Test ideals via algebras of $p^{-e}$-linear maps

Abstract

Continuing ideas of a recent preprint of Schwede arXiv:0906.4313 we study test ideals by viewing them as minimal objects in a certain class of $F$-pure modules over algebras of p^{-e}-linear operators. This shift in the viewpoint leads to a simplified and generalized treatment, also allowing us to define test ideals in non-reduced settings. In combining this with an observation of Anderson on the contracting property of p^{-e}-linear operators we obtain an elementary approach to test ideals in the case of affine k-algebras, where k is an F-finite field. It also yields a short and completely elementary proof of the discreteness of their jumping numbers extending most cases where the discreteness of jumping numbers was shown in arXiv:0906.4679.

29 pages, to appear in Journal of Algebraic Geometry

Cited by in corpus (12)

• A survey of test ideals
• Test ideals of non-principal ideals: Computations, Jumping Numbers, Alterations and Division Theorems
• Depth of $F$-singularities and base change of relative canonical sheaves
• Test ideals in rings with finitely generated anti-canonical algebras
• Supplements to non-lc ideal sheaves
• F-signature of pairs and the asymptotic behavior of Frobenius splittings
• A note on discreteness of $F$-jumping numbers
• An algorithm for producing F-pure ideals
• Functorial Test Modules
• F-singularities via alterations
• Cartier modules on toric varieties
• Castelnuovo-Mumford regularity and the discreteness of $F$-jumping coefficients in graded rings