Generalized Hilbert Functions

2012/02/18 by Claudia Polini, Yu Xie, Polini, Claudia +1
Mathematics · #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC

paper · pdf · doi:10.48550/arxiv.1202.4106

arXiv admin note: text overlap with arXiv:1101.2281

arxiv created 2012/02/18 · arxiv updated 2012/02/21

Abstract

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 0th local cohomology functor. We show that our definition re-conciliates with that of Ciuperc\breve\rm a. By generalizing Singh's formula (which holds in the case of λ(M/IM)<∞), we prove that the generalized Hilbert coefficients j₀,..., j_d-2 are preserved under a general hyperplane section, where d=\rm dim M. We also keep track of the behavior of j_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.

Citations

• j-multiplicity and depth of associated graded modules

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