Explain the explanation of the Rabinowitsch trick

Question

Math Overflow offers this explanation of the Rabinowitsch trick in proving the strong Nullstellensatz from the weak Nullstellensatz.

https://mathoverflow.net/questions/90661/the-rabinowitz-trick

"Perhaps the "Rabinowitz trick" is more clear if one writes down the proof backwards in the following way:

Let $I \subseteq k[x_1,\dotsc,x_n]$ be an ideal and $f \in I(V(I))$, we want to prove $f \in \mathrm{rad}(I)$. In other words, we want to prove that $f$ is nilpotent in $k[x_1,\dotsc,x_n]/I$, or in other words, that the localization $(k[x_1,\dotsc,x_n]/I)_f$ vanishes. By general nonsense this algebra is isomorphic to $k[x_1,\dotsc,x_n,y]/(I,fy-1)$. But, clearly $V(I,fy-1)=\emptyset$ and therefore the Weak Nullstellensatz implies that $(I,fy-1)=(1)$, i.e. that the quotient vanishes."

I don't follow the claim that goes "By general nonsense this algebra is isomorphic to..." Should this be immediately obvious when reading the proof? Can you break it down further for me?

Answer 1

$R_f$ is isomorphic to $R[y]/(fy-1)$, which we can show using the universal property of localizations:

For any ring morphism $\varphi: R\to S$ that inverts $f$, there is a unique extension $\overline{\varphi}:R[y]\to S$ sending $y$ to $\varphi(f)^{-1}$. The kernel contains $fy-1$, and so $\overline{\varphi}$ factors through $R[y]/(fy-1)$.

A morphism $R[y]/(fy-1)\to S$ extending $\varphi$ must send $y$ to $\varphi(f)^{-1}$. In other words, any ring morphism $\varphi: R\to S$ that inverts $f$, factors uniquely through $R\to R[y]/(fy-1)$.

It follows that $R[y]/(fy-1)\cong R_f$.

Answer 2

It's not immediately obvious, at least not to me. But it makes sense at first glance, since the localization $(k[x_1,\ldots,x_n]/I)_f$ is essentially $(k[x_1,\ldots,x_n)/I)[1/f] \approx (k[x_1,\ldots,x_n]/I)[y]/(fy-1) \cong k[x_1,\ldots,x_n]/(I,fy-1)$. I haven't actually checked any of the logic here and some of it isn't even technically right, but just an indication of my thought process, but the idea when someone says it follows by general nonsense is that there should be some sequence of transformations of the algebraic structure to get from one to the other, which are plausible and the claim is that these can be checked to be valid. See Slade's answer for an example of how to check these manipulations.