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Register a new userOn asymptotic depth of integral closure filtration and an application
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Authors
Tony J. Puthenpurakal
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Let $(A,mathfrak{m})$ be an analytically unramified formally equidimensional Noetherian local ring with $ depth A geq 2$. Let $I$ be an $mathfrak{m}$-primary ideal and set $I^*$ to be the integral closure of $I$. Set $G^*(I) = bigoplus_{ngeq 0} (I^n)^*/(I^{n+1})^*$ be the associated graded ring of the integral closure filtration of $I$. We prove that $ depth G^*(I^n) geq 2$ for all $n gg 0$. As an application we prove that if $A$ is also an excellent normal domain containing an algebraically closed field isomorphic to $A/m$ then there exists $s_0$ such that for all $s geq s_0$ and $J$ is an integrally closed ideal emph{strictly} containing $(mathfrak{m}^s)^*$ then we have a strict inequality $mu(J) < mu((mathfrak{m}^s)^*)$ (here $mu(J)$ is the number of minimal generators of $J$).
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The Qth-power algorithm for computing structured global presentations of integral closures of affine domains over finite fields is modified to compute structured presentations of integral closures of ideals in affine domains over finite fields relative to a local monomial ordering. A non-homogeneous version of the standard (homogeneous) Rees algebra is introduced as well.
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