Regularity and h-polynomials of monomial ideals

摘要

Let S=K[x(1),...,x(n)] denote the polynomial ring in n variables over a field K with each deg x(i)=1 and IIS a homogeneous ideal of S with dim S/I=d. The Hilbert series of S/I is of the form h(S/I)(lambda)/(1-lambda)(d), where h(S/I)(lambda)=h(0)+h(1)lambda+h(2)lambda(2)+...+h(s)lambda(s) with h(s)not equal 0 is the h-polynomial of S/I. It is known that, when S/I is Cohen--Macaulay, one has reg(S/I) = deg h(S/I)(lambda), where reg(S/I) is the (Castelnuovo--Mumford) regularity of S/I. In the present paper, given arbitrary integers r and s with r = 1 and s = 1, a monomial ideal I of S=K[x(1),...,x(n)] with nc0 for which reg(S/I) = r and deg h(S/I)(lambda)=s will be constructed. Furthermore, we give a class of edge ideals IIS of Cameron--Walker graphs with reg(S/I) = deg(S/I) (lambda) for which S/I is not Cohen- Macaulay.