Regularity and h‐polynomials of monomial ideals

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 let I ⊂ S be a homogeneous ideal of S with dim S / I = d . The Hilbert series of S / I is of the formh S / I( λ ) / ( 1 − λ ) d , whereh S / I( λ ) = h 0 + h 1 λ + h 2 λ 2 + ⋯ + h s λ swithh s ≠ 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( 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 [ x 1 , … , x n ]with n ≫ 0 for which reg ( S / I ) = r and deg h S / I( λ ) = s will be constructed. Furthermore, we give a class of edge ideals I ⊂ S of Cameron–Walker graphs with reg ( S / I ) = deg h S / I( λ )for which S / I is not Cohen–Macaulay.