Properties of the functor $(X, \mathcal{O}_X) \mapsto (X(k), \mathcal{O}_{X(k)})$

Question

Let $k$ be an algebraically closed field. In Görtz and Wedhorns book one can read about an equivalence of categories

$\{\text{integral schemes of finite type over } k\} \to \{\text{prevarieties over } k\}$

given by $(X, \mathcal{O}_X) \mapsto (X(k), \mathcal{O}_{X(k)})$. $X(k)$ is the set of closed points of $X$ with induced topology and $\mathcal{O}_{X(k)} = \alpha^{-1}\mathcal{O}_X$, where $\alpha: X(k) \to X$ is the inclusion.

As far as I know, Görtz and Wedhorn do not study this functor in detail. But I may be wrong, as I did not read much of this book. Particularly I'm interested in the following questions, which answers could help me a lot to understand why toric varieties can simultaneously be seen as schemes or varieties:

1.: Assume $X$ is normal and seperated. Is this also true for $X(k)$?

2.: Let $X \to Y$ be an open immersion. Is this also true for $X(k) \to Y(k)$?

3.: Is $(X \times_k Y)(k) \cong X(k) \times Y(k)$?

4.: Assume $X(k) = \bigcup_{i \in I}\text{Specm}(A_i)$ and additionally that $\text{Specm}(A_i) \cap \text{Specm}(A_j) = \text{Specm}(A_{i, j})$. Let $\varphi_{j, i}: A_i \to A_{i, j}$ be the corresponding ringhomomorphisms. Then, is it true that $X = \bigcup_{i \in I}\text{Spec}(A_i)$ so that $\text{Spec}(A_i) \cap \text{Spec}(A_j) = \text{Spec}(A_{i, j})$ and do the inclusion maps come from the same ring homomorphisms as before?

Thanks for any help on these questions.

Answer 1

The question have a partial solution in the problem 23, section 3, chapter 2 of hartshorne and here are two solutions of that book, //algebraicgeometry.blogspot.com/, //math.mit.edu/~ssam/, I hope that you get the answer reading this reference via net.

Answer 2

Denote $$\{\text{integral schemes of finite type over } k\} \overset{-(k)}{\underset{t}{\rightleftharpoons}} \{\text{prevarieties over } k\} \qquad (*)$$

For 1, $X(k)$ is normal if $X$ is, since $X$ normal implies $\mathcal{O}_{x,X}$ is a normal domain (i.e., is a domain that is integrally closed in its fraction field) for all $x \in X$, which implies $(\alpha^{-1}\mathcal{O})_{x,X(k)} \cong \mathcal{O}_{\alpha(x),X}$ is a normal domain for all $x \in X(k)$. Note this isomorphism is (2.8.3) in Görtz/Wedhorn.

For separatedness, you'd first need to define what a separated prevariety is. The definition in Görtz/Wedhorn (Ex. 9.17) is basically that $X(k)$ is separated if $X$ is (as a scheme over $\operatorname{Spec} k$), so the claim is automatic.

For 2, if $j\colon X \hookrightarrow Y$ is an open immersion, the underlying continuous map of $j$ is a homeomorphism between $X$ and an open set $U$ of $Y$, which restricts to a homeomorphism between $X(k)$ and $U(k)$ since both $X$ and $U$ have the induced subspace topology. The condition on structure sheaves follows for the same reason as in 1, since stalks don't change under the functor $-(k)$. Here I don't know what the definition for an open immersion of varieties is, so I'm just presuming it is the same as that for schemes.

For 4, a way to see this is that $X(k)$ is the colimit of the diagram with nodes being your maximal spectra (which I assume is what you mean by $\operatorname{Specm}$, as in Milne's notes where they are $\operatorname{Spm}$). Applying the functor $t(-)$, we have that the maps between $\operatorname{Spec} A_i$'s and $\operatorname{Spec} A_{ij}$'s are indeed those induced by inclusions of rings (since the equivalence $(*)$ restricts to that of affine varieties and $k$-algebras; see Görtz/Wedhorn §3.13) and $t(X(k)) \cong X$ is still the colimit of the diagram since $(*)$ is fully faithful.