## Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

## If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

# 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
Preview not available

## 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.

• 95
• 96
• 97
• 98
• 99
• 100
• 101
• 102
• 103
• 104
• 105
• 106
• 107
• 108
• 109
• 110
• 111
• 112
• 113
• 114