Betti numbers of lex-segment ideals (Q1883032)

Consider the polynomial ring \(R = k[x_1,\dots,x_n]\) over a field of arbitrary characteristic. If \(I\) is a lex-segment ideal in \(R\) of degree \(d\), such that \(I_{d'}=0\) for \(d'<d\), then the authors give an explicit basis for the \(k\)-vector space of Koszul homology \(H_i(x_j, \dots, x_n; R/I)_{d+i-1}\). This allows them to write a formula for the lowest degree Betti numbers for lex-segment ideals. In the last section the authors provide a criterion for ideals that are lex-segment in a particular degree to be lex-segment in all degrees. They subsequently use this criterion in order to give another criterion for when a completely lex-segment ideal has a linear resolution, and to compute the Castelnuovo-Mumford regularity.