Abstract.
It is shown that if M is a finite module on a local noetherian ring A which is filtered by an f-good filtration \( \Phi \) = (M n ) where f is a noetherian filtration on A, then the i-th Betti and the i-th Bass numbers of the modules (M n ) and (M / M n ) define quasi-polynomial functions whose period does not depend on i but only of the Rees ring of f. It is proved that the projective and injective dimension of the modules M / M n are perodic for large n. In the particular case where f is a good filtration or a strongly A P filtration it is shown that the projective and injective dimension as well as the depth stabilize. As an application, using a result proved by Brodmann, we give an upper bound of the analytic spread of¶f = (I n ) in terms of the limes inferior of depth (A / I n ).
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Eingegangen am 22. 10. 1999
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Dichi, H. Quasi-polynomials associated with Bass and Betti numbers of filtered modules. Arch. Math. 77, 399–406 (2001). https://doi.org/10.1007/PL00000510
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DOI: https://doi.org/10.1007/PL00000510