Mathematische Annalen

, Volume 362, Issue 1–2, pp 25–42 | Cite as

\(F\)-injectivity and Buchsbaum singularities

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Abstract

Let \((R,\mathfrak {m}, K)\) be a local ring that contains a field. We show that, when \(R\) has equal characteristic \(p>0\) and when \(H_\mathfrak {m}^i(R)\) has finite length for all \(i<\dim R\), then \(R\) is \(F\)-injective if and only if every ideal generated by a system of parameters is Frobenius closed. As a corollary, we show that such an \(R\) is in fact a Buchsbaum ring. This answers positively a question of S. Takagi that \(F\)-injective singularities with isolated non-Cohen–Macaulay locus are Buchsbaum. We also study the characteristic \(0\) analogue of this question and we show that Du Bois singularities with isolated non-Cohen–Macaulay locus are Buchsbaum in the graded case.

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Notes

Acknowledgments

I would like to thank Mel Hochster for many helpful and valuable discussions on this problem. I would like to thank Karl Schwede for answering many of my questions, for carefully reading a preliminary version this manuscript and for his comments. I am grateful to Kazuma Shimomoto for some helpful discussions on Buchsbaum rings, and to Zhixian Zhu for some discussions on Du Bois singularities. I would also like to thank the anonymous referee, whose comments helped improve the paper.

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA

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