Distractions of Shakin rings

Abstract

We study, by means of embeddings of Hilbert functions, a class of rings\ud which we call Shakin rings, i.e. quotients $K[X_1, . . . , X_n]/a$ of a polynomial ring over a field K by ideals a=$L+P$ which are the sum of a piecewise lex-segment ideal $L$, as defined by Shakin, and a pure powers ideal $P$. Our main results extend Abedelfatah’s recent work on the Eisenbud-Green-Harris Conjecture, Shakin’s generalization of Macaulay and Bigatti-Hulett-Pardue Theorems on Betti numbers and, when $\char(K) = 0$, Mermin-Murai Theorem on the Lex-Plus-Power inequality, from monomial regular sequences to a larger class of ideals. We also prove an extremality property of embeddings induced by distractions in terms of Hilbert functions of local cohomology modules

Betti numbers of piecewiselex ideals

“…Hence, we can assume char(K)=0. The characteristic zero result of Caviglia and Sbarra, [CS,Theorem 3.4], gives a lex ideal L such that P + L + L has the same Hilbert function as B and b ij (P + L + L) ≥ b ij (B) for all i, j. Again, since P + L + L is strongly-stable-plus-P , then the Betti numbers do not depend on the characteristic, so the inequality also holds for char (K) arbitrary.…”