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A Few Classical Results on Tensor, Symmetric and Exterior Powers

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A Few Classical Results on Tensor, Symmetric and Exterior Powers A few classical results on tensor, symmetric and exterior powers Darij Grinberg Version 0.3 (June 13, 2017) (not proofread!) Contents 0.1. Version . 1 0.2. Introduction . 1 0.3. Basic conventions . 2 0.4. Tensor products . 2 0.5. Tensor powers of k-modules . 4 0.6. The tensor algebra . 5 0.7. A variation on the nine lemma . 7 0.8. Another diagram theorem about the nine lemma configuration . 8 0.9. Ker (f ⊗ g) when f and g are surjective . 9 0.10. Extension to n modules . 15 0.11. The tensor algebra case . 23 0.12. The pseudoexterior algebra . 29 0.13. The kernel of Exter f ............................ 48 0.14. The symmetric algebra . 53 0.15. The exterior algebra . 58 0.16. The relation between the exterior and pseudoexterior algebras . 74 0.17. The symmetric algebra is commutative . 77 0.18. Some universal properties . 80 0.1. Version ver:S 0.2. Introduction In this note, I am going to give proofs to a few results about tensor products as well as tensor, pseudoexterior, symmetric and exterior powers of k-modules (where k is a commutative ring with 1). None of the results is new, as I have seen them used all around literature as if they were well-known and/or completely trivial. I have not yet found a place where they are actually proved (though I have not looked far), so I am doing it here. 1 This note is not completely new: The first four Subsections (0.4, 0.5, 0.6 and 0.7) as well as the proof of Proposition 38 are lifted from my diploma thesis [3], while Subsections 0.8 and 0.9 are translated from an additional section of [4] which was written by me. 0.3. Basic conventions Before we come to the actual body of this note, let us fix some conventions to prevent misunderstandings from happening: Convention 1. In this note, N will mean the set f0; 1; 2; 3;:::g (rather than the set f1; 2; 3;:::g, which is denoted by N by various other authors). For each n 2 N, we let Sn denote the n-th symmetric group (defined as the group of all permutations of the set f1; 2; : : : ; ng). Convention 2. In this note, a ring will always mean an associative ring with 1. If k is a commutative ring, then a k-algebra will mean a (not necessarily commutative, but necessarily associative) k-algebra with 1. Sometimes we will use the word \alge- bra" as an abbreviation for \k-algebra". If L is a k-algebra, then a left L-module is always supposed to be a left L-module on which the unity of L acts as the identity. Whenever we use the tensor product sign ⊗ without an index, we mean ⊗k. 0.4. Tensor products The goal of this note is not to define tensor products; we assume that the reader already knows what they are. But let us recall one possible way to define the tensor product of several k-modules (assuming that the tensor product of two k-modules is already defined): Definition 3. Let k be a commutative ring. Let n 2 N. Now, by induction over n, we are going to define a k-module V1 ⊗ V2 ⊗ · · · ⊗ Vn for any n arbitrary k-modules V1, V2, :::, Vn: Induction base: For n = 0, we define V1 ⊗ V2 ⊗ · · · ⊗ Vn as the k-module k. Induction step: Let p 2 N. Assuming that we have defined a k-module V1 ⊗ V2 ⊗ · · · ⊗ Vp for any p arbitrary k-modules V1, V2, :::, Vp, we now define a k-module V1 ⊗ V2 ⊗ · · · ⊗ Vp+1 for any p + 1 arbitrary k-modules V1, V2, :::, Vp+1 by the equation V1 ⊗ V2 ⊗ · · · ⊗ Vp+1 = V1 ⊗ (V2 ⊗ V3 ⊗ · · · ⊗ Vp+1) : (1) Here, V1 ⊗ (V2 ⊗ V3 ⊗ · · · ⊗ Vp+1) is to be understood as the tensor product of the k-module V1 with the k-module V2 ⊗V3 ⊗· · ·⊗Vp+1 (note that the k-module V2 ⊗V3 ⊗ · · · ⊗ Vp+1 is already defined because we assumed that we have defined a k-module V1 ⊗ V2 ⊗ · · · ⊗ Vp for any p arbitrary k-modules V1, V2, :::, Vp). This completes the inductive definition. Thus we have defined a k-module V1 ⊗ V2 ⊗ · · · ⊗ Vn for any n arbitrary k-modules V1, V2, :::, Vn for any n 2 N. This k-module V1 ⊗ V2 ⊗ · · · ⊗ Vn is called the tensor product of the k-modules V1, V2, :::, Vn. 2 Remark 4. (a) Definition 3 is not the only possible definition of the tensor prod- uct of several k-modules. One could obtain a different definition by replacing the equation (1) by V1 ⊗ V2 ⊗ · · · ⊗ Vp+1 = (V1 ⊗ V2 ⊗ · · · ⊗ Vp) ⊗ Vp+1: This definition would have given us a different k-module V1 ⊗ V2 ⊗ · · · ⊗ Vn for any n arbitrary k-modules V1, V2, :::, Vn for any n 2 N than the one defined in Definition 3. However, this k-module would still be canonically isomorphic to the one defined in Definition 3, and thus it is commonly considered to be \more or less the same k-module". There is yet another definition of V1 ⊗V2 ⊗· · ·⊗Vn, which proceeds by taking the free k-module on the set V1 × V2 × · · · × Vn and factoring it modulo a certain submodule. This definition gives yet another k-module V1 ⊗ V2 ⊗ · · · ⊗ Vn, but this module is also canonically isomorphic to the k-module V1 ⊗ V2 ⊗ · · · ⊗ Vn defined in Definition 3, and thus can be considered to be \more or less the same k-module". (b) Definition 3, applied to n = 1, defines the tensor product of one k-module V1 as V1 ⊗ k. This takes some getting used to, since it seems more natural to define the tensor product of one k-module V1 simply as V1. But this isn't really different ∼ because there is a canonical isomorphism of k-modules V1 = V1 ⊗ k, so most people consider V1 to be \more or less the same k-module" as V1 ⊗ k. Convention 5. A remark about notation is appropriate at this point: There are two different conflicting notions of a \pure tensor" in a tensor product V1 ⊗ V2 ⊗ · · · ⊗ Vn of n arbitrary k-modules V1, V2, :::, Vn, where n ≥ 1. The one notion defines a \pure tensor" as an element of the form v ⊗ T for some v 2 V1 and 1 some T 2 V2 ⊗ V3 ⊗ · · · ⊗ Vn . The other notion defines a \pure tensor" as an element of the form v1 ⊗ v2 ⊗ · · · ⊗ vn for some (v1; v2; : : : ; vn) 2 V1 × V2 × · · · × Vn. These two notions are not equivalent. In this note, we are going to yield right of way to the second of these notions, i. e. we are going to define a pure tensor in V1 ⊗V2 ⊗· · ·⊗Vn as an element of the form v1 ⊗v2 ⊗· · ·⊗vn for some (v1; v2; : : : ; vn) 2 V1 × V2 × · · · × Vn. The first notion, however, will also be used - but we will not call it a \pure tensor" but rather a \left-induced tensor". Thus we define a left-induced tensor in V1 ⊗ V2 ⊗ · · · ⊗ Vn as an element of the form v ⊗ T for some v 2 V1 and some T 2 V2 ⊗ V3 ⊗ · · · ⊗ Vn. We note that the k-module V1 ⊗V2 ⊗· · ·⊗Vn is generated by its left-induced tensors, but also generated by its pure tensors. We also recall the definition of the tensor product of several k-module homomor- phisms (assuming that the notion of the tensor product of two k-module homomor- phisms is already defined): 1 In fact, if we look at Definition 3, we see that the k-module V1 ⊗ V2 ⊗ · · · ⊗ Vn was defined as V1 ⊗ (V2 ⊗ V3 ⊗ · · · ⊗ Vn), so it is the k-module A ⊗ B where A = V1 and B = V2 ⊗ V3 ⊗ · · · ⊗ Vn. Since the usual definition of a pure tensor in A ⊗ B defines it as an element of the form v ⊗ T for some v 2 A and T 2 B, it thus is logical to say that a pure tensor in V1 ⊗ V2 ⊗ · · · ⊗ Vn means an element of the form v ⊗ T for v 2 V1 and T 2 V2 ⊗ V3 ⊗ · · · ⊗ Vn. 3 Definition 6. Let k be a commutative ring. Let n 2 N. Now, by induction over n, we are going to define a k-module homomorphism f1 ⊗ f2 ⊗· · ·⊗fn : V1 ⊗V2 ⊗· · ·⊗Vn ! W1 ⊗W2 ⊗· · ·⊗Wn whenever V1, V2, :::, Vn are n arbitrary k-modules, W1, W2, :::, Wn are n arbitrary k-modules, and f1 : V1 ! W1, f2 : V2 ! W2, :::, fn : Vn ! Wn are n arbitrary k-module homomorphisms: Induction base: For n = 0, we define f1 ⊗f2 ⊗· · ·⊗fn as the identity map id : k ! k. Induction step: Let p 2 N. Assume that we have defined a k-module homomorphism f1 ⊗f2 ⊗· · ·⊗fp : V1 ⊗V2 ⊗· · ·⊗Vp ! W1 ⊗W2 ⊗· · ·⊗Wp whenever V1, V2, :::, Vp are p arbitrary k-modules, W1, W2, :::, Wp are p arbitrary k-modules, and f1 : V1 ! W1, f2 : V2 ! W2, :::, fp : Vp ! Wp are p arbitrary k-module homomorphisms. Now let us define a k-module homomorphism f1 ⊗ f2 ⊗ · · · ⊗ fp+1 : V1 ⊗ V2 ⊗ · · · ⊗ Vp+1 ! W1 ⊗ W2 ⊗ · · · ⊗ Wp+1 whenever V1, V2, :::, Vp+1 are p + 1 arbitrary k-modules, W1, W2, :::, Wp+1 are p + 1 arbitrary k-modules, and f1 : V1 ! W1, f2 : V2 ! W2, :::, fp+1 : Vp+1 ! Wp+1 are p + 1 arbitrary k-module homomorphisms. Namely, we define this homomorphism f1 ⊗ f2 ⊗ · · · ⊗ fp+1 to be f1 ⊗ (f2 ⊗ f3 ⊗ · · · ⊗ fp+1). Here, f1 ⊗ (f2 ⊗ f3 ⊗ · · · ⊗ fp+1) is to be understood as the tensor product of the k-module homomorphism f1 : V1 ! W1 with the k-module homomorphism f2 ⊗ f3 ⊗ · · · ⊗ fp+1 : V2 ⊗ V3 ⊗ · · · ⊗ Vp+1 ! W2 ⊗ W3 ⊗ · · · ⊗ Wp+1 (note that the k-module homomorphism f2 ⊗ f3 ⊗ · · · ⊗ fp+1 : V2 ⊗ V3 ⊗ · · · ⊗ Vp+1 ! W2 ⊗ W3 ⊗ · · · ⊗ Wp+1 is already defined (because we assumed that we have defined a k-module homomorphism f1 ⊗f2 ⊗· · ·⊗fp : V1 ⊗V2 ⊗· · ·⊗Vp ! W1 ⊗W2 ⊗· · ·⊗Wp whenever V1, V2, :::, Vp are p arbitrary k-modules, W1, W2, :::, Wp are p arbitrary k-modules, and f1 : V1 ! W1, f2 : V2 ! W2, :::, fp : Vp ! Wp are p arbitrary k-module homomorphisms)).
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