understanding schemes finite over Spec $K$

Question

I am following Vakil's FOAG, exercise 7.3.H: Let $X\to $Spec $K$ be a finite morphism, prove that $X$ is a finite union of points with the discrete topology.

I am following the guidance there. If we write $X=$Spec $A$ then $A$ is a finite dimensional vector space over $K$. If $A$ was a domain then it is easy to show that it is a field, and so we get that all primes of $A$ are maximal, and hence $X$ consists only of closed points.

The next part should be to prove that $X$ is discrete, and then finiteness would follow from quasicompactness.

My question is why is $X$ discrete? I will be glad for anything you can say about the general problem, but I am looking to understand how it is possible to show discreteness now, before finiteness, say.

Answer 1

(An answer based on an exchange of comments with the OP.)

As the OP observes, a finite-dimensional $K$-algebra that is a domain is necessarily a field, and so all prime ideals in $A$ are maximal.

Now, by CRT, if $\mathfrak m_1, \ldots,\mathfrak m_k$ are distinct maximal ideals, then $$A/(\mathfrak m_1 \cap \cdots \cap \mathfrak m_k) \cong A/\mathfrak m_1 \times \cdots \times A/\mathfrak m_k, $$ and so $k \leq \dim_K A.$ In particular, $A$ admits no more than $\dim_K A$ maximal ideals, and so Spec $A$ is a finite set of closed points.

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Unfortunately, I don't see how to directly follow the hint (i.e. to first prove discreteness) in a natural way.

Answer 2

Applying the definition goes a long way; assuming that $K$ is a field, $\operatorname{Spec}K$ is a point. For a morphism $f:\ X\ \longrightarrow\ \operatorname{Spec} K$ to be finite means precisely that $f^{-1}(\operatorname{Spec}K)=X=\operatorname{Spec} V$ where $V$ is a $K$-algebra that is a finitely generated as a $K$-module, i.e. it is a finite dimensional $K$-vector space with a ring map $K\ \longrightarrow V$.

EDIT: As Georges Elencwajg points out in the comments below, I was a bit hasty in my conclusions. I won't say too much in attempt to avoid saying more silly things.

Note that $V$ is Artinian, hence its spectrum is finite and all prime ideals are maximal.