Geometric Interpretation of Tight Closure and Test Ideals

We study tight closure and test ideals in rings of characteristic p 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 Q-Gorenstein ring of characteristic p 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. The notion of the tight closure of an ideal in a commutative ring of prime characteristic was defined by Hochster and Huneke [HH1] in terms of the asymptotic behavior of the ideal under iteration of the Frobenius map. Tight closure enables us to define the notions of F -regular rings [HH1] and F -rational rings [FW]. Namely, a ring of characteristic p > 0 is called F -regular (resp. F -rational) if all ideals (resp. all ideals generated by a system of parameters) are tightly closed in all of its local rings. Although these concepts are defined quite ring-theoretically, they have been suspected to have a mysterious correspondence with some classes of singularities in characteristic zero defined via resolution of singularities. Surprisingly, recent results by Smith [S2], Watanabe [W3], Mehta and Srinivas [MS], and the author [Ha] conclude that a ring in characteristic zero has at most rational (resp. log terminal) singularities if and only if its reduction modulo p is F -rational (resp. F -regular and Q-Gorenstein) for p 0. The aim of this paper is to generalize these results to wider classes of singularities. To do this we use a fairly standard technique of “reduction modulo p,” starting from a singularity in characteristic zero. Let (R,m) be a d-dimensional normal local ring which is reduced from characteristic zero to characteristic p 0, together with a resolution of singularities f : X → SpecR. When the non-(F -)rational locus of (R,m) is isolated, we actually proved in [Ha] that the tight closure of the zero Received by the editors July 27, 1999. 2000 Mathematics Subject Classification. Primary 13A35, 14B05; Secondary 13A02, 14B15.