In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in particular in the theory of scheme

Scheme (mathematics)

In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...

s in algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk

Stalk (sheaf)

The stalk of a sheaf is a mathematical construction capturing the behaviour of a sheaf around a given point.-Motivation and definition:Sheaves are defined on open sets, but the underlying topological space X consists of points. It is reasonable to attempt to isolate the behavior of a sheaf at a...

 is a flat map

Flat map

The flat map in differential geometry is a name for the mapping that converts coordinate basis vectors into corresponding coordinate 1-forms.A flat map in ring theory is a homomorphism f from a ring R to a ring S such that S is a flat R-module, where the action of R on S is given by f....

 of rings, i.e.,

f_P: O_Y,f(P) → O_X,P

is a flat map for all P in X. 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

Generic property

In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic...

, and that the failure of flatness occurs on the jumping set of the morphism.

The first of these comes from commutative algebra

Commutative algebra

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on 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

Generic flatness

In algebraic geometry and commutative algebra, the theorems of generic flatness and generic freeness state that under certain hypotheses, a sheaf of modules on a scheme is flat or free...

). Here 'restriction' is interpreted by means of fiber product, applied to f and the inclusion map

Inclusion map

In mathematics, if A is a subset of B, then the inclusion map is the function i that sends each element, x of A to x, treated as an element of B:i: A\rightarrow B, \qquad i=x....

 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

Blowing down

In mathematics, blowing down is a type of geometric modification in algebraic geometry. It is the inverse operation of blowing up.On an algebraic surface, blowing down a curve lying on the surface is a typical effect of a birational transformation...

 in the birational geometry

Birational geometry

In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian...

 of an algebraic surface

Algebraic surface

In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...

, 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

Étale morphism

In algebraic geometry, a field of mathematics, an étale morphism is an algebraic analogue of the notion of a local isomorphism in the complex analytic topology. They satisfy the hypotheses of the implicit function theorem, but because open sets in the Zariski topology are so large, they are not...

 (and so étale cohomology

Étale cohomology

In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...

) depends on the flat morphism concept: an étale morphism being flat, of finite type, and unramified

Ramification

In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign...

.

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 f′^* is an exact functor from the category of quasi-coherent -modules to the category of quasi-coherent -modules.

Assume that and 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 , then the fiber product 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 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 , the pullback of the inclusion map, is an injection, and the image of f^*J in is the annihilator of f^*F on X.
• If f is faithfully flat and if G is a quasi-coherent -module, then the pullback map on global sections is injective.

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

• 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 that 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

If is flat, then it possesses all of the following 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(Y′) dominates Y.
• If Z and Z′ are two irreducible components 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, .

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 X_red to X, then f is a universal homeomorphism, but for X noetherian, f is never flat.

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(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 S_k (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 .

• 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
  Glossary of scheme theory
  This is a glossary of scheme theory. For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme. The concern here is to list the fundamental technical definitions and properties of scheme theory...
  
  , and Y is regular
  Glossary of scheme theory
  This is a glossary of scheme theory. For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme. The concern here is to list the fundamental technical definitions and properties of scheme theory...
  
  , then f is flat.
• If f is faithfully flat, then for each closed subset Z of Y, .
• 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 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 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, f has P if and only if f′ has P.

• Open.
• Universally open.
• Closed.
• Universally closed.
• A homeomorphism.
• A universal homeomorphism.
• Quasi-compact.
• Quasi-compact and universally bicontinuous.
• Quasi-compact and a homeomorphism onto its image.
• Quasi-compact and dominant.
• 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.