On the uniform bound of Frobenius test exponents

Abstract

In this paper we prove the existence of a uniform bound for Frobenius test exponents for parameter ideals of a local ring $(R,\mathfrak{m})$ of prime characteristic in the following cases: (1) $R$ is generalized Cohen-Macaulay. Our proof is much more simpler than the original proof of Huneke, Katzman, Sharp and Yao, (2) The Frobenius actions on all lower local cohomologies $H_{\mathfrak{m}}^i(R)$, $i<\dim R$, are nilpotent.

arXiv commentFinal version, to appear in Journal of Algebra. arXiv admin note: text overlap with arXiv:1801.04827