Test ideals of non-principal ideals: Computations, jumping numbers, alterations and division theorems

Abstract

Abstract. Given an ideal a ⊆ R in a (log) Q-Gorenstein F -finite ring of characteristic p > 0, we study and provide a new perspective on the test ideal τ (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 τ (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 τ (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.

Local 𝔪-adic constancy of F-pure thresholds and test ideals

“…For more effective methods for computing F-pure thresholds in certain cases, we refer the reader to [10,11,12]. Additionally, we refer the reader to [19] for computations of test ideals in a more general (e.g., non-principal) setting.…”

Test Ideals in Rings With Finitely Generated Anti-Canonical Algebras

“…Remark 5.4. It is tempting to try to use Lemma 5.2 to give another proof of discreteness and rationality of F-jumping numbers by appealing to [52]. However, this does not seem to work.…”

Ascending chain condition for -pure thresholds on a fixed strongly -regular germ

“…Let be a triple. Then the following hold.If and , then .[ST14, Lemma 6.1] Assume that is -Cartier. Then there exists a real number such that if , then .…”