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Flat Modules We Recall Here Some Properties of Flat Modules As Flat modules We recall here some properties of flat modules as exposed in Bourbaki, Alg´ebreCommutative, ch. 1. The aim of these pages is to expose the proofs of some of the characterizations of flat and faithfully flat modules given in Matsumura's book Commutative Algebra. This part is mainly devoted to the exposition of a proof of the following Theorem. Let A be a commutative ring with identity and E an A-module. The following conditions are equivalent α β (a) for any exact sequence of A-modules 0 ! M 0 / M / M 00 ! 0 , the sequence 0 1E ⊗α 1E ⊗β 00 0 ! E ⊗A M / E ⊗A M / E ⊗A M ! 0 is exact; ∼ (b) for any finitely generated ideal I of A, the sequence 0 ! E ⊗A I ! E ⊗A A = E is exact; 0 b1 1 0 x1 1 t Pr . r . r (c) If bx = i=1 bixi = 0 for some b = @ . A 2 A and x = @ . A 2 E , there exist a matrix br xr s t A 2 Mr×s(A) and y 2 E such that x = Ay and bA = 0. α β We recall that, for any A-module, N, and for any exact sequence 0 ! M 0 / M / M 00 ! 0 , one gets the exact sequence 0 1N ⊗α 1N ⊗β 00 N ⊗A M / N ⊗A M / N ⊗A M ! 0 : Moreover, if αM 0 is a direct summand in M (which is equivalent to the existence of a morphism πM ! M 0 0 1N ⊗α such that πα = 1M 0 ), then the morphism N ⊗A M / N ⊗A M is injective and the image of 1N ⊗ α is a direct summand in N ⊗A M. Definition. Let A be a commutative ring with identity and E and M two A-modules. We say that E is M-flat if, for any A-module, M 0, and for any injective homomorphism α : M 0 ! M, the homomorphism 0 1E ⊗ α : E ⊗A M ! E ⊗A M is injective. Keeping the notations above, we remark that E is M-flat if the above condition holds for any finitely 0 0 Pr generated A-module M . In fact, for a general A-module, M , and an element z = i=1 ei ⊗yi in ker(1E ⊗α), 0 0 0 one can take the submodule, N , generated by y1; : : : ; yr and the natural homomorphism E⊗AN ! E⊗AM . 0 0 If z = 0 in E ⊗A N , the same holds for its image in E ⊗A M . Lemma 1. Let M and E be A-modules and suppose E to be M-flat. (a) For any submodule, N, of M, E is N-flat. (b) For any quotient module, Q, of M, E is Q-flat. Proof. (a) Let j : N ! M be the natural inclusion map. Starting with the diagram, whose rows and columns are exact, 0 0 0 α 0 1E ⊗α 0 / N / N E ⊗A N / E ⊗A N and tensoring by E, one gets j 1E ⊗j 0 0 / N / M 0 j◦α 0 / E ⊗A N / E ⊗A M 1E ⊗(j◦α) and rows and columns are again exact, because E is M-flat. This implies the injectivity of 1E ⊗ α. 1 (b) Let p : M ! Q be the projection map and let i : K ! M the inclusion of its kernel. Given an injection β : Q0 ! Q, let M 0 = p−1(imβ) and consider the following diagram, whose rows and columns are exact. 0 0 0 K K E ⊗A K E ⊗A K 0 0 i i 1E ⊗i 1E ⊗i j Tensoring by E, one gets 0 0 1E ⊗j 0 / M / M 0 / E ⊗A M / E ⊗A M 0 p p 0 1E ⊗p 1E ⊗p 0 0 / Q / Q 0 β E ⊗A Q / E ⊗A Q 1E ⊗β 0 0 0 0 whose rows and columns are again exact, because E is M-flat. Using the snake lemma, or by a direct check on elements, one deduces that ker(1E ⊗ β) = 0. Lemma 2. Let (Mi)i2I a family of A modules and E be an A-module Mi-flat for each i 2 I. Let M = L i2I Mi, then E is M-flat. i p Proof. Suppose I = f1; 2g and consider the sequence 0 ! M1 / M / M2 ! 0 where M = M1 ⊕M2. 0 0 0 0 0 let M be a submodule of M and set M1 = M \ M1 and M2 = p(M ). In this way one gets the commutative diagram, with exact rows and columns, 0 0 0 0 v1 0 1E ⊗v1 0 / M1 / M1 0 / E ⊗A M1 / E ⊗A M1 0 0 i i 1E ⊗i 1E ⊗i Tensoring by E, one gets 0 v 0 1E ⊗v 0 / M / M E ⊗A M / E ⊗A M 0 p 0 p 1E ⊗p 1E ⊗p 0 0 0 / M / M2 2 v2 0 / E ⊗A M2 / E ⊗A M2 1E ⊗v2 0 0 0 0 whose rows and columns are exact. This allows to conclude that 1E ⊗ v is an injective homomorphism. By induction one can conclude that the claim is true for any finite sum of modules. In order to conclude in the general case, we observe that even if M is a direct sum of a general family of modules, it suffices to verify the condition for a finitely generated submodule M 0 of M and actually M 0 is contained in a finite sum of modules of the family. Now we can prove a characterization of flat modules (one can take as the definition of flatness, that an A module E is flat if, and only if, it satisfies one of the equivalent conditions in the following proposition). 2 Proposition. Let E be an A module. The following facts are equivalent. (a) E is A-flat (i.e. for any ideal I of A, the map I ⊗A E ! A ⊗A E is injective); (b) For any A-module, M, E is M-flat; α β (c) For any exact sequence of A-modules M 0 / M / M 00 , the sequence 0 α⊗1E β⊗1E 00 M ⊗A E / M ⊗A E / M ⊗A E is exact. Proof. (a) ) (b); Any A-module, M, is isomorphic to a quotient of a free A-module. By applying Lemma 1 and Lemma 2, one deduces from (a) that E is M-flat. (c) ) (a); is straightforward. (b) ) (c); is a consequence from the following lemma. α β Lemma 3. Let M 0 / M / M 00 be an exact sequence of A-modules and E be an A module, M 00-flat. Then the sequence 0 α⊗1E β⊗1E 00 M ⊗A E / M ⊗A E / M ⊗A E is exact. α β Proof. Let N 00 = imβ ⊆ M 00 from the exact sequence M 0 / M / N 00 ! 0 one deduces the exactness of 0 α⊗1E β⊗1E 00 M ⊗A E / M ⊗A E / N ⊗A E ! 0 for general properties of tensor product. The fact that E is M 00-flat, implies the injectivity of the map 00 00 N ⊗A E ! M ⊗A E induced by the inclusion. This means precisely that the sequence in the above proclaim is exact. t Pr Corollary. An A-module, E, is flat if, and only if, the following condition holds. If bx = i=1 bixi = 0 for 0 b1 1 0 x1 1 . r . r s some b = @ . A 2 A and x = @ . A 2 E , then there exist a matrix A 2 Mr×s(A) and y 2 E such that br xr x = Ay and tbA = 0. Pr Proof. Let I be the ideal of A generated by b1; : : : ; br. An element, η = i=1 bi ⊗ xi, in the kernel of 0 x1 1 . r t Pr I ⊗A E ! A ⊗A E = E determines an r-uple x = @ . A 2 E such that bx = i=1 bixi = 0. The existence xr s t of a matrix A and y 2 E such that x = Ay and bA = 0, means exactly that η = 0 in I ⊗A E; hence E satisfies the condition (a) of the above Proposition. 0 b1 1 t Pr . r Now suppose that E is a flat module and bx = i=1 bixi = 0 for some b = @ . A 2 A and x = br 0 x1 1 . r r @ . A 2 E . The map A ! A which sends u1; : : : ; ur to b1u1 + ··· + brur, has a kernel, K, and we can xr write the exact sequence K ! Ar ! A, which remains exact after tensoring by E. This means that the Ps r r-uple x = (x1; : : : ; xr) is in the image of K ⊗A E hence x = j=1 αj ⊗ yj for some αj 2 K ⊂ A and Ps Pr yj 2 E. Writing αj = (a1;j; : : : ; ar;j) one has xi = j=1 ai;jyj for i = 1; : : : ; r, and i=1 biai;j = 0 for j = 1; : : : ; s. 3 We have given a complete proof of the Theorem stated at the beginning of the section. We can observe here that if φ : M ! N is a morphism of A-modules, then for any A-module E, one has coker(1E ⊗ φ) = E ⊗A cokerφ, because the sequence M ! N ! cokerφ ! 0 remains exact after tensoring by E. If E is a flat module, even the sequence 0 ! kerφ ! M ! N remains exact after tensoring by E; then ker(1E ⊗ φ) = E ⊗A kerφ.
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