Сақиналар теориясындағы алгебралық құрылымдар

Algebraic structure in ring theory
In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion free modules. Formally, a module M over a ring R is flat if 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 in his paper Géometrie Algébrique et Géométrie Analytique.

Flatness is related to various other module properties, such as being free, projective, or torsion free. In particular, every flat module is torsion free, every projective module is flat, and every free module is projective. There are finitely generated modules that are flat and not projective. However, finitely generated flat modules are all projective over the rings that are most commonly considered. Moreover, a finitely generated module is flat if and only it is locally free, meaning all the localizations at prime ideals are free modules. This is partly summarized in the following graphic.

Every flat module is torsion free. This results from the above characterization in terms of relations by taking 1=m = 1. The converse holds over the integers, and more generally over principal ideal domains and Dedekind rings. An integral domain over which every torsion free module is flat is called a Prüfer domain.

A module M is projective if and only if there is a free module G and two linear maps and such that In particular, every free module is projective (take and ). Every projective module is flat. This can be proven from the above characterizations of flatness and projectivity in terms of linear maps by taking and

Conversely, finitely generated flat modules are projective under mild conditions that are generally satisfied in commutative algebra and algebraic geometry. This makes the concept of flatness useful mainly for modules that are not finitely generated. A finitely presented module (that is the quotient of a finitely generated free module by a finitely generated submodule) that is flat is always projective. This can be proven by taking f surjective and in the above characterization of flatness in terms of linear maps. The condition implies the existence of a linear map such that and thus As f is surjective, one has thus and M is projective. Over a Noetherian ring, every finitely generated flat module is projective, since every finitely generated module is finitely presented. The same result is true over an integral domain, even if it is not Noetherian. On a local ring every finitely generated flat module is free. A finitely generated flat module that is not projective can be built as follows. Let be the set of the infinite sequences whose terms belong to a fixed field F. It is a commutative ring with addition and multiplication defined componentwise. This ring is absolutely flat (that is, every module is flat). The module where I is the ideal of the sequences with a finite number of nonzero terms, is thus flat and finitely generated (only one generator), but it is not projective.

If I is an ideal in a Noetherian commutative ring R, then is not a flat module, except if I is generated by an idempotent (that is an element equal to its square). In particular, if R is an integral domain, is flat only if equals R or is the zero ideal. Over an integral domain, a flat module is torsion free. Thus a module that contains nonzero torsion elements is not flat. In particular and all fields of positive characteristics are non flat modules, where is the ring of integers, and is the field of the rational numbers.

A direct sum of modules is flat if and only if each is flat. A direct limit of flat is flat. In particular, a direct limit of free modules is flat. Conversely, every flat module can be written as a direct limit of finitely generated free modules. Direct products of flat modules need not in general be flat. In fact, given a ring R, every direct product of flat R modules is flat if and only if R is a coherent ring (that is, every finitely generated ideal is finitely presented).

A ring homomorphism is flat if S is a flat R module for the module structure induced by the homomorphism. For example, the polynomial ring R[t] is flat over R, for any ring R.

For any multiplicative subset of a commutative ring , the localization is a flat R algebra (it is projective only in exceptional cases). For example, is flat and not projective over

If is an ideal of a Noetherian commutative ring the completion of with respect to is flat. It is faithfully flat if and only if is contained in the Jacobson radical of (See also Zariski ring.)

Here is one characterization of a faithfully flat homomorphism for a not necessarily flat homomorphism. Given an injective local homomorphism such that is an primary ideal, the homomorphism is faithfully flat if and only if the theorem of transition holds for it; that is, for each primary ideal of ,

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 automorphism. This flat cover conjecture was explicitly first stated in The conjecture turned out to be true, resolved positively and proved simultaneously by L. Bican, R. El Bashir and E. Enochs. 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 and in more recent works focussing on flat resolutions such as .

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.