Dimension of fibre products of $k$-schemes for an arbitrary field $k$

Question

I am trying to get some intuition concerning fibre products of schemes and thus was looking for examples of the following:

Suppose we have schemes $X$ and $Y$ with morphisms $f$ and $g$ from each of them to $S=\operatorname{Spec}(k)$ for an arbitrary field $k$ (I believe these are sometimes called $k$-schemes).

If $f$ and $g$ are of finite type, then the dimension of their fibre product over $k$ is the sum of the dimensions (Prop. 5.37 in Görtz and Wedhorn Algebraic Geometry I).

Otherwise, what can we say about the dimensions of their fibre product over $k$? Can we obtain a bigger dimension than the sum of both? Could this dimension be even infinite if $X$ and $Y$ had both dimension $0$?

I would appreciate examples where possible.

Answer 1

When you leave the realm of finiteness hypotheses, anything can happen.

Theorem (Grothendieck-Sharp): Let $L/k$ and $K/k$ be two field extensions. Then $$\dim_{Krull} L\otimes_k K = \min(\operatorname{trdeg} L/k,\operatorname{trdeg} K/k).$$

This gives you all the examples you want: just let $L=K=k(\{t_i\}_{i\in I})$, so that $L\otimes_k K$ has Krull dimension $|I|$ despite $L$ and $K$ having Krull dimension zero.