Generalized Hilbert Functions |
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Authors: | Claudia Polini Yu Xie |
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Affiliation: | 1. Department of Mathematics , University of Notre Dame , Notre Dame , Indiana , USA cpolini@nd.edu;3. Department of Mathematics , University of Notre Dame , Notre Dame , Indiana , USA |
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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 zeroth local cohomology functor. We show that our definition reconciliates with that of Ciuperc?. 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 = 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. |
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Keywords: | Associated graded rings Hilbert series j-multiplicity |
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