Tight closure, plus closure

and Frobenius closure in cubical cones

Author:
Moira A. McDermott

Journal:
Trans. Amer. Math. Soc. **352** (2000), 95-114

MSC (1991):
Primary 13A35, 13A02, 13H10

DOI:
https://doi.org/10.1090/S0002-9947-99-02396-X

Published electronically:
March 8, 1999

MathSciNet review:
1624198

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider tight closure, plus closure and Frobenius closure in the rings , where is a field of characteristic and . We use a -grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular ring . We show that Frobenius closure is the same as tight closure in certain classes of ideals when . Since , we conclude that for these ideals. Using injective modules over the ring , the union of all th roots of elements of , we reduce the question of whether for -graded ideals to the case of -graded irreducible modules. We classify the irreducible -primary -graded ideals. We then show that for most irreducible -primary -graded ideals in , where is a field of characteristic and . Hence for these ideals.

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Additional Information

**Moira A. McDermott**

Affiliation:
Mathematics and Computer Science Department, Gustavus Adolphus College, 800 W. College Avenue, St. Peter, Minnesota 56082-1498

Email:
mmcdermo@gac.edu

DOI:
https://doi.org/10.1090/S0002-9947-99-02396-X

Keywords:
Tight closure,
characteristic $p$,
Frobenius closure,
plus closure

Received by editor(s):
August 27, 1997

Published electronically:
March 8, 1999

Article copyright:
© Copyright 1999
American Mathematical Society