Note on symbolic powers and going down

Abstract

For primes P ⊆ Q P \subseteq Q of a Noetherian ring R R , we consider when, for all k ≥ 1 k \geq 1 , there is an m m with P ( m ) ⊆ Q ( k ) {P^{(m)}} \subseteq {Q^{(k)}} and reprove a relevant theorem of Schenzel. If R R is a domain, we consider sufficient conditions for P ⊆ Q P \subseteq Q to satisfy going down for all primes Q Q containing P P .