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. A map of rings A\to B is called flat if it is a homomorphism that makes B a flat A-module. A morphism of schemes is called faithfully flat if it is both surjective and flat. 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 Y' of Y, such that f restricted to Y′ is a flat morphism (generic flatness). Here 'restriction' is interpreted by means of the fiber product of schemes, applied to f and the inclusion map of Y' into Y. 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. Flat morphisms are used to define (more than one version of) the flat topos, and flat cohomology of sheaves from it. 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.

Examples/non-examples

Consider the affine scheme morphism induced from the morphism of algebras Since proving flatness for this morphism amounts to computing 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.

Miracle flatness

Other examples of flat morphisms can be found using "miracle flatness" which states that if you have a morphism between a Cohen–Macaulay scheme to a regular scheme with equidimensional fibers, then it is flat. Easy examples of this are elliptic fibrations, smooth morphisms, and morphisms to stratified varieties which satisfy miracle flatness on each of the strata.

Hilbert schemes

The universal examples of flat morphisms of schemes are given by Hilbert schemes. This is because Hilbert schemes parameterize universal classes of flat morphisms, and every flat morphism is the pullback from some Hilbert scheme. I.e., if is flat, there exists a commutative diagram for the Hilbert scheme of all flat morphisms to S. Since f is flat, the fibers all have the same Hilbert polynomial \Phi, hence we could have similarly written for the Hilbert scheme above.

Non-examples

Blowup

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 is a copy of \Complex, i.e., which follows from But for a=b=0, we get the isomorphism The reason this fails to be flat is because of the Miracle flatness lemma, which can be checked locally.

Infinite resolution

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.

Properties of flat morphisms

Let be a morphism of schemes. For a morphism, let and The morphism f is flat if and only if for every g, the pullback f'^* is an exact functor from the category of quasi-coherent -modules to the category of quasi-coherent -modules. Assume and are morphisms of schemes and f is flat at x in X. Then g is flat at f(x) if and only if gf is flat at x. In particular, if f is faithfully flat, then g is flat or faithfully flat if and only if gf is flat or faithfully flat, respectively.

Fundamental properties

Suppose is a flat morphism of schemes. Suppose is flat. Let X and Y be S-schemes, and let X' and Y' be their base change by h.

Topological properties

If is flat, then it possesses all of the following properties: If f is flat and locally of finite presentation, then f is universally open. 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. 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. If 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: If in addition f is proper, then the same is true for each of the following properties:

Flatness and dimension

Assume X and Y are locally noetherian, and let.

Descent properties

Let g : Y′ → Y be faithfully flat. Let F be a quasi-coherent sheaf on Y, and let F′ be the pullback of F to Y′. Then F is flat over Y if and only if F′ is flat over Y′. 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. Suppose f : X → Y is an S-morphism of S-schemes. 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. Additionally, for each of the following properties P, f has P if and only if f′ has P. It is possible for f′ to be a local isomorphism without f being even a local immersion. 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. However, it is not true that f is projective if and only if f′ is projective. 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.