An introduction to abstract algebra, by F. M. Hall PDF

By F. M. Hall

ISBN-10: 0521070554

ISBN-13: 9780521070553

ISBN-10: 0521298628

ISBN-13: 9780521298629

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R is not strongly graded, as R1 Rn−1 = X n K[X n ] = Ro . On the other hand R is a faithful ring, being a commutative domain. Let M = ⊕σ∈G Mσ be an object from the category R-gr. The tM : M → Me , t(m) = me , where m = σ∈G mσ is an element from M is called the trace map. It is clear that tM is Re -linear. Also if a ∈ R, a = σ∈G aσ then we have tM (am) = σ∈G aσ−1 mσ . We denote by rad(tM ) = {m ∈ M |tM (am) = 0 for any a ∈ R}. It is clear that rad(tM ) is a submodule of M . Also if m = σ∈G mσ ∈ rad(tM ), then we have for λ ∈ G that Rλ−1 mλ = 0.

The assumptions then allow to construct the free resolution : → F2 → F1 → F0 → M → 0 where each Fi is a free graded R-module of finite rank. , and using the well-known calculus of right derived functors, the statement follows. A graded R-module M is said to be gr-injective, resp. gr-projective, if M is injective, resp. projective, as an object of R-gr. 2. 3. From loc. cit. it is easy to obtain that M is gr-projective if and only if the functor HomR−gr (M, −) is exact, if and only if the functor HOMR (M, −) is exact, if and only if M is a direct summand in R-gr of a gr-free R-module.

The canonical morphism η : Ind → Coind is a functorial isomorphism. c. The map η(R) : R → Coind(Re ), η(R)(a)(b) = t(ba) is an isomorphism and for every g ∈ G, Rg is projective and finitely generated in Re -mod. 44 2 The Category of Graded Modules d. There exists an isomorphism ϑ : R → Coind(Re ) in R-gr that is also a morphism in mod-Re and for every g ∈ G, Rg is finitely generated and projective in Re -mod Proof a. ⇒ b. Assume that there exists a functorial isomorphism ϕ : Ind → Coind. Hence for any N ∈ Re -mod, ϕ(N ) : Ind(N ) → Coind(N ) is an isomorphism.

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An introduction to abstract algebra, by F. M. Hall

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