In Exercise II 3.22 Hartshorne says let $f:X \to Y$ be a dominant morphism of integral schemes of finite type over a field.
Let $k$ be a field.
I just wanted to verify that this actually means (1) $f$ is dominant (2) $X$ and $Y$ are integral schemes of finite type over $k$?
I was getting slightly confused with the wording, and I just wanted to make sure I had the right meaning. I think it's unlikely but I thought maybe $X$ is an integral schemes of finite type over $k_1$ and $Y$ is an integral scheme of finite type over $k_2$, where $k_1$ and $k_2$ are different fields?. Or possibly that this means $f: X \to Y$ is a morphism of integral schemes and this $f$ is of finite type..
Any clarification would be appreciated. Thank you.
Best Answer
Your interpretation is almost correct: $X$ and $Y$ are each integral schemes of finite type over $k$, and $f:X\to Y$ is a dominant morphism of schemes over $k$ (so, it forms a commutative diagram with the morphisms to $\operatorname{Spec} k$). In general, when one speaks of a "morphism of [...] schemes over $T$", then that means a morphism in the category of schemes over $T$ rather than just the category of schemes.