TY - JOUR

T1 - Geometric interpretation of tight closure and test ideals

AU - Kara, Nobuo

PY - 2001

Y1 - 2001

N2 - We study tight closure and test ideals in rings of characteristic p 3> 0 using resolution of singularities. The notions of F-rational and Fregular rings are defined via tight closure, and they arc 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 3> 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.

AB - We study tight closure and test ideals in rings of characteristic p 3> 0 using resolution of singularities. The notions of F-rational and Fregular rings are defined via tight closure, and they arc 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 3> 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.

KW - Modulo p reduction

KW - Multiplier ideal

KW - Test ideal

KW - Tight closure

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M3 - Article

AN - SCOPUS:23044525810

VL - 353

SP - 1885

EP - 1906

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 5

ER -