# Separable mapping

A dominant morphism $f$ between irreducible algebraic varieties $X$ and $Y$, $f: X \rightarrow Y$, for which the field $K ( X)$ is a separable extension of the subfield $f ^ { * } K ( Y)$( isomorphic to $K ( Y)$ in view of the dominance). Non-separable mappings exist only when the characteristic $p$ of the ground field is larger than 0. If $f$ is a finite dominant morphism and its degree is not divisible by $p$, then it is separable. For a separable mapping there exists a non-empty open set $U \subset X$ such that for all $x \in U$ the differential $( df ) _ {x}$ of $f$ surjectively maps the tangent space $T _ {X,x}$ into $T _ {Y, f ( x) }$, and conversely: If the points $x$ and $f ( x)$ are non-singular and $( df ) _ {x}$ is surjective, then $f$ is a separable mapping.

A morphism $f: X \rightarrow Y$ of schemes $X$ and $Y$ is called separated if the diagonal in $X \times _ {Y} X$ is closed. A composite of separated morphisms is separated; $f: X \rightarrow Y$ is separated if and only if for any point $y \in Y$ there is a neighbourhood $V \ni y$ such that the morphism $f: f ^ { - 1 } ( V) \rightarrow V$ is separated. A morphism of affine schemes is always separated. There are conditions for Noetherian schemes to be separated.

A morphism $f: X \rightarrow Y$ of algebraic varieties or schemes is called dominant if $f( X)$ is dense in $Y$.
Let $A ^ {1}$ be the affine plane, and put $U = A ^ {1} \setminus \{ ( 0, 0) \}$. Let $X$ be obtained by glueing two copies of $A ^ {1}$ along $U$ by the identity. Then $X$ is a non-separated scheme.