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 Ciupercă. 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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