Let $k$ be a field and let $(X, \mathcal{O}_X)$ be a $k$-scheme. Let $K/k$ be a field extension. I don't quite understand what is mean by the scheme $X_K$, formed by "extension of scalars."
One thing seems clear to me - if $U \subset X$ is an open subset, and $\mathcal{O}_X(U)$ is the $k$-algebra of sections on $U$, then the sections of $X_K$ on $U$ should be $\mathcal{O}_X(U) \otimes_k K$.
My question is, what exactly is $X_K$, that is, what is the underlying topological space? Is it just the same as the underlying topological space $X$, and just the structure sheaf changes? If not, how can I describe $X_K$?
If $X$ has an open cover by affine schemes $X_i = \operatorname{spec} A_i$, then the corresponding open subset of $X_K$ should ber $\operatorname{spec} (A_i \otimes_k K)$, so probably $X$ and $X_K$ can't have the same underlying topological space.
When $K = \overline{k}$ is an algebraic closure of $k$, and $X$ is locally of finite type over $k$, it may be easier to consider what is $X$ in terms of $X_{\overline{k}}$, rather than the other way around. The projection $X_{\overline{k}} \rightarrow X$ is a surjective open and closed map, in particular a quotient map. On the level of topological spaces, $X$ is obtained from $X_{\overline{k}}$ by glueing together points which are in the same $\operatorname{Aut}(\overline{k}/k)$ orbit.
This is explained in Mumford's red book, and is also in Stacks project somewhere. There is probably a similar description for when $K$ is an arbitrary algebraic extension of $k$, but I can't immediately find a reference.