Let $R=k[X,Y,U,V]/(XV-YU)$, where $k$ is field of characteristic $0$. Consider $S=R_m$, where $m$ is the maximal ideal $(X,Y,U,V)/(XV-YU)$. How can we find a system of parameters for $S$ and what are they? We know that there are 3 elements in any system of parameters, as $S$ is a 3-dimensional domain.
System of Parameters.
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commutative-algebra
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0@user34377 I would like to know how to choose this set of elements, yes your set of elements is a system of parameters, but can you tell me what would be a good method to find such a system? – 2012-06-24
1 Answers
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Alternatively, $\{x_1, \ldots, x_s\} \subset S$ is a SOP if the ideal $(x_1, \ldots, x_s)$ they generate is $m$-primary, in other words, if the ring $S/(x_1,\ldots,x_s)$ is $0$-dimensional. Then, we see that choosing $\{\overline x,\overline v,\overline y-\overline u\}$ yields $S/(\overline x,\overline v,\overline y-\overline u) \cong K[\overline y]/(\overline y^2),$ which is $0$-dimensional, by the commutativity of localization with quotients. Thus, $\{\overline x,\overline v,\overline y-\overline u\}$ is indeed a SOP for S.
(This definition of SOP is in Matsumura's Commutative Ring Theory, by the way.)
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0Hmm, that's a good question, I definitely went "by intuition." But at least for hypersurfaces this seems like a good thing to try first. – 2012-06-25