Flat morphism
A morphism of schemes such that for any point
the local ring
is flat over
(see Flat module). In general, let
be a sheaf of
-modules; it is called flat over
at a point
if
is a flat module over the ring
. Subject to certain (fairly weak) finiteness conditions, the set of points at which a coherent
-module
is flat over
is open in
. If, moreover,
is an integral scheme, then there exists an open non-empty subset
such that
is a flat sheaf over
at all points lying above
.
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 are locally constant for
). For many geometric properties, the set of points
at which the fibre
of a flat morphism
has this property is open in
. If a flat morphism
is proper (cf. Proper morphism), then the set of points
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 simply by checking this property for the object obtained after a faithfully-flat base change
. In this connection, interest attaches to flatness criteria for a morphism
(or for the
-module
); here
can be regarded as a local scheme. The simplest criterion relates to the case where the base
is one-dimensional and regular: A coherent
-module
is flat if and only if the uniformizing parameter in
has a trivial annihilator in
. In a certain sense the general case is reducible to the one-dimensional case. Let
be a reduced Noetherian scheme and let for any morphism
, where
is a one-dimensional regular scheme, the base change
be a flat morphism; then
is a flat morphism. Another flatness criterion requires that
is universally open, while
and the geometric fibres
are reduced.
References
[1a] | A. Grothendieck, J. Dieudonné, "Eléments de géométrie algébrique" Publ. Math. IHES , 24 (1964) MR0173675 Zbl 0136.15901 |
[1b] | A. Grothendieck, J. Dieudonné, "Eléments de géométrie algébrique" Publ. Math. IHES , 28 (1966) MR0217086 Zbl 0144.19904 |
[2] | D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701 |
[3] | M. Raynaud, L. Gruson, "Critères de platitude et de projectivité. Techniques de "platification" d'un module" Invent. Math. , 13 (1971) pp. 1–89 MR0308104 Zbl 0227.14010 |
Comments
References
[a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 |
Flat morphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Flat_morphism&oldid=12305