Flat morphism
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 → 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:
- flatness is a generic property; and
- the failure of flatness occurs on the jumping set of the morphism.
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 that flatness in morphisms is directly related to controlling this sort of semicontinuity, or one-sided jumping.
Flat morphisms are used to define 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 depends on the flat morphism concept: an étale morphism being flat, of finite type, and unramified.
Examples/non-examples
Consider the affine schemeinduced from the obvious 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 is a non-zero divisor we have a trivial kernel, hence the homology group vanishes.
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.
A simple non-example of a flat morphism is This is because if we compute we have to take a flat resolution of,
and tensor the resolution with, 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. f is flat if and only if for every g, the pullback 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 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
- The composite of two flat morphisms is flat.
- The fiber product of two flat or faithfully flat morphisms is a flat or faithfully flat morphism, respectively.
- Flatness and faithful flatness is preserved by base change: If f is flat or faithfully flat and, then the fiber product is flat or faithfully flat, respectively.
- The set of points where a morphism is flat is open.
- If f is faithfully flat and of finite presentation, and if gf is finite type or finite presentation, then g is of finite type or finite presentation, respectively.
- If F is a quasi-coherent sheaf of finite presentation on Y, and if J is the annihilator of F on Y, then, the pullback of the inclusion map, is an injection, and the image of in is the annihilator of on X.
- If f is faithfully flat and if G is a quasi-coherent -module, then the pullback map on global sections is injective.
- If is quasi-compact and dominant, then its base change is quasi-compact and dominant.
- If h is faithfully flat, then the pullback map is injective.
- Assume is quasi-compact and quasi-separated. Let Z be the closed image of X, and let be the canonical injection. Then the closed subscheme determined by the base change is the closed image of X′.
Topological properties
- For every point x of X and every generization y′ of, there is a generization x′ of x such that.
- For every point x of X,.
- For every irreducible closed subset Y′ of Y, every irreducible component of f−1 dominates Y′.
- If Z and Z′ are two irreducible closed subsets of Y with Z contained in Z′, then for every irreducible component T of f−1, there is an irreducible component T′ of f−1 containing T.
- For every irreducible component T of X, the closure of f is an irreducible component of Y.
- If Y is irreducible with generic point y, and if f−1 is irreducible, then X is irreducible.
- If f is also closed, the image of every connected component of X is a connected component of Y.
- For every pro-constructible subset Z of Y,.
If is faithfully flat, then:
- The topology on Y is the quotient topology relative to f.
- If f is also quasi-compact, and if Z is a subset of Y, then Z is a locally closed pro-constructible subset of Y if and only if f−1 is a locally closed pro-constructible subset of X.
- Serre's condition Sk.
- Geometrically regular.
- Geometrically normal.
- Geometrically reduced.
- Geometrically reduced and having k geometric connected components.
- Geometrically integral.
Flatness and dimension
- Let x be a point of X and. If f is flat, then. Conversely, if this equality holds for all x, X is Cohen–Macaulay, and Y is regular, then f is flat.
- If f is faithfully flat, then for each closed subset Z of Y,.
- Suppose f is flat and F is a quasi-coherent module over Y. If F has projective dimension at most n, then has projective dimension at most n.
Descent properties
- Assume f is flat at x in X. If X is reduced or normal at x, then Y is reduced or normal, respectively, at f. Conversely, if f is also of finite presentation and f−1 is reduced or normal, respectively, at x, then X is reduced or normal, respectively, at x.
- In particular, if f is faithfully flat, then X reduced or normal implies that Y is reduced or normal, respectively. If f is faithfully flat and of finite presentation, then all the fibers of f reduced or normal implies that X is reduced or normal, respectively.
- If f is flat at x in X, and if X is integral or integrally closed at x, then Y is integral or integrally closed, respectively, at f.
- If f is faithfully flat, X is locally integral, and the topological space of Y is locally noetherian, then Y is locally integral.
- If f is faithfully flat and quasi-compact, and if X is locally noetherian, then Y is also locally noetherian.
- Assume f is flat and X and Y are locally noetherian. If X is regular at x, then Y is regular at f. Conversely, if Y is regular at f and f−1 is regular at x, then X is regular at x.
- Assume f is flat and X and Y are locally noetherian. If X is normal at x, then Y is normal at f. Conversely, if Y is normal at f and f−1 is normal at x, then X is normal at x.
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 is an S-morphism of S-schemes. Let 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.
- Open.
- Closed.
- Quasi-compact and a homeomorphism onto its image.
- A homeomorphism.
- Universally open.
- Universally closed.
- A universal homeomorphism.
- Quasi-compact.
- Quasi-compact and dominant.
- Quasi-compact and universally bicontinuous.
- Separated.
- Quasi-separated.
- Locally of finite type.
- Locally of finite presentation.
- Finite type.
- Finite presentation.
- Proper.
- An isomorphism.
- A monomorphism.
- An open immersion.
- A quasi-compact immersion.
- A closed immersion.
- Affine.
- Quasi-affine.
- Finite.
- Quasi-finite.
- Integral.
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.