Let U be the set of prime ideals P of the completion of a Stanley–Reisner ring S, such that the localization at P of the Frobenius algebra of the injective hull of the residue field of S is a finitely generated algebra. We give a partial answer to a conjecture made by M. Katzman about the openness of U. Specifically, we show that U has non-empty interior and we present some sufficient conditions for a principal open set D(f) to be contained in U, and for intersections of closed and principal opens to be contained in the complement of U.
