# Flat morphism

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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) [1b] A. Grothendieck, J. Dieudonné, "Eléments de géométrie algébrique" Publ. Math. IHES , 28 (1966) [2] D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) [3] M. Raynaud, L. Gruson, "Critères de platitude et de projectivité. Techniques de "platification" d'un module" Invent. Math. , 13 (1971) pp. 1–89