“Flat module”的意思、由来-开放百科全书

In homological algebra and algebraic geometry, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact.

Flatness was introduced by {{harvs|txt|authorlink=Jean-Pierre Serre|last=Serre|year=1956}} in his paper Géometrie Algébrique et Géométrie Analytique. See also flat morphism.

Definition

Commutative rings

Let M be an R-module. The following conditions are all equivalent, so M is flat if it satisfies any (thus all) of them:

General rings

When R isn't commutative one needs the more careful statement that, if M is a flat left R-module, the tensor product with M maps exact sequences of right R-modules to exact sequences of abelian groups.

Taking tensor products (over arbitrary rings) is always a right exact functor. Therefore, the R-module M is flat if and only if for any injective homomorphism K → L of R-modules, the induced homomorphism

is also injective.

Examples

Case of commutative rings

Let R be a commutative ring. When M is a finitely-generated R-module, being flat is the same as being locally free in the following sense: M is a flat R-module if and only if for every prime ideal (or even just for every maximal ideal) P of R, the localization

is free as a module over the localization

^[2]

Let R be a local ring with nilpotent maximal ideal (e.g., an artinian local ring) and M a module over it. Then M flat implies M free.^[2]

The significance of this is that S need not be finite over R and we only need to consider the maximal ideal of R instead of an arbitrary ideal of R.

Flat modules over commutative rings are always torsion-free (that is, the multiplication by any regular element of the ring in injective in the module). Projective modules (and thus free modules) are always flat. For certain common classes of rings, these statements can be reversed (for example, every torsion-free module over a Dedekind ring is automatically flat and flat modules over perfect rings are always projective), as is subsumed in the following diagram of module properties:

An integral domain is called a Prüfer domain if every torsion-free module over it is flat.

Faithfully flat ring homomorphism

Let A be a ring (assumed to be commutative throughout this section) and B an A-algebra, i.e., a ring homomorphism

. Then B has the structure of an A-module. Then B is said to be flat over A (resp. faithfully flat over A) if it is flat (resp. faithfully flat) as an A-module.

There is a basic characterization of a faithfully flat ring homomorphism: given a flat ring homomorphism

, the following are equivalent.

Condition 2 implies a flat local homomorphism between local rings is faithfully flat. It follows from condition 5 that

for every ideal

(take

); in particular, if

is a Noetherian ring, then

is a Noetherian ring.

Condition 4 can be stated in the following strengthened form:

is submersive: the topology of

is the quotient topology of

(this is a special case of the fact that a faithfully flat quasi-compact morphism of schemes has this property.^[10]) It compares to an integral extension of an integrally closed domain. See also flat morphism#Properties of flat morphisms for further information.

Example. For a ring

is faithfully flat. More generally, an

-algebra that is free as an

-module is faithfully flat.

is faithfully flat since localizations are flat, their direct sums are then flat and

For a given ring homomorphism

there is an associated complex called the Amitsur complex:^[11]

where the coboundary operators

are the alternating sums of the maps obtained by inserting 1 in each spot; e.g.,

(and f is viewed as the augumentation). Then (Grothendieck) this complex is exact if

is faithfully flat.

Categorical colimits

In general, arbitrary direct sums and direct limits of flat modules are flat, a consequence of the fact that the tensor product commutes with direct sums and direct limits (in fact with all colimits), and that both direct sums and direct limits are exact functors. Submodules and factor modules of flat modules need not be flat in general (e.g.

is not a flat

-module for

). However we have the following result: the homomorphic image of a flat module M is flat if and only if the kernel is a pure submodule of M.

Daniel Lazard proved in 1969 that a module M is flat if and only if it is a direct limit of finitely-generated free modules.^[12] As a consequence, one can deduce that every finitely-presented flat module is projective.

An abelian group is flat (viewed as a

-module) if and only if it is torsion-free.

Homological algebra

Flatness may also be expressed using the Tor functors, the left derived functors of the tensor product. A left R-module M is flat if and only if

for all

(i.e., if and only if

for all

and all right R-modules X). Similarly, a right R-module M is flat if and only if

for all

and all left R-modules X. Using the Tor functor's long exact sequences, one can then easily prove facts about a short exact sequence

If A and B are flat, C need not be flat in general. However, it can be shown that:

Flat resolutions

where the F_i are all flat modules. Any free or projective resolution is necessarily a flat resolution. Flat resolutions can be used to compute the Tor functor.

The length of a finite flat resolution is the first subscript n such that F_n is nonzero and F_i = 0 for i > n. If a module M admits a finite flat resolution, the minimal length among all finite flat resolutions of M is called its flat dimension{{sfn|Lam|1999|loc=p. 183}} and denoted fd(M). If M does not admit a finite flat resolution, then by convention the flat dimension is said to be infinite. As an example, consider a module M such that fd(M) = 0. In this situation, the exactness of the sequence 0 → F₀ → M → 0 indicates that the arrow in the center is an isomorphism, and hence M itself is flat.^[13]

In some areas of module theory, a flat resolution must satisfy the additional requirement that each map is a flat pre-cover of the kernel of the map to the right. For projective resolutions, this condition is almost invisible: a projective pre-cover is simply an epimorphism from a projective module. These ideas are inspired from Auslander's work in approximations. These ideas are also familiar from the more common notion of minimal projective resolutions, where each map is required to be a projective cover of the kernel of the map to the right. However, projective covers need not exist in general, so minimal projective resolutions are only of limited use over rings like the integers.

Flat covers

While projective covers for modules do not always exist, it was speculated that for general rings, every module would have a flat cover, that is, every module M would be the epimorphic image of a flat module F such that every map from a flat module onto M factors through F, and any endomorphism of F over M is an automoprhism. This flat cover conjecture was explicitly first stated in {{harv|Enochs|1981|loc=p 196}}. The conjecture turned out to be true, resolved positively and proved simultaneously by L. Bican, R. El Bashir and E. Enochs.{{sfn|Bican|El Bashir|Enochs|2001}} This was preceded by important contributions by P. Eklof, J. Trlifaj and J. Xu.

Since flat covers exist for all modules over all rings, minimal flat resolutions can take the place of minimal projective resolutions in many circumstances. The measurement of the departure of flat resolutions from projective resolutions is called relative homological algebra, and is covered in classics such as {{harv|MacLane|1963}} and in more recent works focussing on flat resolutions such as {{harv|Enochs|Jenda|2000}}.

In constructive mathematics

Flat modules have increased importance in constructive mathematics, where projective modules are less useful. For example, that all free modules are projective is equivalent to the full axiom of choice, so theorems about projective modules, even if proved constructively, do not necessarily apply to free modules. In contrast, no choice is needed to prove that free modules are flat, so theorems about flat modules can still apply.{{sfn|Richman|1997}}

Definition

Commutative rings

General rings

Examples

Case of commutative rings

Faithfully flat ring homomorphism

Categorical colimits

Homological algebra

Flat resolutions

Flat covers

In constructive mathematics

See also

References