Krull Dimension of $K \otimes_k L$

Question

Let $k$ be a field and $K,L$ are arbitrary extensions of $k$.

What do we know about $dim(K \otimes_k L)$ (as Krull dimension). How does it depend on transcendence degrees $trdeg_k(K), trdeg_k(L)$?

Futhermore: If we cosider by $k$ instead of a field a ring and $K,L$ be it's ring extensions. How does $dim(K \otimes_k L)$ depend on $dim(K), dim(L)$?

Answer 1

In the case where $k$ is a field we have

$$ \dim (K \otimes_k L) = \min(\operatorname{trdeg}_k K, \operatorname{trdeg}_k L) $$ There is a Proof in Görtz's and Wedhorn's Algebraic Geometry 1 (Lemma B.97).

No such thing is possible in the second case: Suppose $k$ is a ring that "happens to be" a field and $K = L = k(x_1, \dots, x_n)$ are the ring extensions of $k$. We have $\dim K = \dim L = 0$ since they are fields but $\dim(K \otimes_k L) = n$. So $\dim(K \otimes_k L)$ does not only depend on $\dim K$ and $\dim L$.