Localizations of a quotient ring

Question

Let $R=\mathbb{C}[x,y,z]/(x^3+y^3+z^3)$ and $f=x^2$. I have to know what is $R_f$ (localization in the element $x^2$). How can I describe $\operatorname{Spec}(R_f) $ and $\operatorname{Spec}(R_{(f)})$? How can I imagine $R$?

Answer 1

You can imagine $R$ as the coordinate ring of an affine surface $X$ in $\mathbb A^3$, defined by the equation $x^3+y^3+z^3=0$. In other words, $R=\Gamma(X,O_X)$ is the ring of functions on this surface. In $X$, you have an open subscheme given by $\textrm{Spec }R_f$: it consists of the points on $X$ at which the function $f=x^2$ does not vanish (indeed elements of $R_f$ are of the form $h/x^{2i}$, with $h\in R$ and $i\geq 0$). I leave it to you to interpret the expression "$f(P)\neq 0$" below, geometrically or algebraically - as you prefer. But of course you can view $R_f\subset K(X)=\textrm{Frac }R$ as a certain class of rational functions on $X$. To sum up, we have inclusions (the first open, the second closed): $$\textrm{Spec }R_f=\{P\in \textrm{Spec }R\,|\,f(P)\neq 0\}\subset \textrm{Spec }R\subset \mathbb A^3$$ corresponding to ring homomorphisms (the first arrow is quotient map, the second is the localization at $f$) $$\mathbb C[x,y,z]\to R=\mathbb C[x,y,z]/(x^3+y^3+z^3)\to R_f.$$ Instead, $R_{(f)}=\{r/f^i\,|\,\deg\,r=2i \}=\{g\in R_f\,|\,\deg\,g =0\}\subset R_f$.

I can only think at $\textrm{Spec }R_{(f)}$ in the projective setting: it is an open subscheme of the projective plane curve $\textrm{Proj }R=V_+(x^3+y^3+z^3)\subset \mathbb P_\mathbb C^2=\textrm{Proj }\mathbb C[x,y,z]$. (We can also regard $\textrm{Spec }R$ as the affine cone over $\textrm{Proj }R$. Added: In this perspective, the map $\textrm{Spec }R_{f}\to \textrm{Spec }R_{(f)}$, which is the restriction of the quotient map $\mathbb A^3\setminus\{0\}\to \mathbb P_\mathbb C^2$, it the morphism corresponding to the inclusion of $\mathbb C$-algebras $R_{(f)}\to R_f$.)

Answer 2

First, the elements of $R$ are the polynomial functions on the variety $X$ cut out by the equation $x^3 + y^3 + z^3 = 0$.

Elements of the ring $R_f$ have the form $r/f^k$ for some $r \in R$ and some integer $k \geq 0$. Note that since we may take $r$ to have a factor of $x$ if we like, the ring $R_{x^2}$ is the same as the ring $R_x$. We can think of $R_f$ as rational functions on $X$ that are allowed to have a "pole" only along the hypersurface cut out by $x = 0$ in $X$, or equivalently rational functions on $X$ that are defined on the open set where $x \neq 0$.

The prime ideals of $R$ correspond to the prime ideals of $\mathbb{C}[x,y,z]$ that contain the ideal $(x^3 + y^3 + z^3)$. The prime ideals of $R_f$ correspond to the prime ideals of $R$ that do not contain $f$ (and so do not contain $x$). Thus, $\text{Spec}(R_f)$ consists of the prime ideals of $\mathbb{C}[x,y,z]$ that contain $(x^3 + y^3 + z^3)$ but do not contain $x$. In particular, the closed points (maximal ideals) correspond to the points of $X$ where $x \neq 0$.