Puneet Varma (Editor)

Flat morphism

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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.,

Contents

fP: OY,f(P) → OX,P

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 f is a faithfully flat morphism if f is a surjective flat morphism.

Two of the basic intuitions are that flatness is a generic property, and that the failure of flatness occurs on the jumping set of the morphism.

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 fiber product, 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

Consider the affine scheme

Spec ( C [ x , y , t ] x 2 + y 2 − t ) ↓ Spec ( C [ t ] )

induced from the obvious morphism of algebras

C [ t ] → C [ x , y , t ] x 2 + y 2 − t t ↦ t

Since proving flatness amounts to computing Tor 1 C [ t ] ( C [ x , y , t ] x 2 + y 2 − t , C ) we resolve the complex numbers

0 → C [ t ] → ⋅ t C [ t ] → 0 ↓ ↓ ↓ ↓ 0 → 0 → C → 0

and tensor by the module representing our scheme giving the sequence of C [ t ] -modules

0 → C [ x , y , t ] x 2 + y 2 − t → ⋅ t C [ x , y , t ] x 2 + y 2 − t → 0

Because t is a non-zero divisor we have a trivial kernel, hence the homology group vanishes.

Properties of flat morphisms

Let f : X → Y be a morphism of schemes. For a morphism g : Y′ → Y, let X′ = X ×Y Y′ and f′ = (f, 1Y′) : X′ → Y′. f is flat if and only if for every g, the pullback f′* is an exact functor from the category of quasi-coherent O Y ′ -modules to the category of quasi-coherent O X ′ -modules.

Assume that f : X → Y and g : Y → Z are morphisms of schemes. Assume furthermore that 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

  • The composite of two flat morphisms is flat.
  • The fibered 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 g : Y′ → Y, then the fiber product f × g : X ×Y Y′ → Y' is flat or faithfully flat, respectively.
  • The set of points where a morphism (locally of finite presentation) 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.

    • Suppose that f: X → Y is a flat morphism of schemes.

  • If F is a quasi-coherent sheaf of finite presentation on Y (in particular, if F is coherent), and if J is the annihilator of F on Y, then f ∗ J → O X , the pullback of the inclusion map, is an injection, and the image of f*J in O X is the annihilator of f*F on X.
  • If f is faithfully flat and if G is a quasi-coherent O Y -module, then the pullback map on global sections Γ ( Y , G ) → Γ ( X , f ∗ G ) is injective.

    • Suppose now that h : S′ → S is flat. Let X and Y be S-schemes, and let X′ and Y′ be their base change by h.

  • If f : X → Y is quasi-compact and dominant, then its base change f′ : X′ → Y′ is quasi-compact and dominant.
  • If h is faithfully flat, then the pullback map HomS(X, Y) → HomS′(X′, Y′) is injective.
  • Assume that f : X → Y is quasi-compact and quasi-separated. Let Z be the closed image of X, and let j : Z → Y be the canonical injection. Then the closed subscheme determined by the base change j′ : Z′ → Y′ is the closed image of X′.

    • Topological properties

    If f : X → Y is flat, then it possesses all of the following properties:

  • For every point x of X and every generization y′ of y = f(x), there is a generization x′ of x such that y′ = f(x′).
  • For every point x of X, f ( Spec O X , x ) = Spec O Y , f ( x ) .
  • For every irreducible closed subset Y′ of Y, every irreducible component of f−1(Y′) 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(Z), there is an irreducible component T′ of f−1(Z′) containing T.
  • For every irreducible component T of X, the closure of f(T) is an irreducible component of Y.
  • If Y is irreducible with generic point y, and if f−1(y) 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, f − 1 ( Z ¯ ) = f − 1 ( Z ) ¯ .

    • 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 noetherian, f is never flat.

    If f : X → Y 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(Z) is a locally closed pro-constructible subset of X.

    • 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:

  • Serre's condition Sk (for any fixed k).
  • Geometrically regular.
  • Geometrically normal.

    • If in addition f is proper, then the same is true for each of the following properties:

  • Geometrically reduced.
  • Geometrically reduced and having k geometric connected components (for any fixed k).
  • Geometrically integral.

    • Flatness and dimension

    Assume that X and Y are locally noetherian, and let f : X → Y.

  • Let x be a point of X and y = f(x). If f is flat, then dimx X = dimy Y + dimx f−1(y). 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, codimY(Z) = codimX(f−1(Z)).
  • Suppose that f is flat and that F is a quasi-coherent module over Y. If F has projective dimension at most n, then f*F 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(x). Conversely, if f is also of finite presentation and f−1(y) 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(x).
  • 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 that f is flat and X and Y are locally noetherian. If X is regular at x, then Y is regular at f(x). Conversely, if Y is regular at f(x) and f−1(f(x)) is regular at x, then X is regular at x.
  • Assume again that f is flat and X and Y are locally noetherian. If X is normal at x, then Y is normal at f(x). Conversely, if Y is normal at f(x) and f−1(f(x)) is normal at x, then X is normal at x.

    • 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 that 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 that 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.

  • Open.
  • Closed.
  • Quasi-compact and a homeomorphism onto its image.
  • A homeomorphism.

    • Additionally, for each of the following properties P, f has P if and only if f′ has P.

  • 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.

    • 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.

    References

    Flat morphism Wikipedia
    (Text) CC BY-SA