Tight closure, plus closure and Frobenius closure in cubical cones.

Abstract

Let R be a Noetherian ring of characteristic p. Given a test element c, we call R strongly bounded relative to c if there exists an R-linear map $R\sp{1/q}\to R\sp{1/pq}$ taking $c\sp{1/q}$ to $c\sp{1/pq}$ for some $q=p\sp{e}.$ It is shown that if R is strongly bounded relative to a test element, then tight closure commutes with localization in R. It is also shown that if R is a one-dimensional F-finite domain then there exists a test element c such that R is strongly bounded relative to c. Let $R=K\lbrack\lbrack x,y,z\rbrack\rbrack/(x\sp3+y\sp3+z\sp3),$ where K is a field of characteristic p and $p\equiv2$ mod 3. It is shown that for most irreducible m-primary $\doubz\sb3$-graded ideals $I\subseteq R,$ we have $I\sp{F}=I\sp*,$ and hence $I\sp*=IR\sp+\cap I.$ It is also shown that $I\sp{F}=I\sp*$ for several classes of not necessarily irreducible $\doubz\sb3$-graded ideals in R. It is shown that the question of whether $I\sp{F}=I\sp*$ in R can be reduced to the case of $\doubz\sb3$-graded irreducible modules.