Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

The asymptotic nature of the analytic spread

M. Brodmanna1

a1 Universität Münster, W. Germany

In (3), corollary, p. 373) Burch gives the following inequality for the analytic spread l(I) of an ideal I of a noetherian local ring (R, m):

S030500410000061XeqnU1

In this paper we shall improve this by showing that the number min depth (R/In) may be replaced by the asymptotic value of depth (R/In) for large n (which exists) (see Section (2)). By its definition (see (6), def. 3)) the analytic spread is of asymptotic nature, i.e. depends on the modules In/mIn = Un only for large n. We shall prove a stronger result, Section (4), which also shows the asymptotic nature of l(I). This result might be interesting for itself, particularly as it is not of local nature. Once Section (4) is proved and once we know that depth (R/In) is asymptotically constant (which turns out to be an easy consequence of (1), (1)), our improved inequality is easily established: Indeed, replacing R by R/xR where x is regular with respect to almost all modules (R/In), we perform a change which affects only finitely many of the modules Un (see Section (8)).

(Received August 21 1978)