Tight closure, plus closure and Frobenius closure in cubical cones期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Tight closure, plus closure and Frobenius closure in cubical cones

Authors:	Moira A McDermott

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

Abstract:	We consider tight closure, plus closure and Frobenius closure in the rings $R = Kx,y,z]]/(x^{3} + y^{3} +z^{3})$ , where is a field of characteristic 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 . 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 , 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 -primary $\mathbb{Z}_3$ -graded ideals. We then show that $I^{F} = I^{}$ for most irreducible -primary $\mathbb{Z}_3$ -graded ideals in , where is a field of characteristic and $p \equiv 2 \text{mod} 3$ . Hence $I^{*} = IR^{+} \cap R$ for these ideals.

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

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