TITLE: Why $K[x,y]$ and $K[x,y,z]/(xz-y^2)$ are not isomorphic as rings?
QUESTION [2 upvotes]: Let $K$ be an algebraically closed field, why $K[x,y]$ and $K[x,y,z]/(xz-y^2)$ are not isomorphic as rings?

REPLY [1 votes]: $K[x,y]$ is a regular ring (all of its localizations are regular). The local ring of $K[x,y,z]/(xz-y^2)$ at the origin $(0,0,0)$ is not regular: The maximal ideal $\mathfrak m$ there cannot be generated by two elements, not even mod $\mathfrak m^2$.
{"set_name": "stack_exchange", "score": 2, "question_id": 2687838}