# Exercise 3.30 from Görtz-Wedhorn: local affine algebras

Here's the Exercise 3.30 from the textbook of Görtz and Wedhorn:

Let $$k$$ be a field, and let $$A$$ be a local $$k$$-algebra of finite type. Prove that $$\operatorname{Spec} A$$ consists of a single point, and that $$A$$ is finite-dimensional as a $$k$$-vector space. In particular $$A$$ is a local Artin ring (why?), and $$\kappa (A)/k$$ is a finite field extension.

I see how this follows from some well-known results:

1. Every finitely generated algebra over a Jacobson ring is Jacobson, so that $$A$$ is Jacobson. As $$A$$ is local, this amounts to saying that its maximal ideal is the only prime ideal.

2. In general, for every maximal ideal $$\mathfrak{m}$$ in a finitely generated $$k$$-algebra $$A$$, the extension $$\kappa (\mathfrak{m})/k$$ is finite (which follows again from the general results about Jacobson rings).

3. A ring is Artinian iff it is Noetherian and every prime ideal is maximal, which is the case here.

4. A finitely generated $$k$$-algebra is Artinian iff it is finite (Atiyah-Macdonald, Exercise 8.3).

However, all this seems to be an overkill, and I think the authors had in mind some direct argument for the very special case when $$A$$ is local. Do you see one?

• Just an opinion, but this doesn't seem like overkill to me. – RghtHndSd Jan 5 at 22:29
• I think there is something wrong with the finite type and I think you need finite(i.e. integrality condition.) For finite type, I assume you mean finitely generated over $k$. However, consider $k[x,y]$ and localize away from $(x,y)=(0,0)$ points. This gives a local ring of dimension 2 and this is finite type over $k$. The spectrum cannot be made of single point. If it is made of 1 point, one would expect this had better be a finite field extension as indicated in the conclusion. – user45765 Jan 10 at 16:07
• @user45765 Every algebra of finite type over a field is a Jacobson ring, and a Jacobson ring is local when its maximal ideal is the only prime ideal. (See e.g. stacks.math.columbia.edu/tag/00FZ for the statements and proofs.) So a local ring of dimension $> 0$ can't be of finite type over a field. – Sr. Tacuacín Jan 10 at 22:25