Strong and weak F -regularity are equivalent for graded rings

Abstract

It is shown that the tight closure of a submodule in a Artinian module is the same as its finitistic tight closure, when the modules are graded over a finitely generated [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]-graded ring over a perfect field. As a corollary, it is deduced that for such a graded ring, strong and weak F -regularity are equivalent. As another application, the following conjecture of Hochster and Huneke is proved: Let ( R, m ) be a finitely generated [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]-graded ring over a field with unique homogeneous maximal ideal m , then R is (weakly) F -regular if and only if R m is (weakly) F -regular.