k is an alegraically closed field and A is a commutative k-algebra. We also know that A is a Noetherian domain and its Krull dimension is one. Are there any necessary and sufficient conditions on A under which A becomes finitely generated module over a polynomial algebra k[c] for some c in A? Does anybody know any papers or books that discuss this?
Thanks guys.
[Math] Commutative Noetherian Domains of Krull Dimension One
ac.commutative-algebra
Best Answer
Dear Amitsur,
It might help you to think geometrically. For example, $k[x,x^{-1}]$ is the ring of functions on a hyperbola $xy = 1$, and the projection from this hyperbola to the line $x = y$ is a finite projection. This corresponds to the fact that $k[x,x^{-1}]$ is finitely generated as a module over $k[x + x^{-1}].$ (If we write $f = x + x^{-1}$, then $x^2 - f x +1 = 0$ and $x^{-2} - f x^{-1} + 1 = 0$.)