Coefficient ideals

Shah, Kishor

doi:10.1090/S0002-9947-1991-1013338-3

Coefficient ideals
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by Kishor Shah PDF

Trans. Amer. Math. Soc. 327 (1991), 373-384 Request permission

Abstract:

Let $R$ be a $d$-dimensional Noetherian quasi-unmixed local ring with maximal ideal $M$ and an $M$-primary ideal $I$ with integral closure $\overline I$. We prove that there exist unique largest ideals ${I_k}$ for $1 \leq k \leq d$ lying between $I$ and $\overline I$ such that the first $k + 1$ Hilbert coefficients of $I$ and ${I_k}$ coincide. These coefficient ideals clarify some classical results related to $\overline I$. We determine their structure and immediately apply the structure theorem to study the associated primes of the associated graded ring of $I$.

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Coefficient ideals of the Hilbert polynomial and integral closures of parameter ideals