Geometric interpretation of tight closure and test ideals

DOI10.1090/S0002-9947-01-02695-2zbMath0976.13003MaRDI QIDQ2701684

Nobuo Hara

Publication date: 19 February 2001

Published in: Transactions of the American Mathematical Society (Search for Journal in Brave)

zbMATH Keywords

test idealtight closuremultiplier idealresolutions of singularitiesmodulo \(p\) reduction

Mathematics Subject Classification ID

Singularities in algebraic geometry (14B05) Integral closure of commutative rings and ideals (13B22) Graded rings (13A02) Local cohomology and algebraic geometry (14B15) Characteristic (p) methods (Frobenius endomorphism) and reduction to characteristic (p); tight closure (13A35)

Related Items (37)

F-singularities of pairs and inversion of adjunction of arbitrary codimension ⋮ F-regular and F-pure rings vs. log terminal and log canonical singularities ⋮ Bernstein–Sato functional equations, V-filtrations, and multiplier ideals of direct summands ⋮ Functorial test modules ⋮ Syzygies of multiplier ideals on singular varieties ⋮ Rational singularities associated to pairs ⋮ A generalization of tight closure and multiplier ideals ⋮ A New Version of α-Tight Closure ⋮ \(F\)-singularities under generic linkage ⋮ An interpretation of multiplier ideals via tight closure ⋮ Legendre polynomials roots and the \(F\)-pure threshold of bivariate forms ⋮ $F$-singularities: applications of characteristic $p$ methods to singularity theory ⋮ Tate algebras and Frobenius non-splitting of excellent regular rings ⋮ Test ideals in diagonal hypersurface rings. II. ⋮ \(F\)-nilpotent rings and permanence properties ⋮ TEST IDEALS IN RINGS WITH FINITELY GENERATED ANTI-CANONICAL ALGEBRAS ⋮ The jumping coefficients of non-\(\mathbb{Q}\)-Gorenstein multiplier ideals ⋮ Finitistic test ideals on numerically \(\mathbb{Q}\)-Gorenstein varieties ⋮ Cohen-Macaulay test ideals over rings of finite and countable Cohen-Macaulay type ⋮ A CharacteristicpAnalog of Multiplier Ideals and Applications^# ⋮ Test ideals via algebras of 𝑝^{-𝑒}-linear maps ⋮ Characteristic-free test ideals ⋮ A characteristic \(p\) analogue of plt singularities and adjoint ideals ⋮ Centers of \(F\)-purity ⋮ Discreteness and rationality of \(F\)-jumping numbers on singular varieties ⋮ Excellence in prime characteristic ⋮ An elementary computation of the \(F\)-pure threshold of an elliptic curve ⋮ Uniform approximation of Abhyankar valuation ideals in function fields of prime characteristic ⋮ Singularities on normal varieties ⋮ On the arithmetic of tight closure ⋮ Singularities in mixed characteristic via perfectoid big Cohen-Macaulay algebras ⋮ Test ideals in non-$\mathbb{Q}$-Gorenstein rings ⋮ Positive characteristic algebraic geometry ⋮ On the behavior of test ideals under finite morphisms ⋮ Comparing multiplier ideals to test ideals on numerically ℚ‐Gorenstein varieties ⋮ A new subadditivity formula for test ideals ⋮ Poincaré series of multiplier and test ideals

Cites Work

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• Pseudo-rational local rings and a theorem of Briancon-Skoda about integral closures of ideals
• \(F\)-regular and \(F\)-pure normal graded rings
• The purity of the Frobenius and local cohomology
• Normal \(\mathbb{Z}_ r\)-graded rings and normal cyclic covers
• The injectivity of Frobenius acting on cohomology and local cohomology modules
• Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism
• Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen
• Residues and duality. Lecture notes of a seminar on the work of A. Grothendieck, given at Havard 1963/64. Appendix: Cohomology with supports and the construction of the \(f^!\) functor by P. Deligne
• The multiplier ideal is a universal test ideal
• F-rational rings have rational singularities
• Tight closure and strong F-regularity
• Tight Closure, Invariant Theory, and the Briancon-Skoda Theorem
• Multiplier ideal sheaves and existence of Kähler-Einstein metrics of positive scalar curvature
• A characterization of rational singularities in terms of injectivity of Frobenius maps
• A Frobenius Characterization of Rational Singularity in 2-Dimensional Graded Rings
• Multiplier Ideals, Vanishing Theorem and Applications
• Some results on test elements
• F-regular and F-pure rings vs. log terminal and log canonical singularities
• Test Ideals in Local Rings
• Strong and weak F -regularity are equivalent for graded rings
• Note on Kodaira-Spencer's proof of Lefschetz theorems
• A characterization of rational singularities

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