In Hartshorne's book, a functor $t$ is defined (II.2.6) from the category of varieties over an algebraically closed field $\kappa$ (Ch I) to the category of schemes over $\operatorname{Spec}\kappa$. I was wondering if it is possible to define in a similar fashion a functor $t'$ from the category of varieties over arbitrary fields $k$ (that is, locally ringed spaces that are locally isomorphic to set of zeros of some polynomials over $k$) to the category of functors over $\operatorname{Spec}k$ such that for every affine variety $V$ with ring of regular functions $A(V)$
$$ t'(V) = \operatorname{Spec}A(V). $$
In particular I am interested in the case when $V$ is a variety defined over some finite field $\mathbb{F}_q$.