A sufficient condition for the finiteness of Frobenius test exponents
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- by Kyle Maddox PDF
- Proc. Amer. Math. Soc. 147 (2019), 5083-5092 Request permission
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 $\operatorname {Fte}(R)<\infty$.References
- Holger Brenner, Bounds for test exponents, Compos. Math. 142 (2006), no. 2, 451–463. MR 2218905, DOI 10.1112/S0010437X05001843
- Robin Hartshorne and Robert Speiser, Local cohomological dimension in characteristic $p$, Ann. of Math. (2) 105 (1977), no. 1, 45–79. MR 441962, DOI 10.2307/1971025
- Craig Huneke, Mordechai Katzman, Rodney Y. Sharp, and Yongwei Yao, Frobenius test exponents for parameter ideals in generalized Cohen-Macaulay local rings, J. Algebra 305 (2006), no. 1, 516–539. MR 2264141, DOI 10.1016/j.jalgebra.2006.06.036
- Mordechai Katzman and Rodney Y. Sharp, Uniform behaviour of the Frobenius closures of ideals generated by regular sequences, J. Algebra 295 (2006), no. 1, 231–246. MR 2188859, DOI 10.1016/j.jalgebra.2005.01.025
- Gennady Lyubeznik, $F$-modules: applications to local cohomology and $D$-modules in characteristic $p>0$, J. Reine Angew. Math. 491 (1997), 65–130. MR 1476089, DOI 10.1515/crll.1997.491.65
- Thomas Polstra and Pham Hung Quy, Nilpotence of Frobenius actions on local cohomology and Frobenius closure of ideals, J. Algebra 529 (2019), 196–225. MR 3938859, DOI 10.1016/j.jalgebra.2019.03.015
- Pham Hung Quy, On the uniform bound of Frobenius test exponents, J. Algebra 518 (2019), 119–128. MR 3872860, DOI 10.1016/j.jalgebra.2018.10.016
- Pham Hung Quy and Kazuma Shimomoto, $F$-injectivity and Frobenius closure of ideals in Noetherian rings of characteristic $p>0$, Adv. Math. 313 (2017), 127–166. MR 3649223, DOI 10.1016/j.aim.2017.04.002
- Rodney Y. Sharp, On the Hartshorne-Speiser-Lyubeznik theorem about Artinian modules with a Frobenius action, Proc. Amer. Math. Soc. 135 (2007), no. 3, 665–670. MR 2262861, DOI 10.1090/S0002-9939-06-08606-0
Additional Information
- Kyle Maddox
- Affiliation: University of Missouri–Columbia, Mathematical Sciences Building, Room 202, Columbia, Missouri 65201
- Email: klmmrb@mail.missouri.edu
- 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
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 5083-5092
- MSC (2010): Primary 13A35, 13D45
- DOI: https://doi.org/10.1090/proc/14673
- MathSciNet review: 4021071