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.

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

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001