Let $Y$ be a scheme of finite type over $\mathbb{C}$ and let $\mathcal{A}$ be a locally free sheaf of $\mathcal{O}_Y$-algebras. (This means that $\mathcal{A}$ is a locally free sheaf of $\mathcal{O}_Y$-modules which is at the same time a sheaf of rings). As an example, let $\mathcal{E}$ be a locally free sheaf on $Y$, and let $\mathcal{A} = S(\mathcal{E})$, i.e the symmetric algebra on $\mathcal{E}$.
Then, from pg. 128 in Hartshorne, there exist a unique scheme $\pi: X=\textbf{Spec}(\mathcal{A}) \to Y$. In this case, $\textbf{Spec} \mathcal{A}$ is of course a vector bundle over $Y$.
Now a vector bundle is the data of the triple $(\textbf{Spec} \mathcal{A}, \pi, Y)$. So, from this definition, it seems like the total space $\textbf{Spec} \mathcal{A}$ is a space existing independent of its morphism to $Y$. But, I am always slight confused as to what category of schemes the total space of the vector bundle $\textbf{Spec} \mathcal{A}$ actually belongs to. I mean this in the sense that, we know that as defined $\textbf{Spec}(\mathcal{A})$ is $Y$-scheme, however, can we also consider $\textbf{Spec} \mathcal{A}$ as a scheme of finite type over $\mathbb{C}$?
I am asking this because I am interested in giving $\textbf{Spec} \mathcal{A}$ the structure of a complex analytic space, but I would need to describe it as a scheme of finite type over $\mathbb{C}$.