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Tight Closure, Plus Closure and Frobenius Closure in Cubical Cones

Moira A. Mcdermott
Transactions of the American Mathematical Society
Vol. 352, No. 1 (Jan., 2000), pp. 95-114
Stable URL: http://www.jstor.org/stable/118147
Page Count: 20
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Tight Closure, Plus Closure and Frobenius Closure in Cubical Cones
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Abstract

We consider tight closure, plus closure and Frobenius closure in the rings R=K[[x,y,z]]/(x3+y3+z3), where K is a field of characteristic p and p≠ 3. We use a Z3-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 $I^{F}\subseteq IR^{+}\cap R\subseteq I^{\ast}$, we conclude that IR+∩ R=I* for these ideals. Using injective modules over the ring R∞, the union of all pεth roots of elements of R, we reduce the question of whether IF=I* for Z3-graded ideals to the case of Z3-graded irreducible modules. We classify the irreducible m-primary Z3-graded ideals. We then show that IF=I* for most irreducible m-primary Z3-graded ideals in K[[x,y,z]]/(x3+y3+z3), where K is a field of characteristic $p\text{and}p\equiv 2\text{mod}3$. Hence I*=IR+∩ R for these ideals.

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