I think of an affine scheme over a field $k$ as an extension of the concept of a (possibly reducible) variety over $k$, by extending the affine $n$-space $k^n$ to $A^n$, where $A$ is a commutative $k$-algebra.

The $k$-points of an affine scheme $S$ correspond to an affine variety $V$. My question is, when are $A$-points of $S$ a variety?

If $S=GL_n$ and $k=\mathbb{R}$ and $A=\mathbb{C}$, both $GL_n\ k$ and $GL_n\ A$ are varieties. I would guess this is not always the case and my question is, when does it hold, and what about general (not necessarily affine) schemes?