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A sufficient condition for the finiteness of Frobenius test exponents


Author: Kyle Maddox
Journal: Proc. Amer. Math. Soc. 147 (2019), 5083-5092
MSC (2010): Primary 13A35, 13D45
DOI: https://doi.org/10.1090/proc/14673
Published electronically: June 10, 2019
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Abstract: The Frobenius test exponent $ \operatorname {Fte}(R)$ of a local ring $ (R,\mathfrak{m})$ of prime characteristic $ p>0$ is the smallest $ e_0 \in \mathbb{N}$ such that for every ideal $ \mathfrak{q}$ generated by a (full) system of parameters, the Frobenius closure $ \mathfrak{q}^F$ has $ (\mathfrak{q}^F)^{\left [p^{e_0}\right ]}=\mathfrak{q}^{\left [ p^{e_0}\right ]}$. We establish a sufficient condition for $ \operatorname {Fte}(R)<\infty $ and use it to show that if $ R$ is such that the Frobenius closure of the zero submodule in the lower local cohomology modules has finite colength, i.e., $ H^j_\mathfrak{m}(R)/0^F_{H^j_\mathfrak{m}(R)}$ is finite length for $ 0 \le j < \dim (R)$, then $ \operatorname {Fte}(R)<\infty $.


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Kyle Maddox
Affiliation: University of Missouri–Columbia, Mathematical Sciences Building, Room 202, Columbia, Missouri 65201
Email: klmmrb@mail.missouri.edu

DOI: https://doi.org/10.1090/proc/14673
Received by editor(s): September 26, 2018
Received by editor(s) in revised form: March 5, 2019
Published electronically: June 10, 2019
Communicated by: Claudia Polini
Article copyright: © Copyright 2019 American Mathematical Society