An algorithm for calculating Betti numbers of manageable modules (Q1809748)

This paper deals with the free resolution of an \(R\)-module \(M\) of finite length \(l\), where \((R,m,k)\) is a noetherian local ring, of any characteristic. More precisely, \(M\) is supposed to be the cokernel of a matrix of the form \[ A=\left( \begin{matrix} {\mathfrak x} & {\mathfrak r}_{12} & \cdots & {\mathfrak r}_{1l} \\ & {\mathfrak r}_{23} & \cdots & {\mathfrak r}_{2l}\\ & &\cdots & \cdots\\ & & & {\mathfrak x}\end{matrix} \right), \] where \({\mathfrak x}= ^t(x_1,\dots,x_n)\) is the column vector of a minimal set of generators of \(m\), defining an \(R\)-regular sequence. Under a condition involving a set of matrices \(A_i\), \(1\leq i\leq n\), built starting from the matrices of the Koszul complex \(K({\mathfrak x},R)\) and the matrix \(A\), the author proves that a (non necessarily minimal) resolution of \(M\) is given by a complex of the form: \[ K.=\dots R^{{n\choose i}l}@>A_i>> R^{{n \choose i-1}l} \to\dots. \] The last part of the paper is devoted to an application of this results in the case: \(M=R/m^{N+1}+(\sum^N_ix_k^N)\), where \({\mathfrak x}=(x_1, \dots,x_n)\) is a minimal set of generators of the maximal ideal of a regular local ring \(R\). This computation leads to verify Horrocks' conjecture on \(r_i(M)\) in this special case.