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Deformation classes of graded modules and maximal Betti numbers

1996; Duke University Press; Volume: 40; Issue: 4 Linguagem: Inglês

10.1215/ijm/1255985937

1945-6581

Keith Pardue,

Rings, Modules, and Algebras

graded free S-module of rank r.Fix a basis e er of F where each ei has degree di and dl < < dr.Definitions 1.A monomial of F is an element x uei where x u x' .xnU" is a monomial of S. The lexicographic order on monomials of S is the order in whichx u > x if #s > Vs and #i 1)i for every < s.The lexicographic order on monomials of F is the order in which xZei > x vej if < j, or j and x u > x .A monomial subspace of Fd, the vector space of homogeneous elements of F of degree d, is a subspace spanned by a set of monomials.The lexicographic subspace of Fd of dimension e is the monomial subspace spanned by the first e monomials of F in lexicographic order.A submodule L of F is a lexicographic submodule if it is graded and L is a lexicographic subspace of F for every d.Note that in the definition of lexicographic order, I do not compare the degrees of monomials as one would with degree-lexicographic order.PROPOSITION 2 (Macaulay [Ma], Hulett [Hu 1,3]).Let N be a graded submodule of F. Then there is a lexicographic submodule L of F such that dim Lt dim Nd for every d.Macaulay proved Proposition 2 in the case that F S so that L is a lexicographic ideal.Hulett proved the theorem in the form stated above.Macaulay also proved that among all homogeneous ideals with the same Hilbert function, that is with the same dimension in every degree, the lexicographic ideal has the largest number of minimal generators of each degree.It is not difficult to compute the minimal generators for the lexicographic ideal for a given Hilbert function, so it is easy to bound the number of generators that an ideal requires in each degree if we know its Hilbert function.Bigatti and Hulett proved a remarkable generalization of this theorem when k has characteristic zero.If M is a finitely generated graded S-module with minimal graded free resolution 0--F, -+ ---Fo-+ M -+'0, then iij (M) is the number of degree j minimal generators of Fi.These numbers are the graded Betti numbers of M. They are well defined; in fact ij(M) dim Tor/S(M, S/m)j where m (x xn).See [EvGr].THEOREM 3 (Bigatti [Bi], Hulett ).If k has characteristic zero, N is a graded submodule of F and L is the lexicographic submodule of F such that FIN and F/L have the same Hilbert function, then Iij(F/N) < ij(F/L) for every and j.Bigatti and Hulett independently proved this theorem for F S and Hulett later proved this theorem in the form above.The theorem is a generalization of Macaulay's because the number of minimal generators of N of degree j is lj(F/N).