Abstract
In this paper, using ultra-Frobenii, we introduce a variant of Schoutens’ non-standard tight closure (Schoutens in Manuscr Math 111:379–412, 2003), ultra-tight closure, on ideals of a local domain R essentially of finite type over \(\mathbb {C}\). We prove that the ultra-test ideal \(\tau _{\textrm{u}}(R,\mathfrak {a}^t)\), the annihilator ideal of all ultra-tight closure relations of R, coincides with the multiplier ideal \(\mathcal {J}({\text {Spec}}R,\mathfrak {a}^t)\) if R is normal \(\mathbb {Q}\)-Gorenstein. As an application, we study a behavior of multiplier ideals under pure ring extensions.
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Acknowledgements
The author wishes to express his gratitude to his supervisor Professor Shunsuke Takagi for his encouragement, valuable advice and suggestions. The author is also grateful to Tatsuro Kawakami, Kenta Sato and Shou Yoshikawa for useful conversations. Without their continuous help, this paper would not have been possible. He also thanks the referee who provided useful comments and suggestions.
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Yamaguchi, T. A characterization of multiplier ideals via ultraproducts. manuscripta math. 172, 1153–1168 (2023). https://doi.org/10.1007/s00229-022-01446-3
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DOI: https://doi.org/10.1007/s00229-022-01446-3