Test ideals of non-principal ideals: Computations, Jumping Numbers, Alterations and Division Theorems

Abstract

Given an ideal $a\subseteq R$ in a (log) $Q$-Gorenstein $F$-finite ring of characteristic $p>0$, we study and provide a new perspective on the test ideal $\tau (R,a^t)$ for a real number $t>0$. Generalizing a number of known results from the principal case, we show how to effectively compute the test ideal and also describe $\tau (R,a^t)$ using (regular) alterations with a formula analogous to that of multiplier ideals in characteristic zero. We further prove that the $F$-jumping numbers of $\tau (R,a^t)$ as $t$ varies are rational and have no limit points, including the important case where $R$ is a formal power series ring. Additionally, we obtain a global division theorem for test ideals related to results of Ein and Lazarsfeld from characteristic zero, and also recover a new proof of Skoda's theorem for test ideals which directly mimics the proof for multiplier ideals.

36 pages, typos corrected. To appear in Journal de Mathématiques Pures et Appliquées

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Cited by in corpus (11)

• Test ideals in rings with finitely generated anti-canonical algebras
• Frobenius Splitting in Commutative Algebra
• Ascending chain condition for $F$-pure thresholds on a fixed strongly $F$-regular germ
• The F-pure threshold of a Calabi-Yau hypersurface
• Seshadri Constants and Fujita's Conjecture via Positive Characteristic Methods
• Local m-adic constancy of F-pure thresholds and test ideals
• $D$-modules, Bernstein-Sato polynomials and $F$-invariants of direct summands
• $F$-pure thresholds and $F$-Volumes of some non principal ideals
• On the superadditivity of anticanonical Iitaka dimension
• Discreteness of $F$-jumping numbers at isolated non-Q-Gorenstein points
• On accumulation points of $F$-pure thresholds on regular local rings