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Let $f:A\rightarrow B$ be a morphism of commutative unital rings. Assume that there exists a positive integer $n$ and a surjective map of $A$-modules $A^n\rightarrow B$.

Let $\mathfrak{p}\subset A$ be a prime ideal with the residue field $k$. Then $B\otimes_A k$ is a finite-dimensional $k$-algebra. As such it splits into the direct product of finitely many non-zero local $k$-algebras.

I believe the factors are in bijection with the number of prime ideals of $B$ whose inverse image is $\mathfrak{p}$. How can I express the factors in terms of $f$? They definitely are not intrinsic to $A$ (project parabola onto different lines, the non-reduced point will be in different places depending on where you are projecting to).

jon
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  • The second sentence means that $B$ is a finitely generated $A$-module. – user26857 Jun 09 '19 at 13:22
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    And the third sentence is a bit off. If $P\subset A$ is a prime ideal (and not maximal), the residue ring is not a field unless you meant the fraction field of the residue ring. Also, in the fourth sentence, there is no reason for $B\otimes_A k$ to be non-zero in general. – Mohan Jun 09 '19 at 13:25
  • I don't understand what do you mean by expressing the factors in terms of $f$. What if $A\subset B$ as in many concrete examples? – user26857 Jun 09 '19 at 13:27
  • Maybe this could be helpful somehow: https://math.stackexchange.com/questions/753042/finitely-many-prime-ideals-lying-over-mathfrakp?r=SearchResults – user26857 Jun 09 '19 at 13:28
  • @Mohan I do not think anyone claimed that $B\otimes_A k$ is non-zero in general (zero is also a finite number). – jon Jun 09 '19 at 14:12
  • @Mohan regarding the third sentence I thought that "residue field" was the standard terminology for $A_{\mathfrak{p}}/\mathfrak{p}A_{\mathfrak{p}}$ so I do not think that it is off. If you can prove that it is not standard, I will admit the sentence was off. – jon Jun 09 '19 at 14:15
  • @user26857 indeed, the second sentence does mean that. – jon Jun 09 '19 at 14:15
  • @user26857 I mean to give any characterization other than "a local factor of $B\otimes_A k$". I hope there is some useful characterization though I can not define "useful". – jon Jun 09 '19 at 14:17

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