اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe s-multiplicity function of 2x2-determinantal rings
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This article generalizes joint work of the first author and I. Swanson to the $s$-multiplicity recently introduced by the second author. For $k$ a field and $X = [ x_{i,j}]$ a $m times n$-matrix of variables, we utilize Grobner bases to give a closed form the length $lambda( k[X] / (I_2(X) + mathfrak{m}^{ lceil sq rceil} + mathfrak{m}^{[q]} ))$ where $s in mathbf{Z}[p^{-1}]$, $q$ is a sufficiently large power of $p$, and $mathfrak{m}$ is the homogeneous maximal ideal of $k[X]$. This shows this length is always eventually a {it polynomial} function of $q$ for all $s$.
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Let k be an arbitrary field (of arbitrary characteristic) and let X = [x_{i,j}] be a generic m x n matrix of variables. Denote by I_2(X) the ideal in k[X] = k[x_{i,j}: i = 1, ..., m; j = 1, ..., n] generated by the 2 x 2 minors of X. We give a recurs
ive formulation for the lengths of the k[X]-module k[X]/(I_2(X) + (x_{1,1}^q,..., x_{m,n}^q)) as q varies over all positive integers using Grobner basis. This is a generalized Hilbert-Kunz function, and our formulation proves that it is a polynomial function in q. We give closed forms for the cases when m is at most 2, %as well as the closed forms for some other special length functions. We apply our method to give closed forms for these Hilbert-Kunz functions for cases $m le 2$.
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