# Flat morphism

A morphism of schemes $f: X\rightarrow Y$ such that for any point $x \in X$ the local ring ${\mathcal O} _ {X,x}$ is flat over ${\mathcal O} _ {Y,f( x) }$( see Flat module). In general, let ${\mathcal F}$ be a sheaf of ${\mathcal O} _ {X}$- modules; it is called flat over $Y$ at a point $x \in X$ if ${\mathcal F} _ {x}$ is a flat module over the ring ${\mathcal O} _ {Y,f( x) }$. Subject to certain (fairly weak) finiteness conditions, the set of points at which a coherent ${\mathcal O} _ {X}$- module ${\mathcal F}$ is flat over $Y$ is open in $X$. If, moreover, $Y$ is an integral scheme, then there exists an open non-empty subset $U \subset Y$ such that ${\mathcal F}$ is a flat sheaf over $Y$ at all points lying above $U$.
A flat morphism of finite type corresponds to the intuitive concept of a continuous family of varieties. A flat morphism is open and equi-dimensional (i.e. the dimensions of the fibres $f ^ { - 1 } ( y)$ are locally constant for $y \in Y$). For many geometric properties, the set of points $x \in X$ at which the fibre $f ^ { - 1 } ( f( x))$ of a flat morphism $f: X \rightarrow Y$ has this property is open in $X$. If a flat morphism $f$ is proper (cf. Proper morphism), then the set of points $y \in Y$ for which the fibres over them have this property is open .
Flat morphisms are used also in descent theory. A morphism of schemes is called faithfully flat if it is flat and surjective. Then, as a rule, one may check any property of a certain object over $Y$ simply by checking this property for the object obtained after a faithfully-flat base change $f: C\rightarrow Y$. In this connection, interest attaches to flatness criteria for a morphism $f: X\rightarrow Y$( or for the ${\mathcal O} _ {X}$- module ${\mathcal F}$); here $Y$ can be regarded as a local scheme. The simplest criterion relates to the case where the base $Y$ is one-dimensional and regular: A coherent ${\mathcal O} _ {X}$- module ${\mathcal F}$ is flat if and only if the uniformizing parameter in $Y$ has a trivial annihilator in ${\mathcal F}$. In a certain sense the general case is reducible to the one-dimensional case. Let $Y$ be a reduced Noetherian scheme and let for any morphism $Z \rightarrow Y$, where $Z$ is a one-dimensional regular scheme, the base change $f _ {Z} : X _ {Y} \times Z \rightarrow Z$ be a flat morphism; then $f$ is a flat morphism. Another flatness criterion requires that $f: X \rightarrow Y$ is universally open, while $Y$ and the geometric fibres $f ^ { - 1 } ( \overline{y}\; )$ are reduced.