49,159 research outputs found
Generalized Hilbert Functions
Let be a finite module and let be an arbitrary ideal over a
Noetherian local ring. We define the generalized Hilbert function of on
using the 0th local cohomology functor. We show that our definition
re-conciliates with that of Ciuperc. By generalizing Singh's
formula (which holds in the case of ), we prove that the
generalized Hilbert coefficients are preserved under a
general hyperplane section, where . We also keep track of the
behavior of . Then we apply these results to study the generalized
Hilbert function for ideals that have minimal -multiplicity or almost
minimal -multiplicity. We provide counterexamples to show that the
generalized Hilbert series of ideals having minimal or almost minimal
-multiplicity does not have the `expected' shape described in the case where
. Finally we give a sufficient condition such that the
generalized Hilbert series has the desired shape.Comment: arXiv admin note: text overlap with arXiv:1101.228
Generalized stretched ideals and Sally Conjecture
We introduce the concept of -stretched ideals in a Noetherian local ring.
This notion generalizes to arbitrary ideals the classical notion of stretched
-primary ideals of Sally and Rossi-Valla, as well as the concept
of ideals of minimal and almost minimal -multiplicity introduced by
Polini-Xie. One of our main theorems states that, for a -stretched ideal,
the associated graded ring is Cohen-Macaulay if and only if two classical
invariants of the ideal, the reduction number and the index of nilpotency, are
equal. Our second main theorem, presenting numerical conditions which ensure
the almost Cohen-Macaulayness of the associated graded ring of a -stretched
ideal, provides a generalized version of Sally's conjecture. This work, which
also holds for modules, unifies the approaches of Rossi-Valla and Polini-Xie
and generalizes simultaneously results on the Cohen-Macaulayness or almost
Cohen-Macaulayness of the associated graded module by several authors,
including Sally, Rossi-Valla, Wang, Elias, Corso-Polini-Vaz Pinto, Huckaba,
Marley and Polini-Xie.Comment: 25 pages (modified the presentation of the material and added
examples). Comments are welcom
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