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On a generalization of test ideals

Nobuo Hara·Shunsuke Takagi
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Abstract

The test ideal $τ(R)$ of a ring $R$ of prime characteristic is an important object in the theory of tight closure. In this paper, we study a generalization of the test ideal, which is the ideal $τ(\a^t)$ associated to a given ideal $\a$ with rational exponent $t \ge 0$. We first prove a key lemma of this paper, which gives a characterization of the ideal $τ(\a^t)$. As applications of this key lemma, we generalize the preceding results on the behavior of the test ideal $τ(R)$. Moreover, we prove an analog of so-called Skoda's theorem, which is formulated algebraically via adjoint ideals by Lipman in his proof of the "modified Briançon--Skoda theorem."

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Nobuo Hara, Shunsuke Takagi. 2003-12-03. On a generalization of test ideals. https://arxiv.org/abs/math/0210131

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