Geometric interpretation of tight closure and test ideals

Abstract

We study tight closure and test ideals in rings of characteristic $p \gg 0$ using resolution of singularities. The notions of $F$-rational and $F$-regular rings are defined via tight closure, and they are known to correspond with rational and log terminal singularities, respectively. In this paper, we reformulate this correspondence by means of the notion of the test ideal, and generalize it to wider classes of singularities. The test ideal is the annihilator of the tight closure relations and plays a crucial role in the tight closure theory. It is proved that, in a normal $\mathbb Q$-Gorenstein ring of characteristic $p \gg 0$, the test ideal is equal to so-called the multiplier ideal, which is an important ideal in algebraic geometry. This is proved in more general form, and to do this we study the behavior of the test ideal and the tight closure of the zero submodule in certain local cohomology modules under cyclic covering. We reinterpret the results also for graded rings.