Commutative Algebra/Torsion-free, flat, projective and free modules

Free modules The following definitions are straightforward generalisations from linear algebra. We begin by repeating a definition we already saw in chapter 6. 0 100% #FFFFFF Definition 6.1 (generators of modules): Let be a module over the ring . A generating set of is a subset such that . We also have: 0 100% #FFFFFF Definition 11.1: Let be an -module. A subset of is called linearly independent if and only if, whenever , we have . 0 100% #FFFFFF Definition 11.2: A free -module is a module over where there exists a basis, that is, a subset of that is a linearly independent generating set. Theorem 11.3: Let be free modules. Then the direct sum is free. Proof: Let bases of the be given. We claim that is a basis of . Indeed, let an arbitrary element be given. Then by assumption, each of the has a decomposition for suitable . By summing this, we get a decomposition of in the aforementioned basis. Furthermore, this decomposition must be unique, for otherwise projecting gives a new composition of one of the particular . The converse is not true in general! Theorem 11.4: Let be free -modules, with bases and respectively. Then is a free module, with basis , where we wrote for short (note that it is quite customary to use this notation). Proof: We first prove that our supposed basis forms a generating system. Clearly, by summation it suffices to show that elements of the form , can be written in terms of the . Thus, write and , and obtain by the rules of computing within the tensor product, that . On the other hand, if is a linear combination (i.e. all but finitely many summands are zero), then all the must be zero. The argument is this: Fix and define a bilinear function , where , are the coefficients of , in the decomposition of and respectively. According to the universal property of the tensor product, we obtain a linear map with , where is the canonical projection on the quotient space. We have the equations , and inserting the given linear combination into this map therefore yields the desired result. Projective modules The following is a generalisation of free modules: 0 100% #FFFFFF Definition 11.5: Let be an -module. is called projective if and only if for a fixed module and a fixed surjection every other module morphism with codomain (call ) has a factorisation 200px. Theorem 11.6: Every free module is projective. Proof: Pick a basis of , let be surjective and let be some morphism. For each pick with . Define where . This is well-defined since the linear combination describing is unique. Furthermore, it is linear, since we have , where the right hand side is the sum of the linear combinations coinciding with and respectively, which is why . By linearity of and definition of the , it has the desired property. There are a couple equivalent definitions of projective modules. Theorem 11.7: A module is projective if and only if there exists a module such that is free. Proof: : Define the module (this obviously is a free module) and the function . is a surjective morphism, whence we obtain a commutative diagram 130px; that is, . We claim that the map is an isomorphism. Indeed, if , then and thus also (injectivity) and further , where , which is why (surjectivity). : Assume is a free module. Assume is a surjective morphism, and let be any morphism. We extend to via . This is still linear as the composition of the linear map and the linear inclusion . Now is projective since it's free. Hence, we get a commutative diagram 150px where satisfies . Projecting to gives the desired diagram for . Definition 11.8: An exact sequence of modules is called split exact iff we can augment it by three isomorphisms such that 400px commutes. Theorem 11.9: A module is projective iff every exact sequence is split exact. Proof: : The morphism is surjective, and thus every other morphism with codomain lifts to . In particular, so does the projection . Thus, we obtain a commutative diagram 400px where we don't know yet whether is an isomorphism, but we can use to define the function , which is an isomorphism due to injectivity: Let , that is . Then first and therefore second . And surjectivity: Let . Set . Then and hence for a suitable , thus . We thus obtain the commutative diagram 400px and have proven what we wanted. : We prove that is free for a suitable . We set , where is defined as in the proof of theorem 11.7 . We obtain an exact sequence which by assumption splits as 400px which is why is isomorphic to the free module and hence itself free. Theorem 11.10: Let and be projective -modules. Then is projective. Proof: We choose -modules such that and are free. Since the tensor product of free modules is free, is free. But , and thus occurs as the summand of a free module and is thus projective. Theorem 11.11: Let be -modules. Then is projective if and only if each is projective. Proof: Let first each of the be projective. Then each of the occurs as the direct summand of a free module, and summing all these free modules proves that is the direct summand of free modules. On the other hand, if is the summand of a free module, then so are all the s. Flat modules The following is a generalisation of projective modules: 0 100% #FFFFFF Definition 11.12: An -module is called flat if and only if tensoring by it preserves exactness: exact implies exact. The morphisms in the right sequence induced by any morphism are given by the bilinear map . Theorem 11.13: The module is a flat -module. Proof: This follows from theorems 9.10 and 10.?. Theorem 11.14: Flatness is a local property. Proof: Exactness is a local property. Furthermore, for any multiplicatively closed by theorem 9.11. Since every -module is the localisation of an -module (for instance itself as an -module via ), the theorem follows. Theorem 11.15: A projective module is flat. Proof: We first prove that every free module is flat. This will enable us to prove that every projective module is flat. Indeed, if is a free module and a basis of , we have via , where all but finitely many of the summands on the left are nonzero. Hence, by distributivity of direct sum over tensor product, if we are given any exact sequence , to show that the sequence is exact, all we have to do is to prove that is exact, since we may then augment the latter sequence by suitable isomorphisms Theorem 11.16: direct sum flat iff all summands are Theorem 11.17: If are flat -modules, then is as well. Proof: Let be an exact sequence of modules. Torsion-free modules The following is a generalisation of flat modules: 0 100% #FFFFFF Definition 11.18: Let be an -module. The torsion of is defined to be the set . Lemma 11.19: The torsion of a module is a submodule of that module. Proof: Let , . Obviously (just multiply the two annihilating elements together), and further if (we used commutativity here). We may now define torsion-free modules. They are exactly what you think they are. 0 100% #FFFFFF Definition 11.20: Let be a module. is called torsion-free if and only if . Theorem 11.21: A flat module is torsion-free. To get a feeling for the theory, we define -torsion for a multiplicatively closed subset . Definition 11.22: Let be a multiplicatively closed subset of a ring , and let be an -module. Then the -torsion of is defined to be . Theorem 11.23: Let be a multiplicatively closed subset of a ring , and let be an -module. Then the -torsion of is precisely the kernel of the canonical map .