Connected scheme - Imported Knowlpedia Content

Definition

Why does connectedness distinguish a Galois field extension from a product of fields? Because

\operatorname{Spec}(K_1\times K_2) \cong \operatorname{Spec}K_1\amalg\operatorname{Spec}K_2

has two separated pieces, whereas the spectrum of a field has one point.

A scheme $X$ is connected if its underlying topological space in the Zariski topology is connected: there do not exist disjoint nonempty open subsets $U,V\subseteq X$ with $X=U\cup V$ . Equivalently, the only subsets that are both open and closed are $\varnothing$ and $X$ .

For an affine scheme $X=\operatorname{Spec}A$ , this is equivalent to $A$ having no idempotents other than $0$ and $1$ :

e^2=e\quad\Longrightarrow\quad e\in\{0,1\}.