I was reading https://www.math.columbia.edu/~dejong/courses/deJongNotes.pdf. (Observe you don't have to look at the link as the question is self contained)
See Example 1 in the beginning. There I could not understand why $k[x_1] \to A$ (inclusion map) is not finite [$A$ is definitely finitely generated $k[x_1]$-module having generators $1, x_2$], where $A=k[x_1,x_2]/(x_1x_2-1)$. Moreover, I need help in understanding why $\phi: k[y] \to A$, $\phi(y)=x_1+x_2$ is finite?
Edit: From the comment (thanks to Samir Canning), I got that because of the variable $x_2$ and as $k[x_1]$ is a polynomial ring we have that $A$ is not finitely generated. Can you help me in the 2nd question?