# Localization Notation in Hartshorne

This is a question about notation in Hartshorne's Algebraic Geometry. According to my understanding $k[x_0,\cdots,x_n]_{(x_i)}$, (see e.g. page 18), consists of the elements of degree zero in the localization of $k[x_0,\cdots,x_n]$ by the prime ideal $(x_i)$. That is, denominators must belong to the complement of $(x_i)$. So why is $g/x_i^N$ an element of $k[x_0,\cdots,x_n]_{(x_i)}$, for $g \in k[x_0,\cdots,x_n]$ (see e.g. bottom of page 18)?

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The notation $k[x_1,\ldots,x_n]_{(x_i)}$ here is equivalent to $k[x_1,\ldots,x_n][x_i^{-1}]_0$, meaning the subring of degree $0$ elements of the localization at the element $x_i.$ It does not simply denote the localization $k[x_1,\ldots,x_n]_{\frak p},$ where $\frak p$$=\langle x_i\rangle is a prime ideal, nor is it simply the localization k[x_1,\ldots,x_n][x_i^{-1}] at the multiplicative set generated by x_i. That is why any f\in S(Y)_{(x_i)} can be written f=g_i/x_i^N with g_i having degree N, where S(Y) is the homogeneous coordinate ring mentioned on that page; g_i and x_i^N both have degree N, meaning f has degree zero. If you have read a book like Shafarevich before this, you may have seen this localization as "dehomogenization" of a homogeneous polynomial, and it will help to keep this in mind. For example, to dehomogenize f(x,y,z)=y^2z+xyz with respect to x, we divide by x^3 since it has degree 3, getting$$\dfrac{f(x,y,z)}{x^3}=\dfrac{y^2z+xyz}{x^3}=(\dfrac{y}{x})^2(\dfrac{z}{x})+(\dfrac{y}{x})(\dfrac{z}{x}).$$We consider$\dfrac{y}{x},\dfrac{z}{x}$as coordinates on$\Bbb A_x^2$, where$\Bbb A_x^2$is a standard open subset of$\Bbb P^2,$where the original variety lived, and then this dehomogenization of$f$with respect to$x$cuts out locally the projective variety in this open chart. (Note that the dehomogenization need not be homogeneous. We have an affine variety here!) - Regarding your comment about$f = g_i / x_i^N$i can see why$g_i \in S(Y)$must have degree$N$. But why must it be homogeneous? – Manos Oct 9 '12 at 15:33 Dear @Manos, if$g_i$were not homogeneous, then$g_i/x_i^N$would also not be homogeneous, and in particular, would not be homogeneous of degree$0,$as was assumed. – Andrew Oct 9 '12 at 19:20 As far as I can tell, the notation$k[x_0,\ldots,x_n]_{(x_i)}$means the degree zero part of the localization of the polynomial ring at (the multiplicative set generated by)$x_i$, whereas Hartshorne would write$k[x_0,\ldots,x_n]_{((x_i))}$to mean elements of degree zero in the localization by the ideal$(x_1)$. See how he writes$k[x_0,\ldots,x_n]_{((0))}\$, for example.