Test Ideals in Rings With Finitely Generated Anti-Canonical Algebras

Abstract

Many results are known about test ideals and F -singularities for Q-Gorenstein rings. In this paper we generalize many of these results to the case when the symbolic Rees algebra OX ⊕ OX (−KX) ⊕ OX (−2KX ) ⊕ . . . is finitely generated (or more generally, in the log setting for −KX − ∆). In particular, we show that the F -jumping numbers of τ (X, a t ) are discrete and rational. We show that test ideals τ (X) can be described by alterations as in Blickle-Schwede-Tucker (and hence show that splinters are strongly F -regular in this setting -recovering a result of Singh). We demonstrate that multiplier ideals reduce to test ideals under reduction modulo p when the symbolic Rees algebra is finitely generated. We prove that Hartshorne-Speiser-Lyubeznik-Gabber type stabilization still holds. We also show that test ideals satisfy global generation properties in this setting.

Discreteness of 𝐹-jumping numbers at isolated non-ℚ-Gorenstein points

“…If K R is Cartier, it follows from [HS77, Lyu97, Gab04] that these images stabilize, giving a canonical scheme structure to the non-F -pure locus of X = Spec R. Blickle and Böckle also proved a related stabilization result for arbitrary rings (and even more) [BB11,Bli09] but their result does not seem to imply that J e = J e+1 for e ≫ 0 (Blickle obtained another result which implies stabilization of a different set of smaller ideals). However, as a corollary of our work, we obtain the following generalization, also see [CEMS14,Proposition 3.7].…”

Finitistic test ideals on numerically Q-Gorenstein varieties

“…It follows from[23, Theorem 4.3] and [13, Theorem 3.3] that there exists an effective Q-Weil divisor ∆ on X such that K X + ∆ is Q-Cartier and (X, ∆) is klt. Then i 0 O X (⌊iD⌋) is Noetherian by[7, Lemma 2.27]. Applying Lemma 3.5, we see that D is numerically Q-Cartier at x if and only if it is Q-Cartier at x.…”

“…What if R is normal but not Q-Gorenstein? Various results on τ b (R) in the Q-Gorenstein setting were generalized in [7] to the case where the anti-canonical ring i 0 O X (−iK X ) of X = Spec R (that is, the symbolic Rees algebra of the anti-canonical ideal of R) is finitely generated. In particular, they gave an affirmative answer to Conjecture 1.1 in this case.…”

Test Ideals in Rings With Finitely Generated Anti-Canonical Algebras – Corrigendum

“…

In the article by Chiecchio et al [1] there was an error. This error is explained and corrected in this notice, in particular, the statement of Lemma 2.10 is incorrect.

The corrected version of the statement, appearing as Lemma 2.10 here, is slightly weaker.

…”

“…In the article by Chiecchio et al [1] there was an error. This error is explained and corrected in this notice, in particular, the statement of Lemma 2.10 is incorrect.…”