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 N-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 N-graded ring over a field with unique homogeneous maximal ideal m, then R is (weakly) F-regular if and only if Rm is (weakly) F-regular.
|Original language||English (US)|
|Number of pages||12|
|Journal||American Journal of Mathematics|
|State||Published - Dec 1 1999|