Some questions on the fundamentals of algebraic varieties Fix an algebraically closed field $k$. The affine varieties over $k$ and their morphisms, by definition, form a full subcategory of the category $\mathrm{LRS}$ of locally ringed spaces. Does it make sense to think of the full subcategory of $\mathrm{LRS}$ whose objects are the affine varieties over any algebraically closed field? That is, to think of morphisms of LRS between affine varieties over different fields?
Further, an algebraic variety is a LRS with an open covering of affine varieties (satisfying some extra conditions to prevent the case of a general prevariety), do the affine varieties of the covering need to be over the same field?
I feel like the answer to both questions is no, because in Gathmann’s notes the only LRS considered are those of $k$-valued functions, and the field $k$ stays fixed while defining varieties.
So call $\mathrm{Var}$ the full subcategory of $\mathrm{LRS}$ whose objects are varieties over a fixed field $k$. Also, let $\mathrm {LRS}_k \subset \mathrm{LRS}$ be the subcategory of LRS of $k$-valued functions; is this inclusion full? Most likely yes, because otherwise the induced inclusion of categories $(\mathrm{Var} \cap \mathrm{LRS}_k)\subset \mathrm{Var}$ wouldn’t be an equivalence, and reasonably Gathmann is restricting to a category at least equivalent to the usual category $\mathrm{Var}$ of varieties over $k$. However I wouldn’t know how to even start a proof of this fact.
(Maybe it’s not clear: $\mathrm{LRS}$ means the category of locally ringed spaces, LRS is just a shorthand for locally ringed space/spaces).
 A: Question 1: “Fix an algebraically closed field k. The affine varieties over k and their morphisms, by definition, form a full subcategory of the category LRS of locally ringed spaces. Does it make sense to think of the full subcategory of LRS whose objects are the affine varieties over any algebraically closed field? That is, to think of morphisms of LRS between affine varieties over different fields?”
Answer: Yes, this makes sense. If $k \rightarrow L$ is an embedding of fields and $A$ a $k$-algebra, $B$ a $L$ algebra, it makes sense to speak of maps
$$\phi: X:=\mathrm{Spec}(B) \rightarrow S:=\mathrm{Spec}(A).$$
Whenever you have a point $x \in X$ with image $f(x):=s \in S$
you get an induced extension of residue fields  $\kappa(s) \subseteq \kappa(x)$,
hence for this to make sense, your fields must have the same characteristic.
Note that if $k,L$ are field with $char(k) \neq char(L)$ and if $X$ is a scheme over $k$, $Y$ a scheme over $L$, there are no morphisms $f: X \rightarrow Y$. Hence $X$ cannot be a scheme over $k$ and $L$. Hence if $Aff(k)$ (resp $Aff(L)$) are the "categories of affine schemes over $k$" (resp $L$) there is no object $X \in Ob(Aff(k))$ and $Ob(Aff(L))$. The two categories have no "objects in common".
Question 2: “Further, an algebraic variety is a LRS with an open covering of affine varieties (satisfying some extra conditions to prevent the case of a general prevariety), do the affine varieties of the covering need to be over the same field?”
If a variety/scheme $X$ is defined over a field $k$ it follows any open subscheme (or subvariety) $U:=\mathrm{Spec}(A) \subseteq X$ is defined over $k$.
