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Abstract
Many concepts of analytic spread for filtrations exist in the current literature. As far as we know, none of them is explicitly of asymptotic nature. Here we give a definition of analytic spread for filtrations which is shown to be of asymptotic nature.
On the other hand, it is known that, in a noetherian local ring, the analytic spread of ideals is invariant under the integral closure operation. In this paper we investigate too the behaviour of the analytic spread of a filtration f under some closure operations, the analytic spread of f= (In ) being indifferently one of the following numbers:
Where J is an ideal of
the Rees rings of the filtration f,X an indeterminate and u=X-1.Here we give also in a noetherian local ring (A, m) a class of filtrations f= (In
), containing I-good filtrations, such that the analytic spread of fcoincides with that of In
for all n ≥1, in answer to a question raised by J.S.Okon in 1984 which was shown to be false for general noetherian filtrations. We give equally an upper bound of the analytic spread of a strongly noetherian filtration In
in terms of the asymptotic value of depth
which is the analogue of a result of Brodmann.
