Koszul homology and extremal properties of Gin and Lex

where the subscript j on the right of a graded module denotes, throughout the paper, the degree j component of that module. There are two monomial ideals canonically attached to I: the generic initial ideal Gin(I) with respect to the revlex order and the lex-segment ideal Lex(I). They play a fundamental role in the investigation of many algebraic, homological, combinatorial and geometric properties of the ideal I itself. By definition, the generic initial ideal Gin(I) is the initial ideal of I with respect to the revlex order after performing a generic change of coordinates. The ideal Lex(I) is defined as follows. For every vector space V of forms of degree, say, d one defines Lex(V ) to be the vector space generated by the largest, in the lexicographic order, dimV monomials of degree d. For a homogeneous ideal I one sets Lex(I) = ⊕d Lex(Id). By the very definition, Lex(I) is simply a graded vector space but Macaulay’s theorem on Hilbert functions, see for instance [V, Sect.1], says that Lex(I) is indeed an ideal. By construction, it is clear that Lex(I) only depends on the Hilbert function of I. The graded Betti numbers of I,Gin(I) and Lex(I) satisfy the following inequalities: