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I am trying to solve Ex. 5.4.M in Vakil's notes (which has been discussed here) my question is different than that, I want to solve:

If $A$ is a integral domain which is also $k-$algebra, and $l$ is a finite extension of $k$ ,and assume $A\otimes_k l$ is a normal integral domain. I want to show that $K(A) \otimes_k l$ is an integral domain.I have thought it for a while and don't have idea to prove it

some post shows that tensor product of integral domain over algebraic closed field is a domain assuming one of them is finite generated. However there is no algebraic closed condition here.Tensor product of domains is a domain

(And I try to show it by direct computation(it seems not to be a good idea so you can ignore it), assume $b_i$ are $k$- linear basis for $l$. then each element in $K(A)\otimes_k l$ has the form $\sum a_i\otimes b_i$ if $$(\sum a_i\otimes b_i)(\sum c_j\otimes b_j) = \sum a_ic_j\otimes b_ib_j = 0$$

but I don't know how to preceed then)

yi li
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  • See here shows tensor product of two fields needs not to be integral domain without algebraic closed condition :https://math.stackexchange.com/a/2520807/360262 – yi li Jan 22 '23 at 09:48

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Suppose that $$ (\sum_i \frac{a_i}{s_i}\otimes l_i)(\sum_j \frac{a'_j}{s'_j}\otimes l'_j)=0, $$ then in particular, if $s:=\left(\prod_i s_i\right)\cdot\left(\prod_j s'_j\right)$, and $b_i=(s/s_i)a_i\in A$, $b'_j=(s/s'_j)a'_j\in A$, we have $$ 0=(s^2\otimes 1)(\sum_i \frac{a_i}{s_i}\otimes l_i)(\sum_j \frac{a'_j}{s'_j}\otimes l'_j)=(\sum_i \frac{b_i}{1}\otimes l_i)(\sum_j \frac{b'_j}{1}\otimes l'_j). $$ So as the map $A\otimes_k l\to K(A)\otimes_k l$ is injective (because $l$ is flat over $k$) we obtain that $$ (\sum_i {b_i}\otimes l_i)(\sum_j {b'_j}\otimes l'_j)=0 $$ inside $A\otimes_k l$, which by assumption is an integral domain. So one of them is equal to $0$; assume WLOG that it is $\sum_i {b_i}\otimes l_i=0$. But then we have $$ 0=(\frac{1}{s}\otimes 1)(\sum_i \frac{b_i}{1}\otimes l_i)=\sum_i \frac{a_i}{s_i}\otimes l_i. $$ So $K(A)\otimes_k l$ is an integral domain.

imtrying46
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  • thank you imtrying46 , I have the same idea as you did, the key point is that $A\otimes_k l $ is integral domain without this condition it may not holds. for example $\Bbb{C}\otimes_{\Bbb{R}}\Bbb{C}$ – yi li Jan 22 '23 at 10:19