Approximations and Endomorphism Algebras of Modules - download pdf or read online

By Ruediger Goebel, Jan Trlifaj

ISBN-10: 3110110792

ISBN-13: 9783110110791

This monograph offers an intensive remedy of 2 vital components of latest module conception: approximations of modules and their purposes, significantly to limitless dimensional tilting conception, and realizations of algebras as endomorphism algebras of teams and modules. cognizance is usually given to E-rings and unfastened modules with extraordinary submodules. The monograph begins from easy evidence and progressively develops the speculation to its current frontiers. it really is appropriate for graduate scholars drawn to algebra in addition to specialists in module and illustration conception.

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Example text

B) Let n < ω, B ∈ R–Mod–S, and let C be an injective right S–module. Then ExtnR (A, HomS (B, C)) ∼ = HomS (TorR n (A, B), C). (c) Assume A is finitely presented. Let B ∈ S–Mod–R and C an injective left S–module. Then there is a natural isomorphism A ⊗R HomS (B, C) ∼ = HomS (HomR (A, B), C). (d) Let m < ω. Assume A has a projective resolution . . → Pn → . . → P0 → A → 0 such that Pi is finitely generated for each i ≤ m + 1. Moreover, let B ∈ S–Mod–R and C an injective left S–module. Then i ∼ TorR i (A, HomS (B, C)) = HomS (ExtR (A, B), C) for each i ≤ m.

6. Let R be a ring and F be a finitely presented module. Let D = (Mi , fij | i ≤ j ∈ I) be a direct system of modules with direct limit (M, fi (i ∈ I)). Then any R–homomorphism from F to M has a factorization through some fi (i ∈ I). Proof. Take x ∈ HomR (F, M ). Since X = Im(x) is finitely generated, there exists i ∈ I such that X ⊆ Im(fi ) = Fi . Consider the pullback of x and fi : z P −−−−→ F −−−−→ 0 ⏐ ⏐ ⏐ ⏐ y x fi Mi −−−−→ Fi −−−−→ 0 There is a finitely generated submodule P ⊆ P such that z = z P is surjective.

9. Let R be a ring and C be a class of finitely presented modules closed under finite direct sums. Then the following are equivalent for a module M. 26 1 Some useful classes of modules (a) M ∈ lim C. −→ (b) There is a pure epimorphism f : of modules in C. i∈I Ci → M for a sequence (Ci | i ∈ I) (c) Every homomorphism h : F → M , where F is finitely presented, has a factorization through a module in C. Moreover, lim C is closed under direct limits, pure submodules and pure epimor−→ phic images, and the finitely presented modules in lim C are exactly the direct −→ summands of modules in C.

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Approximations and Endomorphism Algebras of Modules by Ruediger Goebel, Jan Trlifaj


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