# Relation between a generalization of Weil's abstract varieties and algebraic schemes

We would like to generalize this question when the base field $k$ is not necessarily algbraically closed.

We fix an algebraically closed field $\Omega$ which has infinite trancendence dimension over the prime subfield. Let $k$ be a subfield of $\Omega$ such that tr.dim $\Omega/k = \infty$. Let $X$ be a topological space. We denote by $\mathcal{F}_X$ the sheaf of $\Omega$-valued functions on $X$. We regard $\mathcal{F}_X$ as a sheaf of $k$-algebras in the obvious way. Let $\mathcal{O}_X$ be a $k$-algebra subsheaf of $\mathcal{F}_X$. Namely, $\Gamma(U, \mathcal{O}_X)$ is a $k$-subalgebra of $\Gamma(U, \mathcal{F}_X)$ for every open subset $U$ of $X$ and the application $U \rightarrow \Gamma(U, \mathcal{O}_X)$ defines a subsheaf of $\mathcal{F}_X$. We call the pair $(X, \mathcal{O}_X)$ a $k$-space. By abuse of notation, we usually say $X$ is a $k$-space. Let $X, Y$ be $k$-spaces. Let $f\colon X \rightarrow Y$ be a continuous map. Let $U$ be any open subset of $Y$. Suppose $\psi\circ f \in \Gamma(f^{-1}(U), \mathcal{O}_X)$ for any $\psi \in \Gamma(U, \mathcal{O}_Y)$. Then $f$ is called a morphism of $k$-spaces. $k$-spaces form a category.

Let $E$ be a subset of a polynomial ring $k[x_1,\dots,x_n]$. We denote by $V(E)$ the common zeros of $E$ in $\Omega^n$. It is easy to see that by taking subsets of the form $V(E)$ as closed sets, we can define a topology in $\Omega^n$. We call this topology $k$-topology. A closed(resp. open) subset of $\Omega^n$ with respect to $k$-topology is called $k$-closed(resp. $k$-open) subset

Let $X$ be a $k$-closed subset of $\Omega^n$. Let $I(X) = \{f \in k[x_1,\dots,x_n]| f(p) = 0$ for every $p \in X\}$. Let $A = k[x_1,\dots,x_n]/I(X)$. Let $U$ be a $k$-open subset of $X$. Let $f\colon U \rightarrow \Omega$ be a function. We say $f$ is $k$-regular at a point $p$ of $U$ if there exist a $k$-open neighborhood $V$ of $p$ contained in $U$ and $g, h \in A$ such that $h$ does not vanish at every point of $V$ and $f(x) = g(x)/h(x)$ for every $x \in V$. We say $f$ is $k$-regular on $U$ if $f$ is $k$-regular at every point of $U$. Let $\Gamma_k(U)$ be the set of $k$-regular functions on $U$. $\Gamma_k(U)$ can be regarded as $k$-algebra in the obvious way. Clearly $U \rightarrow \Gamma_k(U)$ defines a $k$-algebra subsheaf $\mathcal{O}_X$ of $\mathcal{F}_X$. A $k$-space which is isomorphic to $(X, \mathcal{O}_X)$ is called an affine $k$-variety.

Let $(X, \mathcal{O}_X)$ be a $k$-space satisfying the following conditions.

(1) $X$ is covered by a finite number of open subsets $U_i$.

(2) Each $(U_i, \mathcal{O}_X|U_i)$ is an affine $k$-variety.

Then $X$ is called a $k$-prevariety. A morphism of $k$-prevarieties defined to be that of $k$-spaces. It is also called a $k$-morhism.

Let $X, Y$ be $k$-prevarieties. $X\times Y$ becomes a $k$-prevariety in the obvious way. Suppose the diagonal subset $\Delta_X = \{(x, x)|\ x \in X\}$ is closed in $X\times X$. Then $X$ is called a $k$-variety. $k$-prevarieties(resp. $k$-varieties) form a category.

Are the following assertions true?

(1) For every $k$-variety $V$, $(t(V), \alpha_*(\mathcal{O}_V)$ is a reduced separated scheme of finite type over $k$, where $t(V)$ and $\alpha$ are those defined in this question.

(2) There is a fully faithful and essentially epimorpfic functor $t\colon Var(k) \rightarrow Sch(k)$, where $Var(k)$ is the category of $k$-varieties and $Sch(k)$ is the category of reduced separated schemes of finite type over $k$.

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