Minimal Monomial Reductions and the Reduced Fiber Ring of an Extremal Ideal

Details Minimal Monomial Reductions and the Reduced Fiber Ring of an Extremal Ideal Minimal Monomial Reductions and the Reduced Fiber Ring of an Extremal Ideal

2005-12-20

arxiv.org/abs/math/0512456v1

Mathematics

Commutative Algebra

Scientific paper

Let $I$ be a monomial ideal in a polynomial ring $A=K[x_1,...,x_n]$. We call a monomial ideal $J$ to be 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 $J$.

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