On the commutation of the test ideal with localization and completion

DOI10.1090/S0002-9947-01-02643-5zbMath0977.13002MaRDI QIDQ2716133

Karen E. Smith, Gennady Lyubeznik

Publication date: 6 June 2001

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

zbMATH Keywords

Cohen-Macaulay ringcharacteristic \(p\)test idealtight closurelocalizationFrobenius actionexcellent regular local ring of prime characteristicnon-Gorenstein rings

Mathematics Subject Classification ID

Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) (13H10) Characteristic (p) methods (Frobenius endomorphism) and reduction to characteristic (p); tight closure (13A35) Excellent rings (13F40) Theory of modules and ideals in commutative rings (13C99)

Related Items (61)

Global parameter test ideals ⋮ Counting skew monomials in the Frobenius skew polynomial ring ⋮ The Frobenius complexity of a local ring of prime characteristic ⋮ SOME CONDITIONS FOR THE EQUIVALENCE OF WEAK AND STRONG F-REGULARITY ⋮ A sufficient condition for strong $F$-regularity ⋮ Local Cohomology—An Invitation ⋮ On the discreteness and rationality of \(F\)-jumping coefficients ⋮ Graded annihilators and tight closure test ideals ⋮ On the existence of 𝐹-thresholds and related limits ⋮ $F$-singularities: applications of characteristic $p$ methods to singularity theory ⋮ Tight closure and strongly F-regular rings ⋮ On the Frobenius complexity of determinantal rings ⋮ \(D\)-modules, Bernstein-Sato polynomials and \(F\)-invariants of direct summands ⋮ \(F\)-jumping and \(F\)-Jacobian ideals for hypersurfaces ⋮ Compatible ideals in ℚ-Gorenstein rings ⋮ Tate algebras and Frobenius non-splitting of excellent regular rings ⋮ A refinement of sharply \(F\)-pure and strongly \(F\)-regular pairs ⋮ On the equality of test ideals ⋮ Frobenius splitting, strong F-regularity, and small Cohen-Macaulay modules ⋮ Openness of splinter loci in prime characteristic ⋮ Big tight closure test elements for some non-reduced excellent rings ⋮ TEST IDEALS IN RINGS WITH FINITELY GENERATED ANTI-CANONICAL ALGEBRAS ⋮ On the Frobenius complexity of Stanley–Reisner rings ⋮ Finitistic test ideals on numerically \(\mathbb{Q}\)-Gorenstein varieties ⋮ A Koszul complex over skew polynomial rings ⋮ Test ideals of non-principal ideals: computations, jumping numbers, alterations and division theorems ⋮ Tight closure with respect to a multiplicatively closed subset of an \(F\)-pure local ring ⋮ Strong \(F\)-regularity and generating morphisms of local cohomology modules ⋮ A CharacteristicpAnalog of Multiplier Ideals and Applications^# ⋮ Closure-interior duality over complete local rings ⋮ A dual to tight closure theory ⋮ On a generalization of test ideals ⋮ A non-finitely generated algebra of Frobenius maps ⋮ Frobenius and Cartier algebras of Stanley-Reisner rings ⋮ \(F\)-signature of pairs and the asymptotic behavior of Frobenius splittings ⋮ \(F\)-signature exists ⋮ Relative test elements for tight closure ⋮ Fiber cones, analytic spreads of the canonical and anticanonical ideals and limit Frobenius complexity of Hibi rings ⋮ Globally \(F\)-regular and log Fano varieties ⋮ Centers of \(F\)-purity ⋮ Discreteness and rationality of \(F\)-jumping numbers on singular varieties ⋮ Extension of weakly and strongly F-regular rings by flat maps ⋮ Observations on the \(F\)-signature of local rings of characteristic \(p\) ⋮ An excellent $F$-pure ring of prime characteristic has a big tight closure test element ⋮ Excellence in prime characteristic ⋮ Behavior of test ideals under smooth and étale homomorphisms. ⋮ Strong test ideals ⋮ The Frobenius complexity of Hibi rings ⋮ Uniform approximation of Abhyankar valuation ideals in function fields of prime characteristic ⋮ Addendum to ``Frobenius and Cartier algebras of Stanley-Reisner rings ⋮ Test ideals in non-$\mathbb{Q}$-Gorenstein rings ⋮ Positive characteristic algebraic geometry ⋮ On the behavior of test ideals under finite morphisms ⋮ Frobenius and Cartier algebras of Stanley-Reisner rings. II ⋮ Nakayama closures, interior operations, and core-hull duality -- with applications to tight closure theory ⋮ Generating functions associated to Frobenius algebras ⋮ The F-rational signature and drops in the Hilbert-Kunz multiplicity ⋮ On the topological structure of the (non-)finitely generated locus of Frobenius algebras emerging from Stanley–Reisner rings ⋮ Graded annihilators and uniformly \(F\)-compatible ideals ⋮ Rings of Frobenius operators ⋮ Test ideals and base change problems in tight closure theory

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Some basic results on canonical modules
• Tight closure of parameter ideals
• Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism
• Applications of the existence of big Cohen-Macaulay algebras
• F-rational rings have rational singularities
• Tight closure and strong F-regularity
• Tight Closure, Invariant Theory, and the Briancon-Skoda Theorem
• General Néron desingularization and approximation
• On Noetherian Rings of Characteristic p
• Test ideals in quotients of $F$-finite regular local rings
• F-Regularity, Test Elements, and Smooth Base Change
• F-modules: applications to local cohomology and D-modules in characteristic p>0.
• Some results on test elements
• Strong F-regularity in images of regular rings
• Test Ideals in Local Rings
• Strong and weak F -regularity are equivalent for graded rings
• General Néron desingularization

This page was built for publication: On the commutation of the test ideal with localization and completion