Distractions of Shakin rings

We study, by means of embeddings of Hilbert functions, a class of rings 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.