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Rings of Frobenius operators

Published online by Cambridge University Press:  24 April 2014

MORDECHAI KATZMAN
Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield, S3 7RH. e-mail: M.Katzman@sheffield.ac.uk
KARL SCHWEDE
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, U.S.A. e-mail: schwede@math.psu.edu
ANURAG K. SINGH
Affiliation:
Department of Mathematics, University of Utah, 155 S 1400 E, Salt Lake City, UT 84112, U.S.A. e-mail: singh@math.utah.edu
WENLIANG ZHANG
Affiliation:
Department of Mathematics, University of Nebraska, Lincoln, NE 68505, U.S.A. e-mail: wzhang15@unl.edu

Abstract

Let R be a local ring of prime characteristic. We study the ring of Frobenius operators ${\mathcal F}(E)$, where E is the injective hull of the residue field of R. In particular, we examine the finite generation of ${\mathcal F}(E)$ over its degree zero component ${\mathcal F}^0(E)$, and show that ${\mathcal F}(E)$ need not be finitely generated when R is a determinantal ring; nonetheless, we obtain concrete descriptions of ${\mathcal F}(E)$ in good generality that we use, for example, to prove the discreteness of F-jumping numbers for arbitrary ideals in determinantal rings.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2014 

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References

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