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I have a question about an argument used in Tamas Szamuely's "Galois Groups and Fundamental Groups" in following excerpt (see page 102):

We start with affine variety $X$ of dimension $n$. According Noether Normalization Lemma there exist an injection $\mathcal{O}(\mathbb{A}^n_k) = k[x_1,...,x_n] \to \mathcal{O}(X)$.

The contravariant category equivalence between affine varieties over $k$ and that of finitely generated reduced $k$-algebras induce a surjection $\phi: X \to \mathbb{A}^n_k$.

Let $M_p=(x_1-a_1,..., x_n -a_n)$ an maximal ideal of $\mathbb{A}^n_k$.

I have following two questions:

  1. Why is the fiber $\mathcal{O}(X)/M_P \mathcal{O}(X)$ a finite dimensional $k$-algebra?

  2. If 1. holds: Why does it imply that a finite dimensional $k$-algebra has finitely many maximal ideals?

user267839
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    $k[x]$ is not a finite dimensional $k$-algebra. – user1728 Feb 13 '19 at 20:21
  • See also here: https://math.stackexchange.com/questions/511597/finite-dimensionality-and-maximal-ideals?rq=1 – user1728 Feb 13 '19 at 20:23
  • @DistractedKerl: ah yes I see the dimension is not the Krull dimension but the vector space dimension. I deleted this false conter example – user267839 Feb 13 '19 at 20:24
  • @DistractedKerl: ok so 2. holds because it's Artin. btw: DO you have a nice reference for the proof that a finite dimensional $k$-algebra is Artin? Futhermore do you see why $\mathcal{O}(X)/M_P \mathcal{O}(X)$ has this property? – user267839 Feb 13 '19 at 20:30
  • is (one version) of the Nullstellensatz (more precisely Zariskis lemma: see here https://en.wikipedia.org/wiki/Zariski%27s_lemma ) 2. Is the link in the first comment. Hope that helps you :)
    • – user1728 Feb 13 '19 at 20:31
  • @DistractedKerl: ad 1: Do you mean this in sense that $Frac(\mathcal{O}(X)/M_P \mathcal{O}(X))$ is a finite $k$-extension? – user267839 Feb 13 '19 at 20:41
  • The quotient is already a field. :) – user1728 Feb 13 '19 at 21:55
  • @DistractedKerl: Why is the quotient $\mathcal{O}(X)/M_P \mathcal{O}(X)$ a field? – user267839 Feb 13 '19 at 22:02
  • @DistractedKerl: ...hmm guess that there could exist some statement that for a finite map maximal ideals are mapped to maximal ideals – user267839 Feb 13 '19 at 22:05
  • Part 1 is that finite morphisms are stable under fiber products (and taking the fiber is a fiber product) and part 2 is well-known, and could be found easily by searching: https://math.stackexchange.com/questions/919463/a-finite-dimensional-algebra-over-a-field-has-only-finitely-many-prime-ideals-an . Please try harder to find answers on your own before coming here next time - most basic results like these are already up somewhere on MSE. – KReiser Feb 13 '19 at 22:20