The asymptotic nature of the analytic spread

Source abstract

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 ): In this paper we shall improve this by showing that the number min depth ( R / I n ) may be replaced by the asymptotic value of depth ( R / I n ) 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 I n / mI n = U n 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 / I n ) 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 / I n ), we perform a change which affects only finitely many of the modules U n (see Section (8)).