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).

  • $\begingroup$ The second sentence means that $B$ is a finitely generated $A$-module. $\endgroup$ – user26857 Jun 9 at 13:22
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    $\begingroup$ 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. $\endgroup$ – Mohan Jun 9 at 13:25
  • $\begingroup$ 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? $\endgroup$ – user26857 Jun 9 at 13:27
  • $\begingroup$ Maybe this could be helpful somehow: math.stackexchange.com/questions/753042/… $\endgroup$ – user26857 Jun 9 at 13:28
  • $\begingroup$ @Mohan I do not think anyone claimed that $B\otimes_A k$ is non-zero in general (zero is also a finite number). $\endgroup$ – jon Jun 9 at 14:12

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