Linear equivalence of ideal topologies

It is proved that whenever P is a prime ideal in a commutative Noethe-rian ring such that the P-adic and the P-symbolic topologies are equivalent, then the two topologies are equivalent linearly. Several explicit examples are calculated, in particular for all prime ideals corresponding to non-torsion points on nonsingular elliptic cubic curves. There are many examples of prime ideals P in commutative Noetherian rings for which the symbolic Rees algebra S(P) = n P (n) , where P (n) is the nth symbolic power of P, is not a Noetherian ring. The rst such example was found by Rees in Re4], and later Roberts Ro1], Ro2] and Goto, Nishida and Watanabe GNW] found examples in regular rings. Noetherianness of symbolic Rees algebras has been studied by many other authors, for example Even in the case of space curves, it is not yet known what distinguishes the primes whose symbolic Rees algebras are Noetherian. When a symbolic Rees algebra is Noetherian, then certainly S(P) is generated over the base ring in degrees up to some integer k. It is easy to show that then for all positive integers n, P (kn) P n. Thus perhaps this \linear" equivalence of P-symbolic and P-adic topologies distinguishes the primes whose symbolic Rees algebras are Noetherian? The consequence of the main result of this paper is that this is not at all the case. In fact, I prove that as long as the P-symbolic and P-adic topologies are equivalent, the two topologies are equivalent \linearly". The assumption that the two topologies be equivalent is deenitely necessary, yet it is satissed in great many cases, say if the ring is a regular local domain and P has dimension one (see Schenzel's Theorem 2.2 below). In fact, equivalence of adic and symbolic topologies has been studied by many people, for example by Huckaba, Throughout all rings will be commutative with identity. The main result of this paper, Theorem 3.3, says that whenever I and J are two ideals in a Noetherian ring R such that the I-adic topology is equivalent to the topology deened To appear in Mathematische Zeitschrift. Copyrighted by Springer-Verlag.