A
separated morphism is a morphism f such that the
fiber product of f with itself along f has its
diagonal as a closed subscheme — in other words, the diagonal morphism is a
closed immersion. As a consequence, a scheme X is
separated when the diagonal of X within the
scheme product of X with itself is a closed immersion. Emphasizing the relative point of view, one might equivalently define a scheme to be separated if the unique morphism X \rightarrow \textrm{Spec} (\mathbb{Z}) is separated. Notice that a
topological space Y is
Hausdorff iff the diagonal embedding :Y \stackrel{\Delta}{\longrightarrow} Y \times Y, \, y \mapsto (y, y) is closed. In algebraic geometry, the above formulation is used because a scheme which is a Hausdorff space is necessarily empty or zero-dimensional. The difference between the topological and algebro-geometric context comes from the topological structure of the fiber product (in the category of schemes) X \times_{\textrm{Spec} (\mathbb{Z})} X, which is different from the product of topological spaces. Any
affine scheme
Spec A is separated, because the diagonal corresponds to the surjective map of rings (hence is a closed immersion of schemes): :''A \otimes_{\mathbb Z} A \rightarrow A, a \otimes a' \mapsto a \cdot a'''. Let S be a scheme obtained by identifying two affine lines through the
identity map except at the origins (see gluing scheme#Examples). It is not separated. Indeed, the image of the diagonal morphism S \to S \times S image has two origins, while its closure contains four origins. == Use in intersection theory ==