By Robert R. Colby

ISBN-10: 0521838215

ISBN-13: 9780521838214

This ebook presents a unified method of a lot of the theories of equivalence and duality among different types of modules that has transpired during the last forty five years. extra lately, many authors (including the authors of this booklet) have investigated relationships among different types of modules over a couple of earrings which are triggered by way of either covariant and contravariant representable functors, particularly, by way of tilting and cotilting theories. amassing and unifying the elemental result of those investigations with leading edge and simply comprehensible proofs, the authors' supply an reduction to extra study in this important subject in summary algebra.

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This publication presents a unified method of a lot of the theories of equivalence and duality among different types of modules that has transpired over the past forty five years. extra lately, many authors (including the authors of this e-book) have investigated relationships among different types of modules over a couple of earrings which are brought on by way of either covariant and contravariant representable functors, particularly, via tilting and cotilting theories.

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**Additional info for Equivalence and Duality for Module Categories with Tilting and Cotilting for Rings **

**Sample text**

3 Tilting Modules Tilting modules were originally introduced in [8], [9], and [46] as a tool in the representation (or module) theory of finite dimensional algebras. They, and more general versions of tilting modules, have found many important applications to this theory. Our concern here is their application to the module theory of more general rings. 1. Generalized Tilting Modules We begin with a generalization of the notion of tilting modules that was introduced by R. Colpi and J. Trlifaj in [32].

3, and since U R generates E(R R ) ∈ T , U R is faithful. (c) ⇒ (a). 3. 2. Since n R is right artinian, U R = ⊕i=1 Ui where each Ui is indecomposable. From the n set {Ui }i=1 choose a subset that is minimal with respect to the property that the direct sum of its members generates Gen(U R ). Renumbering the Ui , we can k k where k ≤ n and let W R = ⊕i=1 Ui . To comassume that this set is {Ui }i=1 ⊥ plete the proof it suffices to show that Gen(W R ) ⊆ W R = U1⊥ ∩ · · · ∩ Uk⊥ . 36 Tilting Modules Let 1 ≤ m ≤ k and suppose Ext1R (Um , M) = 0 where M ∈ Gen(W R ).

3. A finitely generated module VR is a tilting module if Gen(VR ) = VR⊥ . If VR is a progenerator Gen(VR ) = Mod-R = VR⊥ , so tilting modules also generalize progenerators. 5 we shall see that progenerators, quasi-progenerators, and tilting modules are indeed ∗-modules, and we shall provide criteria for a ∗-module to be of each of these types. 4. If V is projective relative to each Mi for i ∈ I and generates all submodules of each Mi , then V generates every submodule of ⊕ I Mi . Proof. It suffices to prove that V generates every cyclic submodule of ⊕ I Mi , so we may assume that I = {1, 2}.

### Equivalence and Duality for Module Categories with Tilting and Cotilting for Rings by Robert R. Colby

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