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Let $A$ be a commutative ring with identity and $B$ a finitely generated $A$-algebra that is integral over $A$. If $\mathfrak{p}$ is a prime ideal of $A$, there are only finitely many prime ideals $P$ of $B$ such that $P\cap A=\mathfrak{p}$.

Let me say that I am aware of this answer, but I can't follow through the hint. Also, I don't know how to extend to work for algebras rather than extension rings.

Community
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Dick Squizer
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2 Answers2

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Let $f:A\to B$ be a finite ring homomorphism and $\mathfrak p$ a prime ideal of $A$. Consider $S=A\setminus\mathfrak p$ and the ring homomorphism $S^{-1}f:S^{-1}A\to S^{-1}B$. This is also a finite ring homomorphism and all primes of $B$ lying over $\mathfrak p$ "survive" in $S^{-1}B$. This shows that one can assume $A$ local.

If $A\to B$ is a finite ring homomorphism and $(A,\mathfrak m)$ is a local ring, then all prime ideals of $B$ lying over $\mathfrak m$ contain $\mathfrak mB$. This allows us to replace $A\to B$ by $A/\mathfrak m\to B/\mathfrak mB$ (which is also a finite ring homomorphism), and therefore one can assume that $B$ is a finite algebra over a field. Now we have to prove that any finite algebra over a field has only finitely many prime (maximal) ideals. But such algebra is an artinian ring...

user26857
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One very useful tool is:

For a ring map $A\to B$, and $\mathfrak{p}$ a prime ideal of $A$, the prime ideals which contract to $\mathfrak{p}$ are in $1:1$ correspondence to the prime ideals of $\kappa(\mathfrak{p})\otimes_AB$, where $\kappa(p)=Q(A/\mathfrak{p})$ is the quotient field of the domain $A/\mathfrak{p}$.

Now we come back to the question. Since $\kappa(\mathfrak{p})\otimes_AB$ is integral over $\kappa(\mathfrak{p})$, all prime ideals of $\kappa(\mathfrak{p})\otimes_AB$ must be maximal (in fancy words, the Krull dimension is zero). And note that $\kappa(\mathfrak{p})\otimes_AB$ is finitely generated over $\kappa(\mathfrak{p})$, then $\kappa(\mathfrak{p})\otimes_AB$ is noetherian. A noetherian ring whose prime ideals are all maximal is an Artinian ring. An Artinian ring has only finitely many maximal ideals.

Alan Yan
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user119882
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    Is there any reference for proof of the fact ? Because I don't know if there is any characterization for prime ideal in the tensor product of two rings. – user371231 Nov 05 '18 at 17:45