What means “top exterior power of the sheaf of differentials $\Omega^1_{X/Y}$” when the rank of $\Omega^1_{X/Y}$ is globally unbounded?

Question

In Conrad's Grothendieck Duality and Base Change, p. 6, it is said:

When $f:X\to Y$ is a smooth map of schemes, then $\omega_{X/Y}$ denotes the top exterior power of the locally free finite rank sheaf $\Omega_{X/Y}^1$ on $X$.

My question is: for $k$ a field, what is “the top power of $\Omega_{X/Y}^1$” when $f$ equals $\coprod_{n\geq 0}\mathbb{A}_k^n\to\operatorname{Spec}k$?

Answer 1

On $\Bbb A^n_k$, it is $\bigwedge^n \Omega_{\Bbb A^n_k/k}^1$. If you have a smooth morphism, $\Omega_{X/Y}^1$ is locally free, and so $X$ is a disjoint union of open sets where $\Omega_{X/Y}^1$ is of constant rank. Take the appropriate exterior power on each of these sets. If you schemes are nice, you can reduce this to the appropriate exterior power on each connected component, and that's a morally instructive picture.