As the title describes, how to show that $(k[x,y,z]/(xz,yz))_z$, $k[x,y,z]/(xz,yz)$ localized at $(1,z,z^2,...)$, is isomorphic to $k[z]_z$?
I have tried using a theorem saying that if the sequence $M'\to M\to M''$ is exact at $M$ as $R$ modules, then $S^{-1}M'\to S^{-1}M\to S^{-1}M''$ is also exact, with $S$ being a multiplicative set (in this scenario, I tried setting $R=k[x,y,z]$, $M'=(xz,yz)$, $M=k[x,y,z]$, $M''=k[x,y,z]/(xz,yz)$. I don't know what to do next because I can't find an easy expression for $(xz,yz)_z$.)
$$\begin{align}\left(k[x,y,z]/(xz,yz)\right)_z&\cong k[x,y,z,t]/(xz,yz,1-zt)\\&\cong k[x,y,z,t]/(x,y,1-zt)\\&\cong k[z,t]/(x,y,1-zt)\\&\cong \left(k[z]\right)_z.\end{align}$$