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I have some questions concerning Hanno's proof of the following proposition:

A point of a locally finite type $k$-scheme is a closed point if and only if the residue field of the stalk of the structure sheaf at that point is a finite extension of $k$

(1)

"$\Rightarrow$": We may assume $X=\text{Spec}(A)$ affine, so $x={\mathfrak m}$ is a maximal ideal. Then $A/{\mathfrak m}$ is a field extension of finite type, hence finite.

I don't know how he concluded $k(x)/k$ is a finite type extension. The residue field we are looking at should be the residue field of the local ring $\mathcal O_{X,x}=A_{\mathfrak m}$ instead of the field $A/{\mathfrak m}$, right (Vakil Definition 4.3.6)? Why is $A_{\mathfrak m}$(as well as its residue field) a finitely generated $k$-algebra?

(2)

"$\Leftarrow$": Since $X$ is covered by affine open subsets of the form $\text{Spec}(A)$ with $A$ of finite type over $k$, and since being closed can be checked on open coverings, we may assume that $X=\text{Spec}(A)$. Then $x={\mathfrak p}$ is a prime in $A$ such that $\text{Quot}(A/{\mathfrak p})/k$ is finite. In particular, $A/{\mathfrak p}$ is algebraic over $k$, and hence $A/{\mathfrak p}$ is a field.

I am not sure if he had typos in this answer. From the finite type hypothesis we can conclude that $\text{Quot}(A_{\mathfrak p})/k$ is finite. How do we conclude that $A/{\mathfrak p}$ is a field?

Any other approach to this problem is also welcomed here.

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  1. Exercise: $A_\mathfrak{m}/\mathfrak{m}_\mathfrak{m}\cong A/\mathfrak{m}$ for any ring $A$ and maximal ideal $\mathfrak{m}\subset A$.

  2. Second part means that the field of fractions of $A/\mathfrak{p}$ is a finite extension of $k$, so because every subring of an algebraic field extension which contains the base field is again a field, we see that $A/\mathfrak{p}$ must be a field (reference).

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