On the Hartshorne-Speiser-Lyubeznik theorem about Artinian modules with a Frobenius action

Author:
Rodney Y. Sharp

Journal:
Proc. Amer. Math. Soc. **135** (2007), 665-670

MSC (2000):
Primary 13A35, 13E10, 16S36; Secondary 13D45

Published electronically:
August 28, 2006

MathSciNet review:
2262861

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a commutative Noetherian local ring of prime characteristic. The purpose of this paper is to provide a short proof of G. Lyubeznik's extension of a result of R. Hartshorne and R. Speiser about a module over the skew polynomial ring (associated to and the Frobenius homomorphism , in the indeterminate ) that is both -torsion and Artinian over .

**1.**Maurice Auslander and David A. Buchsbaum,*Codimension and multiplicity*, Ann. of Math. (2)**68**(1958), 625–657. MR**0099978****2.**Robin Hartshorne and Robert Speiser,*Local cohomological dimension in characteristic 𝑝*, Ann. of Math. (2)**105**(1977), no. 1, 45–79. MR**0441962****3.**Melvin Hochster and Craig Huneke,*Localization and test exponents for tight closure*, Michigan Math. J.**48**(2000), 305–329. Dedicated to William Fulton on the occasion of his 60th birthday. MR**1786493**, 10.1307/mmj/1030132721**4.**Craig L. Huneke and Rodney Y. Sharp,*Bass numbers of local cohomology modules*, Trans. Amer. Math. Soc.**339**(1993), no. 2, 765–779. MR**1124167**, 10.1090/S0002-9947-1993-1124167-6**5.**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**, 10.1016/j.jalgebra.2005.01.025**6.**Ernst Kunz,*Characterizations of regular local rings for characteristic 𝑝*, Amer. J. Math.**91**(1969), 772–784. MR**0252389****7.**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**, 10.1515/crll.1997.491.65**8.**Hideyuki Matsumura,*Commutative ring theory*, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR**879273****9.**R. Y. Sharp,*Tight closure test exponents for certain parameter ideals*, Michigan Math. J., to appear (arXiv math.AC/0508214).**10.**K. E. Smith,*Tight closure of parameter ideals*, Invent. Math.**115**(1994), no. 1, 41–60. MR**1248078**, 10.1007/BF01231753**11.**Karen E. Smith,*Test ideals in local rings*, Trans. Amer. Math. Soc.**347**(1995), no. 9, 3453–3472. MR**1311917**, 10.1090/S0002-9947-1995-1311917-0

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
13A35,
13E10,
16S36,
13D45

Retrieve articles in all journals with MSC (2000): 13A35, 13E10, 16S36, 13D45

Additional Information

**Rodney Y. Sharp**

Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom

Email:
R.Y.Sharp@sheffield.ac.uk

DOI:
http://dx.doi.org/10.1090/S0002-9939-06-08606-0

Keywords:
Commutative Noetherian ring,
prime characteristic,
Frobenius homomorphism,
Artinian module,
Frobenius skew polynomial ring

Received by editor(s):
September 28, 2005

Published electronically:
August 28, 2006

Additional Notes:
The author was partially supported by the Engineering and Physical Sciences Research Council of the United Kingdom (grant number EP/C538803/1).

Communicated by:
Bernd Ulrich

Article copyright:
© Copyright 2006
American Mathematical Society