Abstract. Let (R, m , k) be an excellent, local, normal ring of characteristic p with a perfect residue field and dim R = d. Let M be a finitely generated R-module. We show that there exists
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 = 3. We use a 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 ≡ 2 mod 3. Since I F ⊆ IR + ∩ R ⊆ I * , we conclude that IR + ∩ R = I * for these ideals. Using injective modules over the ring R ∞ , the union of all p e th roots of elements of R, we reduce the question of whether I F = I * for Z 3 -graded ideals to the case of Z 3 -graded irreducible modules. We classify the irreducible m-primary Z 3 -graded ideals. We then show that I F =
Abstract. Regular closure is an operation performed on submodules of arbitrary modules over a commutative Noetherian ring. The regular closure contains the tight closure when both are defined, but in general, the regular closure is strictly larger. Regular closure is interesting, in part, because it is defined a priori in all characteristics, including mixed characteristic. We show that one can test regular closure in a Noetherian ring R by considering only local maps to regular local rings. In certain cases, it is necessary only to consider maps to certain affine algebras. We also prove the equivalence of two variants of regular closure for a class of rings that includesOne of our main motivations for studying regular closure is that, unlike the current state of affairs in tight closure theory, regular closure is defined in all characteristics, including mixed characteristic. Also, when an element is in the tight closure of a certain submodule or ideal it is also in the corresponding regular closure. The regular closure contains the tight closure when both are defined, but in general, the regular closure is strictly larger. So, for example, if we could prove that regular closure "captures colons" (see [HH1, (7.6) and (7.14)] for example), then we would be able to prove theorems about maps of Tor vanishing (see . This was due, in part, to the lack of a good theory of test elements at that time. One consequence of the theory of test elements is that one can prove that tight closure is preserved by certain base changes [HH5]. This allows one to show that the tight closure of an ideal is contained in one notion of the regular closure of the ideal. We would like to determine when the two notions of regular closure coincide, if at all.
Despite the apparent power of tight closure techniques, the tight closure operation itself is quite difficult to handle in practice. For example, it is generally difficult to find the tight closure of an arbitrary ideal. Also, it is not known whether tight closure behaves well under localization. We y1 Ž y1 . would like to know whether it is true that I*W R s IW R * where I is an ideal of a ring R, W is an arbitrary multiplicative system, and I* denotes the tight closure of I. It is not even known that if all of the ideals of R are tightly closed, then all of the ideals of R are tightly closed.
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