At the moment I'm trying to understand the concept of localizations of rings / modules. I have done some exercises (using the book of Atiyah / MacDonald) and I will do some more, but a more practical question took my attention.
I started with an easy example and considered the ring $k[x]_x$. Localizing at $x$ means that $x$ and its powers become units in the localization. Hence
$$k[x]_x\cong k[x,x^{-1}]\cong k[x,y]/\langle xy-1\rangle,$$
am I right there for a start?
Then I wanted to compute the localization of $k[x,y]/\langle xy-1\rangle$ at $\langle x-1,y-1\rangle$. This should be a maximal ideal in the quotient, since $\langle xy-1\rangle\subset\langle x-1,y-1\rangle$. (Geometrically, what does this mean here? The maximal ideal corresponds to the point $p=(1,1)$ on the hyperbola $xy-1=0$, so we "gather only local information around $p$" in some way?)
Localization commutes with the quotient, thus
$$(k[x,y]/\langle xy-1\rangle)_{\langle x-1,y-1\rangle}\cong k[x,y]_{\langle x-1,y-1\rangle}/\langle xy-1\rangle_{\langle x-1,y-1\rangle},$$
and here already I am stuck. Is there a general way to compute localized rings of this form, or at least some plan that often works in a case like this? Edit: Can we guess what this should be isomorphic to when looking at it geometrically?
Thank you for your help / hints in advance!
Best Answer
Everything you wrote is correct and you have the right attitude : aggressively attacking simple examples and trying to see what is going on geometrically.
As for localization, you can use your last isomorphism if you want: localization does indeed commute with quotients.
But it is simpler to exploit your preceding isomorphism $ k[x,y]/\langle xy-1\rangle \cong k[x,x^{-1}]$ instead and to write $$(k[x,y]/\langle xy-1\rangle)_{\langle x-1,y-1\rangle}\cong k[x,x^{-1}]_{{\langle x-1,x^{-1}-1\rangle}}=k[x,x^{-1}]_{{\langle x-1\rangle}}=k[x]_{{\langle x-1\rangle}} =k[x-1]_{{\langle x-1\rangle}}$$
Geometrical interpretation (very important!)
You are studying a hyperbola in the plane near the point $(1,1)$ by projecting it on the $x$-axis and you obtain an isomorphism with the affine line punctured at zero.
The projection sends $(1,1)$ to the point $x=1$ on the punctured line and the local ring $(k[x,y]/\langle xy-1\rangle)_{\langle x-1,y-1\rangle}$ of the hyperbola at $(1,1)$ isomorphically to the local ring of the line at $1$, namely $k[x-1]_{{\langle x-1\rangle}}$.
Note carefully that the puncture of the affine line plays no role in these local questions: the point $x=1$ only sees its immediate vicinity and doesn't care what happens at $x=0$.