Given:
- A Noetherian local ring $(A,\mathfrak{m})$.
- Another Noetherian local ring $(B,\mathfrak{n})$ such that $f: A\to B$ is faithfully flat
- Residue field $k = A/\mathfrak{m} \cong B/\mathfrak{n}$.
To prove:
$$ \dim (B \otimes_A k) = 0$$
My attempt:
I know that:
- $\dim(B\otimes_B k) = 0$ since $B\otimes_B k \cong k$.
- Since $B$ is faithfully flat over $A$, each prime ideal $\mathfrak{p}$ of $A$ is contraction $f^{-1}(\mathfrak{q})$ of a prime ideal $\mathfrak{q}$ of $B$.
Now, I should be able to derive some relation between $B \otimes_A k$ and $B\otimes_B k$ using this information.
This might be a dumb question, but I am just stuck here and would really appreciate some hint.
Edit: Source of this statement is the proof of Lemma 2.26, chapter 4 (p. 132), Q. Liu's Algebraic Geometry and Arithmetic Curves.
Edit2: I think the above claim holds when $B=\widehat{A}$, the $\mathfrak{m}$-adic completion of $A$. Follows from Corollary 3.14, chapter 1 (p. 21), Liu's book.