Connected scheme - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

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\}.

Example

The decomposition

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

shows that the spectrum of a product of two fields is disconnected, whereas the one-point spectrum of a field is connected.

Warning

Connected does not mean irreducible, path connected, or connected in a Euclidean topology. Schemes are being tested in their Zariski topology, whose open sets are usually much larger than familiar analytic open sets.