Tight closure, plus closure and Frobenius closure in cubical cones
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- by Moira A. McDermott PDF
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Abstract:
We consider tight closure, plus closure and Frobenius closure in the rings $R = K[[x,y,z]]/(x^{3} + y^{3} +z^{3})$, where $K$ is a field of characteristic $p$ and $p \neq 3$. We use a $\mathbb {Z}_3$-grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular ring $K[[x,y]]$. We show that Frobenius closure is the same as tight closure in certain classes of ideals when $p \equiv 2 \text {mod} 3$. Since $I^{F} \subseteq IR^{+} \cap R \subseteq I^{*}$, we conclude that $IR^{+} \cap R = I^{*}$ for these ideals. Using injective modules over the ring $R^{\infty }$, the union of all ${p^{e}}$th roots of elements of $R$, we reduce the question of whether $I^{F} = I^{*}$ for $\mathbb {Z}_3$-graded ideals to the case of $\mathbb {Z}_3$-graded irreducible modules. We classify the irreducible $m$-primary $\mathbb {Z}_3$-graded ideals. We then show that $I^{F} = I^{*}$ for most irreducible $m$-primary $\mathbb {Z}_3$-graded ideals in $K[[x,y,z]]/(x^3+y^3+z^3)$, where $K$ is a field of characteristic $p$ and $p \equiv 2 \text {mod} 3$. Hence $I^{*} = IR^{+} \cap R$ for these ideals.References
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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
- Received by editor(s): August 27, 1997
- Published electronically: March 8, 1999
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1624198