Flat and smooth morphisms

The concept of flatness is a riddle that comes out of algebra, but which technically is the answer to many prayers. Let s recall the basic algebra first  

It is clear that a free 12-module has this property (i.e., take the m - s to be part of a basis of M) and that any direct limit of flat 12-modules again has this property, hence is flat. Conversely, it was recently proven that any flat 12-module is a direct limit of free 12-modules. Intuitively, one should consider flatness to be an abstraction embodying exactly that part of the concept of freeness which can be expressed in terms of linear equations. Or, one may say that flatness means that the linear structure of M preserves accurately that of R itself. On the other hand, to see an example of flat but not free modules, suppose 12 is a domain and let M be its quotient field. Since 

The defining property can be souped up without much difficulty to give the following  

This is the form in which one usually uses the definition. In the special case where M is flat over R, N2 = R and N is an ideal I C R, this implies that the natural map I r M -y I M is injective. Conversely, this special case implies that M is flat over R. Here are some of the basic facts  

If M is an R-module, then M is flat over R if and only if for all prime ideals P C R, the localization Mp is flat over Rp. 

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Theorem 3 . Let f X -tY be a morphism of finite type. Then f is smooth of relative dimension k if and only if f is flat and its geometric fibres are disjoint unions of k-dimensional non-singular varieties. 

The following result explains the connection between smooth and flat morphisms. 

Theorem 28.11. Let I be a finite ordered category, and f, X, —> Yt a morphism in Vil. Schl. Assume that Y, is noetherian with flat arrows, and /, is separated cartesian smooth of finite type. Assume that /, has a constant relative dimension d. Then for any F G L)Y (Yt), there is a functorial isomorphism... 

First assume that the characteristic is p > 0. Then there is some r 0 such that the scheme theoretic image of the Frobenius map f ( —> is reduced (or equivalently, A -smooth) and agrees with Note that the induced morphism G —> is flat, since the flat locus is a G-stable open... 

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