Illinois Journal of Mathematics

Minimal monomial reductions and the reduced fiber ring of an extremal ideal

Pooja Singla

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Abstract

Let $I$ be a monomial ideal in a polynomial ring $A= K[x_1,\ldots,x_n]$. We call a monomial ideal $J$ a minimal monomial reduction ideal of $I$ if there exists no proper monomial ideal $L \subset J$ such that $L$ is a reduction ideal of~$I$. We prove that there exists a unique minimal monomial reduction ideal $J$ of $I$ and we show that the maximum degree of a monomial generator of $J$ determines the slope $p$ of the linear function $\reg(I^t)=pt+c$ for $t\gg 0$. We determine the structure of the reduced fiber ring $\mathcal{F}(J)_{\red}$ of $J$ and show that $\mathcal{F}(J)_{\red}$ is isomorphic to the inverse limit of an inverse system of semigroup rings determined by convex geometric properties of $J$.

Article information

Source
Illinois J. Math., Volume 51, Number 4 (2007), 1085-1102.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258138534

Digital Object Identifier
doi:10.1215/ijm/1258138534

Mathematical Reviews number (MathSciNet)
MR2417417

Zentralblatt MATH identifier
1151.13011

Subjects
Primary: 13C15: Dimension theory, depth, related rings (catenary, etc.)
Secondary: 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)

Citation

Singla, Pooja. Minimal monomial reductions and the reduced fiber ring of an extremal ideal. Illinois J. Math. 51 (2007), no. 4, 1085--1102. doi:10.1215/ijm/1258138534. https://projecteuclid.org/euclid.ijm/1258138534


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