5,364 research outputs found
Variation of Hilbert Coefficients
For a Noetherian local ring (\RR, \m), the first two Hilbert coefficients,
and , of the -adic filtration of an \m-primary ideal are
known to code for properties of \RR, of the blowup of \spec(\RR) along
, and even of their normalizations. We give estimations for these
coefficients when is enlarged (in the case of in the same integral
closure class) for general Noetherian local rings
Star Operations on Numerical Semigroups
It is proved that the number of numerical semigroups with a fixed number n of star operations is finite if n\ua0>\ua01. The result is then extended to the class of analytically irreducible residually rational one-dimensional Noetherian rings with finite residue field and integral closure equal to a fixed discrete valuation domain
Non-compact subsets of the Zariski space of an integral domain
Let be a minimal valuation overring of an integral domain and let
be the Zariski space of the valuation overrings of .
Starting from a result in the theory of semistar operations, we prove a
criterion under which the set is not compact.
We then use it to prove that, in many cases, is not a
Noetherian space, and apply it to the study of the spaces of Kronecker function
rings and of Noetherian overrings.Comment: To appear in the Illinois Journal of Mathematic
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