# Tensor a finite algebra with the residue field

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).

• The second sentence means that $B$ is a finitely generated $A$-module. – user26857 Jun 9 at 13:22
• 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 9 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 9 at 13:27
• Maybe this could be helpful somehow: math.stackexchange.com/questions/753042/… – user26857 Jun 9 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 9 at 14:12