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2015•arXiv (Cornell University)Open access

On the vanishing of local cohomology of the absolute integral closure in\n positive characteristic

Phạm Hùng Quý

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Abstract

The aim of this paper is to extend the main result of C. Huneke and G.\nLyubeznik in [Adv. Math. 210 (2007), 498--504] to the class of rings that are\nimages of Cohen-Macaulay local rings. Namely, let $R$ be a local Noetherian\ndomain of positive characteristic that is an image of a Cohen-Macaulay local\nring. We prove that all local cohomology of $R$ (below the dimension) maps to\nzero in a finite extension of the ring. As a direct consequence we obtain that\nthe absolute integral closure of $R$ is a big Cohen-Macaulay algebra. Since\nevery excellent local ring is an image of a Cohen-Macaulay local ring, this\nresult is a generalization of the main result of M. Hochster and Huneke in\n[Ann. of Math. 135 (1992), 45--79] with a simpler proof.\n

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The aim of this paper is to extend the main result of C. Huneke and G.\nLyubeznik in [Adv. Math. 210 (2007), 498--504] to the class of rings that are\nimages of Cohen-Macaulay local rings. Namely, let $R$ be a local Noetherian\ndomain of positive characteristic that is an image of a Cohen-Macaulay local\nring. We prove that all local cohomology of $R$ (below the dimension) maps to\nzero in a finite extension of the ring. As a direct consequence we obtain that\nthe absolute integral closure of $R$ is a big Cohen-Macaulay algebra. Since\nevery excellent local ring is an image of a Cohen-Macaulay local ring, this\nresult is a generalization of the main result of M. Hochster and Huneke in\n[Ann. of Math. 135 (1992), 45--79] with a simpler proof.\n

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Available abstract

The aim of this paper is to extend the main result of C. Huneke and G.\nLyubeznik in [Adv. Math. 210 (2007), 498--504] to the class of rings that are\nimages of Cohen-Macaulay local rings. Namely, let $R$ be a local Noetherian\ndomain of positive characteristic that is an image of a Cohen-Macaulay local\nring. We prove that all local cohomology of $R$ (below the dimension) maps to\nzero in a finite extension of the ring. As a direct consequence we obtain that\nthe absolute integral closure of $R$ is a big Cohen-Macaulay algebra. Since\nevery excellent local ring is an image of a Cohen-Macaulay local ring, this\nresult is a generalization of the main result of M. Hochster and Huneke in\n[Ann. of Math. 135 (1992), 45--79] with a simpler proof.\n

Key concepts: Local cohomology, Local ring, Mathematics, Noetherian, Closure (psychology), Pure mathematics, Regular local ring, Ring (chemistry)

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