In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e., is a flat map for all P in X.
[2] Two basic intuitions regarding flat morphisms are: The first of these comes from commutative algebra: subject to some finiteness conditions on f, it can be shown that there is a non-empty open subscheme
Here 'restriction' is interpreted by means of the fiber product of schemes, applied to f and the inclusion map of
For the second, the idea is that morphisms in algebraic geometry can exhibit discontinuities of a kind that are detected by flatness.
For instance, the operation of blowing down in the birational geometry of an algebraic surface can give a single fiber that is of dimension 1 when all the others have dimension 0.
It turns out (retrospectively) that flatness in morphisms is directly related to controlling this sort of semicontinuity, or one-sided jumping.
This is a deep-lying theory, and has not been found easy to handle.
The concept of étale morphism (and so étale cohomology) depends on the flat morphism concept: an étale morphism being flat, of finite type, and unramified.
Consider the affine scheme morphism induced from the morphism of algebras Since proving flatness for this morphism amounts to computing[3] we resolve the complex numbers and tensor by the module representing our scheme giving the sequence of
-modules Because t is not a zero divisor we have a trivial kernel, hence the homology group vanishes.
Other examples of flat morphisms can be found using "miracle flatness"[4] which states that if you have a morphism
Easy examples of this are elliptic fibrations, smooth morphisms, and morphisms to stratified varieties which satisfy miracle flatness on each of the strata.
This is because Hilbert schemes parameterize universal classes of flat morphisms, and every flat morphism is the pullback from some Hilbert scheme.
One class of non-examples are given by blowup maps One easy example is the blowup of a point in
If we take the origin, this is given by the morphism where the fiber over a point
, we get the isomorphism The reason this fails to be flat is because of the Miracle flatness lemma, which can be checked locally.
A simple non-example of a flat morphism is
This is because is an infinite complex, which we can find by taking a flat resolution of k, and tensor the resolution with k, we find that showing that the morphism cannot be flat.
Another non-example of a flat morphism is a blowup since a flat morphism necessarily has equi-dimensional fibers.
is an exact functor from the category of quasi-coherent
is flat, then it possesses all of the following properties: If f is flat and locally of finite presentation, then f is universally open.
[26] However, if f is faithfully flat and quasi-compact, it is not in general true that f is open, even if X and Y are noetherian.
[27] Furthermore, no converse to this statement holds: If f is the canonical map from the reduced scheme Xred to X, then f is a universal homeomorphism, but for X non-reduced and noetherian, f is never flat.
is faithfully flat, then: If f is flat and locally of finite presentation, then for each of the following properties P, the set of points where f has P is open:[31] If in addition f is proper, then the same is true for each of the following properties:[32] Assume
[44] Assume f is faithfully flat and quasi-compact.
Let G be a quasi-coherent sheaf on Y, and let F denote its pullback to X.
Then F is finite type, finite presentation, or locally free of rank n if and only if G has the corresponding property.
Let g : S′ → S be faithfully flat and quasi-compact, and let X′, Y′, and f′ denote the base changes by g. Then for each of the following properties P, if f′ has P, then f has P.[46] Additionally, for each of the following properties P, f has P if and only if f′ has P.[47] It is possible for f′ to be a local isomorphism without f being even a local immersion.
[48] If f is quasi-compact and L is an invertible sheaf on X, then L is f-ample or f-very ample if and only if its pullback L′ is f′-ample or f′-very ample, respectively.
It is not even true that if f is proper and f′ is projective, then f is quasi-projective, because it is possible to have an f′-ample sheaf on X′ which does not descend to X.