Let $k$ be a field, $R:=k[x_1, \cdots , x_n]$ and $\mathfrak m$ be a maximal ideal such that $R/\mathfrak m$ is a finite separable field extension of $k$. Consider the algebraic closure $\overline k$ of $k$ and let $\mathfrak m_1, \dots , \mathfrak m_r$ be the maximal ideals of the ring $$\overline R :=\overline k \otimes_k R = \overline k[x_1, \cdots , x_n]$$ lying over $\mathfrak m$.

In an article I have been reading, the following isomorphism is used: $$\overline k \otimes_k R/\mathfrak m \cong \overline R \big/ \bigcap_{1 \leq j \leq r} \mathfrak m_j$$

Since I have not been able to find any reference for this, I have trying to justify the same myself:

In the forward direction, I could make the usual map work, namely, the one sending each element $$(\alpha, f + \mathfrak m) \in \overline k \times R/\mathfrak m \text{ (for }f \in R\text{) }\hspace{2mm}\text{ to }\hspace{2mm} \alpha f+ \bigcap_{i=1}^r \mathfrak m_i \in R \big/ \bigcap_{1 \leq j \leq r} \mathfrak m_j.$$ Indeed it turns out to be well-defined since each $\mathfrak m_i$ lies over $\mathfrak m$ and $k$-bilinearity is checked easily, whereupon the universal property of the tensor product applies.

However it is while showing the inverse isomorphism that I have gotten stuck. My idea was to try and get a map $$\phi: \overline R \longrightarrow \overline k \otimes_k R/\mathfrak m$$ whose kernel contains the intersection $\bigcap_{j=1}^r \mathfrak m_j$, and to do the same I tried sending each $\alpha \otimes f \in \overline k \otimes_k R = \overline R$ to $\alpha \otimes \overline f = \alpha \otimes (f + \mathfrak m) \in \overline k \otimes_k R/\mathfrak m$. However, in order to be able to use the Mapping Property and get my desired map $\phi$, I seem to need to have $\bigcap_{i=1}^r \mathfrak m_i \subset \mathfrak m$, which in this case becomes equivalent to $\bigcap_{i=1}^r \mathfrak m_i = \mathfrak m$).

And it is not clear to me why this last equality should hold; perhaps I am missing something obvious or some property of the ring extension $R \subset \overline R$ is at work here. I tried to use the weak Nullstellensatz to note that each $\mathfrak m_i$ must be of the form $(x_1-a_{i1}, \cdots , x_n-a_{in})$ for some $(a_{i1}, \cdots , a_{in}) \in \overline k^n$, but this hasn't proven to be useful yet. I am not sure whether the condition of $R/\mathfrak m$ being separable over $k$ will be helpful here.

I would really appreciate a proof (perhaps one using the above ideas?) or reference. Thank you.

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