Test ideals in non-Q-Gorenstein rings

By Karl Schwede. 2009

79 citations recorded by the source corpus.

Suppose that $X = \Spec R$ is an $F$-finite normal variety in characteristic $p > 0$. In this paper we show that the big test ideal $\tau_b(R) = \tld \tau(R)$ is equal to $\sum_{\Delta} \tau(R; \Delta)$ where the sum is over $\Delta$ such that $K_X + \Delta$ is $\bQ$-Cartier. This affirmatively answers a question asked by various people, including Blickle, Lazarsfeld, K. Lee and K. Smith. Furthermore, we have a version of this result in the case that $R$ is not even necessarily normal.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/332d1eab866e0f90aa07609d64eddbb60ef19362

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

A survey of test ideals (2011, background): “7.7. [Sch11b, Deﬁnition 3.16] The big test ideal τ(R,D) of a pair (R,D), if it exists, is the unique smallest ideal Jthat satisﬁes two conditions: (1) Jis D-compatible, and (2) J6= {0}. Exercise 7.8. [Sch11b] Suppose that D has a non-zero homogeneous element φ∈De, e>0. Prove that τ(R,D) exists by”
F-signature and Hilbert–Kunz multiplicity : a combined approach and comparison (2016, unknown): “ Moreover, it may well be the case that answers to the questions above require the use of such pairs (particularly to remove Gorenstein or Q-Gorensetein hypotheses). In this direction, and in view of [Sch11], one could imagine a positive and constructive answer to the question below could prove quit”