(x_n, \dots, x_1\) is proper: For such modules the author are able to describe a basis of \(H(M)\) and, as a consequence, to apply their results. Finally, they give an explicit resolution of the \(k\)-algebras for which \(x_n,\dots,x_1\) is ``strongly proper''.
scientific article; zbMATH DE number 883764