Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $k$.

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.