TY - JOUR

T1 - Generalized Hilbert Functions

AU - Polini, Claudia

AU - Xie, Yu

N1 - Funding Information:
The first author was partially supported by the NSF and the NSA.

PY - 2014/6

Y1 - 2014/6

N2 - Let M be a finite module, and let I be an arbitrary ideal over a Noetherian local ring. We define the generalized Hilbert function of I on M using the zeroth local cohomology functor. We show that our definition reconciliates with that of Ciupercǎ. By generalizing Singh's formula (which holds in the case of λ(M/IM) < ∞), we prove that the generalized Hilbert coefficients j{fraktur}0,., j{fraktur}d-2 are preserved under a general hyperplane section, where d = dim M. We also keep track of the behavior of j{fraktur}d-1. Then we apply these results to study the generalized Hilbert function for ideals that have minimal j-multiplicity or almost minimal j-multiplicity. We provide counterexamples to show that the generalized Hilbert series of ideals having minimal or almost minimal j-multiplicity does not have the 'expected' shape described in the case where λ(M/IM) < ∞. Finally, we give a sufficient condition such that the generalized Hilbert series has the desired shape.

AB - Let M be a finite module, and let I be an arbitrary ideal over a Noetherian local ring. We define the generalized Hilbert function of I on M using the zeroth local cohomology functor. We show that our definition reconciliates with that of Ciupercǎ. By generalizing Singh's formula (which holds in the case of λ(M/IM) < ∞), we prove that the generalized Hilbert coefficients j{fraktur}0,., j{fraktur}d-2 are preserved under a general hyperplane section, where d = dim M. We also keep track of the behavior of j{fraktur}d-1. Then we apply these results to study the generalized Hilbert function for ideals that have minimal j-multiplicity or almost minimal j-multiplicity. We provide counterexamples to show that the generalized Hilbert series of ideals having minimal or almost minimal j-multiplicity does not have the 'expected' shape described in the case where λ(M/IM) < ∞. Finally, we give a sufficient condition such that the generalized Hilbert series has the desired shape.

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U2 - 10.1080/00927872.2012.756884

DO - 10.1080/00927872.2012.756884

M3 - Article

AN - SCOPUS:84893443348

SN - 0092-7872

VL - 42

SP - 2411

EP - 2427

JO - Communications in Algebra

JF - Communications in Algebra

IS - 6

ER -