flat module

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.

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. \sum_{i=1}^m r_i x_i = 0 with r_i \in R and x_i \in M , there exist elements y_j\in M and a_{i,j}\in R, such that \sum_{i=1}^m r_ia_{i,j}=0\qquad for j=1, \ldots, n , and x_i=\sum_{j=1}^n a_{i,j} y_j\qquad for i=1, \ldots, m. It is equivalent to define elements of a module, and a linear map from R^n to this module, which maps the standard basis of R^n to the elements. This allows rewriting the previous characterization in terms of homomorphisms, as follows. An -module is flat if and only if the following condition holds: for every map f : F \to M, where F is a finitely generated free -module, and for every finitely generated -submodule K of \ker f, the map f factors through a map to a free -module G such that g(K)=0: A module is projective if and only if there is a free module and two linear maps i:M\to G and p:G\to M such that p\circ i = \mathrm{id}_M. In particular, every free module is projective (take G=M 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 g=i\circ f and h=p. 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 surjective and K=\ker f in the above characterization of flatness in terms of linear maps. The condition g(K)=0 implies the existence of a linear map i:M\to G such that i\circ f = g, and thus h\circ i \circ f =h\circ g = f. As is surjective, one has thus h\circ i=\mathrm{id}_M, and 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 R=F^\mathbb N be the set of the infinite sequences whose terms belong to a fixed field . It is a commutative ring with addition and multiplication defined componentwise. This ring is absolutely flat (that is, every module is flat). The module R/I, where 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. == Local property== In this section, denotes a commutative ring. If \mathfrak p is a prime ideal of , the localization at \mathfrak p is, as usual, denoted with \mathfrak p as an index. That is, R_{\mathfrak p} = (R\setminus \mathfrak p)^{-1}R, and, if is an -module, M_{\mathfrak p} = (R\setminus \mathfrak p)^{-1}M = R_{\mathfrak p}\otimes_R M. If is an -module the three following conditions are equivalent: M is a flat R -module; M_\mathfrak p is a flat R_\mathfrak p -module for every prime ideal \mathfrak p; M_\mathfrak m is a flat R_\mathfrak m -module for every maximal ideal \mathfrak m. This property is fundamental in commutative algebra and algebraic geometry, since it reduces the study of flatness to the case of local rings. They are often expressed by saying that flatness is a local property. For example, consider the flat \mathbb{C}[t] -algebra R = \mathbb{C}[t,x,y]/(xy-t) (see below). The inclusion \mathbb{C}[t] \hookrightarrow R induces the flat morphism \pi : \operatorname{Spec}(R) \to \operatorname{Spec}(\mathbb C[t]). The (geometric) fiber \pi^{-1}(t) is the curve given by the equation xy = t. (See also flat degeneration and deformation to normal cone.) Let S = R[x_1, \dots, x_r] be a polynomial ring over a commutative Noetherian ring R and f \in S a nonzerodivisor. Then S/fS is flat over R if and only if f is primitive (the coefficients generate the unit ideal). An example is \mathbb{C}[t,x,y]/(xy-t), which is flat (and even free) over \mathbb{C}[t] (see also below for the geometric meaning). Such flat extensions can be used to yield examples of flat modules that are not free and do not result from a localization. For every prime ideal \mathfrak{p} of R, there is a prime ideal \mathfrak{P} of S such that \mathfrak{p} = f^{-1}(\mathfrak P). In other words, the map f^*\colon \operatorname{Spec}(S) \to \operatorname{Spec}(R) induced by f on the spectra is surjective. f, is injective, and R is a pure subring of S; that is, M \to M \otimes_R S is injective for every R -module M . The second condition implies that a flat local homomorphism of local rings is faithfully flat. It follows from the last condition that I = I S \cap R for every ideal I of R (take M = R/I ). In particular, if S is a Noetherian ring, then R is also Noetherian. The second-last condition can be stated in the following strengthened form: \operatorname{Spec}(S) \to \operatorname{Spec}(R) is submersive, which means that the Zariski topology of \operatorname{Spec}(R) is the quotient topology of that of \operatorname{Spec}(S) (this is a special case of the fact that a faithfully flat quasi-compact morphism of schemes has this property. ). See also . 0 \to A \overset{f}\to B \overset{\delta^0}\to B \otimes_A B \overset{\delta^1}\to B \otimes_A B \otimes_A B \to \cdots where the coboundary operators \delta^n are the alternating sums of the maps obtained by inserting 1 in each spot; e.g., \delta^0(b) = b \otimes 1-1 \otimes b . Then (Grothendieck) this complex is exact if f is faithfully flat. Here is one characterization of a faithfully flat homomorphism for a not-necessarily-flat homomorphism. Given an injective local homomorphism (R, \mathfrak m) \hookrightarrow (S, \mathfrak n) such that \mathfrak{m} S is an \mathfrak{n} -primary ideal, the homomorphism S \to B is faithfully flat if and only if the theorem of transition holds for it; that is, for each \mathfrak m -primary ideal \mathfrak q of R , \operatorname{length}_S (S/ \mathfrak q S) = \operatorname{length}_S (S/ \mathfrak{m} S) \operatorname{length}_R(R/\mathfrak q). \operatorname{Tor}_n^R (X, M) = 0 for all n \ge 1 and all right R -modules X ). In fact, it is enough to check that the first Tor term vanishes, i.e., M is flat if and only if \operatorname{Tor}_1^R (N, M) = 0 for any R -module N or, even more restrictively, when N=R/I and I\subset R is any finitely generated ideal. Using the Tor functor's long exact sequences, one can then easily prove facts about a short exact sequence 0 \to A \overset{f}{\longrightarrow} B \overset{g}{\longrightarrow} C \to 0 If A and C are flat, then so is B . Also, if B and C are flat, then so is A . If A and B are flat, C need not be flat in general. However, if A is pure in B and B is flat, then A and C are flat. 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 and denoted \operatorname{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 \operatorname{fd}(M)=0 . In this situation, the exactness of the sequence 0 \to F_0 \to M \to 0 indicates that the arrow in the center is an isomorphism, and hence M itself is flat. 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.

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