On page 134, Weil divisors, example 6.5.2, he said: "The divisor of $y$ is $2Y$, because $y=0$ implies $z^2=0$, and $z$ generate the maximal ideal of the local ring at the generic point of $Y$." I was stupid and can not figure this out. Can someone give a down to earth computation what is the generic point of $Y$(Depict it using prime ideals), and what is the local ring at the generic point of $Y$? Further, you are give a closed subset of $X$, cut out by several polynomials, how can you compute the generic point of this subset at once?
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1Please don't double post: http://mathoverflow.net/questions/124151/a-question-on-generic-point-and-a-question-on-hartshorne Since questions about Hartshorne are not "research level" it is safest to ask here first and if you don't get a satisfactory answer after a significant time repost noting clearly that you also have it posted here. – Matt Mar 10 '13 at 18:04
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Your advise is reasonable, I should be more careful about this. Thank you! – Yoshinobu Osawa Mar 11 '13 at 01:28
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$\bullet$ The coordinate ring of the cone is $k[\rm{ X, Y ,Z}] / (XY - Z^2)$ ;
$\bullet$ The prime ideal representing $\rm Y$ is $\mathfrak p = \rm (Y, Z)$ ;
so the local ring you want is $(k[\mathrm{X, Y, Z}]/\rm (XY-Z^2))_{(Y,Z)}$.
Now in this local ring $\rm X$ is invertible and $\rm XY - Z^2 = 0$ which implies $\rm Y = X^{-1} Z^2$;
so the maximal ideal $\mathfrak p (k[\mathrm{X, Y, Z}]/(\rm XY - Z^2))_{\mathfrak p}$ which is by definition generated by $\rm Y$ and $\rm Z$, is only generated by $\rm Z$.
Damien L
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Dear Damien, you should be consistent in your use of the lower case and upper case typographical characters $X,Y,Z,x,y,z$. – Georges Elencwajg Mar 10 '13 at 20:13
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Dear Georges, I think I am perfectly consistent in my choice for lower cases and upper cases typographical characters. – Damien L Mar 10 '13 at 21:33
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Dear Damien: how can $(xy-z^2)$ (lower case ) be an ideal in $k[X,Y,Z]$ (upper case), since you may not write $xy-z^2\in k[X,Y,Z]$ ? – Georges Elencwajg Mar 10 '13 at 23:08
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Your answer is clear enough for a non math major such as me, thanks a lot! – Yoshinobu Osawa Mar 11 '13 at 01:29
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