What affine line are we considering when defining a geometric vector bundle over a scheme $X$?

Question

I recently began studying algebraic geometry, and in an attempt to learn about Abelian Varieties, I needed to learn the definition of a geometric vector bundle. I began reading this book, and in the notation section, it states:

By a geometric vector bundle of rank $d$ on $X$ we mean a group scheme $π: \mathbb V \to X$ over $X$ for which there exists an affine open covering $X = \bigcup U_\alpha$ such that the restriction of $\mathbb V$ to each $U_\alpha$ is isomorphic to $\mathbb G^d_a$ over $U_\alpha$.

I assume the $\mathbb G_a$ is the additive group of $\mathbb A^1$, but I am not sure what field $\mathbb A^1$ comes from. Could someone enlighten me?

Thanks ever so much! :)

Answer 1

The beginning of that example begins

(0.4) If $k$ is a field then by a variety over $k$ ... Recall that a $k$-scheme is

But in general, the affine line is $\mathbb{A}^1 = \operatorname{Spec} \mathbb{Z}[x]$, and over any scheme $X$, the trivial line bundle is given by the projection $\mathbb{A}^1 \times X \to X$.

e.g. if $k$ is a field, then $\mathbb{A}^1_k \cong \operatorname{Spec} k[x] \cong \mathbb{A}^1 \times \operatorname{Spec} k$, and for a $k$-scheme $X$, $$ \mathbb{A}^1_k \times_k X \cong \mathbb{A}^1 \times \operatorname{Spec} k \times_k X = \mathbb{A}^1 \times X$$

Answer 2

The affine spaces are based over $X$, or more generally over open subsets of $X$.