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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0What does it mean, a "system of parameters"? – 2012-06-24
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0@Gerry Myerson Let $(A,m)$ be a Noetherian local ring and $M$ be a finitely generated $A$ module of dimension $n$. A family $(x_1,x_2,...,x_s)$ is called a system of parametes(SOP) for $M$ if $M/(x_1,x_2,...,x_s)M$ is of finite length and if $s=n$. – 2012-06-24
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0The way i have computed SOP's for some examples is to find an element outside the minimal primes and then go modulo the ideal generated by this element and repeating the process, but here i am unable to do that. If someone can go through this process and find all the 3 elements of a system of parameters, it would be very instructive. Thanks – 2012-06-24
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0@Gerry Myerson In other words, finding an $m$-primary ideal. – 2012-06-24
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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
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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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0thanks, but how did you choose those set of elements? I mean, what would be a method to do this? I – 2012-06-24
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0This is a great answer. – 2012-06-24
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0Thanks! I tried beginning with a regular sequence for $K[x,y,u,v]$ (at the origin), such that when I kill off those variables (e.g. $\{x,v\}$) I might have a chance at finding a $0$-dimensional quotient. I'm not sure how systematic this can be, but it seems to work with basic examples. Thanks for the question! – 2012-06-24
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0Sorry, I noticed I used $K,x,y,u,v$ where you used $k,X,Y,U,V.$ Hopefully that doesn't cause confusion. – 2012-06-24
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0it did not, i understood what you meant. I mean if there is another complicated looking example, how can we go about finding a SOP by the above method? Here it seemed somehow easy(ok only after i saw the solution) that adding y-u to x and v would do the job, but in general how can we do this? – 2012-06-24
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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