Regularity and h‐polynomials of monomial ideals

Let S=K[x1,…,xn] denote the polynomial ring in n variables over a field K with each degxi=1 and let I⊂S be a homogeneous ideal of S with dimS/I=d. The Hilbert series of S/I is of the form hS/I(λ)/(1−λ)d, where hS/I(λ)=h0+h1λ+h2λ2+⋯+hsλs with hs≠0 is the h‐polynomial of S/I. It is known that, when S/I is Cohen–Macaulay, one has reg(S/I)=deghS/I(λ), 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[x1,…,xn] with n≫0 for which reg(S/I)=r and deghS/I(λ)=s will be constructed. Furthermore, we give a class of edge ideals I⊂S of Cameron–Walker graphs with reg(S/I)=deghS/I(λ) for which S/I is not Cohen–Macaulay.