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!