Tight closure with respect to a multiplicatively closed subset of an F-pure local ring

Abstract

Let R be a (commutative Noetherian) local ring of prime characteristic that is F-pure. This paper studies a certain finite set I of radical ideals of R that is naturally defined by the injective envelope E of the simple R-module. This set I contains 0 and R, and is closed under taking primary components. For a multiplicatively closed subset S of R, the concept of tight closure with respect to S, or S-tight closure, is discussed, together with associated concepts of S-test element and S-test ideal. It is shown that an ideal a of R belongs to I if and only if it is the S′-test ideal of R for some multiplicatively closed subset S′ of R. When R is complete, I is also ‘closed under taking test ideals’, in the following sense: for each proper ideal c in I, it turns out that R/c is again F-pure, and if g and h are the unique ideals of R that contain c and are such that g/c is the (tight closure) test ideal of R/c and h/c is the big test ideal of R/c, then both g and h belong to I. The paper ends with several examples.