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''of schemes''

The concept of a family of non-singular algebraic varieties (cf. [[Algebraic variety|Algebraic variety]]) generalized to the case of schemes. In the classical case of a morphism of complex algebraic varieties this concept reduces to the concept of a regular mapping (a submersion) of complex manifolds. A finitely-represented (local) morphism of schemes  $  f:  X \rightarrow Y $
is called a smooth morphism if  $  f $
is a [[Flat morphism|flat morphism]] and if for any point  $  y \in Y $
the fibre  $  f ^ { - 1 } ( y) $
is a [[Smooth scheme|smooth scheme]] (over the field  $  k( y) $). 
A scheme  $  X $
is called a smooth scheme over a scheme  $  Y $, 
or a smooth  $  Y $-
scheme, if the structure morphism  $  f:  X \rightarrow Y $
is a smooth morphism.

An example of a smooth  $  Y $-
scheme is the affine space  $  A _ {Y}  ^ {n} $. 
A special case of the concept of a smooth morphism is that of an [[Etale morphism|étale morphism]]. Conversely, any smooth morphism  $  f:  X \rightarrow Y $
can be locally factored with respect to  $  X $
into a composition of an étale morphism  $  X \rightarrow A _ {Y}  ^ {n} $
and a projection  $  A _ {Y}  ^ {n} \rightarrow Y $.

A composite of smooth morphisms is again a smooth morphism; this is also true for any base change. A smooth morphism is distinguished by its differential property: A flat finitely-represented morphism  $  f:  X \rightarrow Y $
is a smooth morphism if and only if the sheaf of relative differentials is a locally free sheaf of rank  $   \mathop{\rm dim} _ {x}  f $
at a point  $  x $.

The concept of a smooth morphism is analogous to the concept of a [[Serre fibration|Serre fibration]] in topology. E.g., a smooth morphism of complex algebraic varieties is a locally trivial differentiable fibration. In the general case the following analogue of the covering homotopy axiom is valid: For any [[Affine scheme|affine scheme]]  $  Y  ^  \prime  $, 
any closed subscheme  $  Y _ {0}  ^  \prime  $
of it which is definable by a nilpotent ideal and any morphism  $  Y  ^  \prime  \rightarrow Y $, 
the canonical mapping  $   \mathop{\rm Hom} _ {Y} ( Y  ^  \prime  , X) \rightarrow  \mathop{\rm Hom} _ {Y} ( Y _ {0}  ^  \prime  , X) $
is surjective.

If  $  f:  X \rightarrow Y $
is a smooth morphism and if the [[Local ring|local ring]]  $  {\mathcal O} _ {Y,y} $
at the point  $  y \in Y $
is regular (respectively, normal or reduced), then the local ring  $  {\mathcal O} _ {X,x} $
of any point  $  x \in X $
with  $  f( x) = y $
will also have this property.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck, "Eléments de géométrie algébrique IV. Etude locale des schémas et des morphismes des schémas" ''Publ. Math. IHES'' : 32 (1967) {{MR|0238860}} {{ZBL|0144.19904}} {{ZBL|0135.39701}} {{ZBL|0136.15901}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> A. Grothendieck (ed.) et al. (ed.) , ''Revêtements étales et groupe fondamental. SGA 1'' , ''Lect. notes in math.'' , '''224''' , Springer (1971) {{MR|0354651}} {{ZBL|1039.14001}} </TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR>
</table>