Test ideals in non-Q-Gorenstein rings

Details Test ideals in non-Q-Gorenstein rings Test ideals in non-Q-Gorenstein rings

2009-06-23

arxiv.org/abs/0906.4313v2

Trans. Amer. Math. Soc. 363 (2011), no. 11, 5925--5941

Mathematics

Commutative Algebra

18 pages, minor changes, to appear in Transactions of the American Mathematical Society

Scientific paper

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.

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