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]]/(x3 + y3 + Z3), where K is a field of characteristic p and p 74 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 =2 mod 3. Since IF C IR+ n R C 1*, we conclude that IR+ n R = 1* for these ideals. Using injective modules over the ring R?, the union of all peth roots of elements of R, we reduce the question of whether IF = I* for 23-graded ideals to the case of 23-graded irreducible modules. We classify the irreducible m-primary Z3-graded ideals. We then show that jF 1* for most irreducible rn-primary Z3-graded ideals in K[[x, y, z]]/(x3+y3+z3), where K is a field of characteristic p and p _ 2 mod 3. Hence 1* = IRMf n R for these ideals. In this paper we discuss the conjecture that I* = IR+ n R, where R+ denotes the integral closure of a domain R of characteristic p in an algebraic closure of its fraction field and I* denotes the tight closure of I. The ring R+ is characterized by the property that it is a domain integral over R and every monic polynomial with coefficients in R? factors into monic linear factors. This characterization can be used to prove the following property of R+: If W is a multiplicatively closed set of R, then (W-lR)+ W-1R+. Aside from providing a much more concrete description of tight closure, proving that I* = IR+ n R would solve the localization problem for tight closure. It is known that I* -IR+ n R for parameter ideals [Sml] and for rings in which every ideal of the normalization is tightly closed. Also, for those ideals I of an excellent local domain R such that R/I has finite phantom projective dimension, it is known that I* IR+ n R [Ab]. However, the conjecture is open even for two-dimensional normal Gorenstein domains. In particular, the conjecture is open for the cubical cone K[[x, y, z]]/(x3 + y3 + z3), where K is a field of characteristic p and p 5L 3, and more generally for rings of the form K[[x, y, z]]/(F(x, y, z)) where F is a homogeneous cubic polynomial. 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 5L 3. In Section 1 we use a Z3-grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular rings K[[x, y]]. In Section 2 we show that the Frobenius closure of an ideal I, denoted IF, is the same as the tight closure in certain classes of ideals when p _ 2 mod 3. Since IF C IR+ n R C I*, Received by the editors August 27, 1997. 1991 Mathematics Subject Classification. Primary 13A35, 13A02, 13HI10.