Tight closure, plus closure and Frobenius closure in cubical cones

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.