Cover of: Equivalence and Duality for Module Categories with Tilting and Cotilting for Rings (Cambridge Tracts in Mathematics) | Robert R. Colby

Equivalence and Duality for Module Categories with Tilting and Cotilting for Rings (Cambridge Tracts in Mathematics)

  • 152 Pages
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Cambridge University Press
Fields & rings, Science/Mathematics, Mathematics, Algebra - General, Mathematics / Algebra / General, Duality theory (Mathematics), Modules (Algebra), Rings (Alg
The Physical Object
FormatHardcover
ID Numbers
Open LibraryOL7766013M
ISBN 100521838215
ISBN 139780521838214

More recently, many authors (including the authors of this book) have investigated relationships between categories of modules over a pair rings that are induced by both covariant and contravariant representable functors, in particular, by tilting and cotilting theories.

Collecting Cited by: Equivalence and duality for module categories: with tilting and cotilting for rings Robert R. Colby, Kent R. Fuller This book provides a unified approach to much of the theories of equivalence and duality between categories of modules that has transpired over the last 45 years.

Equivalence and Duality for Module Categories with Tilting and Cotilting for Rings (Cambridge Tracts in Mathematics) Author Robert R. Colby, Kent R. Fuller Format/binding Hardcover Book condition Used:Good Quantity available 1 Binding Hardcover ISBN 10 ISBN 13 Publisher Cambridge University Press Place of Publication.

In particular, during the past dozen or so years many authors (including the authors of this book) have investigated relationships between categories of modules over a pair of rings that are induced by both covariant and contravariant representable functors, in particular by tilting and cotilting theories.

Equivalence and Duality for Module Categories with Tilting and Cotilting for Rings (Cambridge Tracts in Mathematics) by Colby, Robert R., Fuller, Kent R.

and a great selection of related books, art and collectibles available now at Get this from a library. Equivalence and duality for module categories: with tilting and cotilting for rings. [Robert R Colby; Kent R Fuller;] -- This book provides a unified approach to much of the theories of equivalence and duality between categories of modules that has transpired over the last 45 years.

In particular, during the past dozen.

Details Equivalence and Duality for Module Categories with Tilting and Cotilting for Rings (Cambridge Tracts in Mathematics) PDF

and Mod-R and R-Mod represent the categories of right and left R-modules and homomorphisms, while mod-R and R-mod denote their subcategories of finitely generated modules. The Kernel of Ext1 R (V,) Forany R-module V wedenotethekernelofExt1 R (V,)byV⊥.Closureprop-erties of V⊥are related to both homological and module-theoretic properties of V.

Following a brief outline of set-theoretic and categorical foundations, the text begins with the basic definitions and properties of rings, modules and homomorphisms and ranges through comprehensive treatments of direct sums, finiteness conditions, the Wedderburn-Art in Theorem, the Jacobson radical, the hom and tensor functions, Morita equivalence and duality, de- composition theory of injective and projective modules, and semiperfect and perfect rings.

Homological tilting modules of finite projective dimension are investigated. They generalize both classical and good tilting modules of projective dimension at most one, and produce recollements of derived module categories of rings in which generalized localizations of rings are by: 1.

So one expects certain dualities for the dual notion, the cotilting modules. However it is well known that there are no dualities between full module categories and one has to restrict to nitely closed subcategories.

We will see in and that under some niteness conditions cotilting modules yield such dualities. Providing a unified approach to much of the theories of equivalence and duality between categories of modules that has transpired over the last 45 years, the authors investigate relationships between categories of modules over a pair of rings that are induced by covariant and contravariant representable functors.

The purpose of this section is to construct 1-tilting modules generating those classes, and hence classify all 1-tilting modules over commutative rings up to equivalence. Of course we can always construct such modules using the Small Object Argument (see Cited by: Module categories ; module theory in a category-theoretic context; Morita equivalence and duality Rings and algebras arising under various constructions 16S90 Torsion theories; radicals on module categories Homological methods 16E30 Homological functors on modules (Tor, Ext, etc.) Representation theory of rings and algebras 16G10Cited by: Find Equivalence and Duality For Module Categories With Tilting and Cotilting For Rings by Colby, Robert R Fuller, Kent R at Biblio.

Uncommonly good collectible and. We prove that given a Grothendieck category G with a tilting object of finite projective dimension, the induced triangle equivalence sends an injective cogenerator of G to a big cotilting module. Moreover, every big cotilting module can be constructed like that in an essentially unique by: for the category of finitely presented modules by using a module which induces a quotient duality TheoremPropositionand Corollary Furthermore, we apply it to cotilting bimodules, and in particular, to.

rings with finite self-injective dimension Propositions; Corollaries, Good tilting modules and recollements of derived module categories, II. Hongxing Chen and Changchang Xi Abstract Homological tilting modules of finite projective dimension are in.

DUALITIES AND EQUIVALENCES INDUCED BY ADJOINT FUNCTORS 2 However, starting with a pair of adjoint functors between some abelian categories, in particular Grothendieck categories, we can construct other useful pairs of adjoint functors. It would be also nice to know if concepts developed for module categories work in this case.

A triangular matrix ring A is defined by a triplet (R,S,M) where R and S are rings and M is an S-R-bimodule. In the main theorem of this paper we show that if T is a tilting S-module, then under certain homological conditions on M as an S-module, one can extend T to a tilting complex over A inducing a derived equivalence between A and another triangular matrix ring specified by (S',R,M Cited by:   module (plural modules) A self-contained component of a system, often interchangeable, which has a well-defined interface to the other components.

(architecture) A standard unit of measure used for determining the proportions of a building. Further, we show that there are bijections between the following four classes (1) equivalent classes of AIR-cotilting (resp., cosilting, quasi-cotilting) modules, (2) equivalent classes of 2-term.

The basic property - existence of a maximal category equivalence repre- sented by the tilting module - forces the module to be finitely generated, cf. (31, §2), (85).Author: Silvana Bazzoni. good tilting modules of projective dimension at most one, and produce recollements of derived module categories of rings in which generalized localizations of rings are involved.

To decide whether a good tilting module is homological, a sufficient and necessary condition is presented in terms of the internal properties of the given tilting module. Cotilting theory (for arbitrary modules over arbitrary unital rings) extends Morita duality in analogy to the way tilting theory extends Morita equivalence.

In particular, cotilting modules generalize injective cogenerators similarly as tilting modules generalize progenerators. Here, right R-module UR is cotilting if UR has injective dimension Cited by: Equivalence and duality for module categories: with tilting and cotilting for rings.

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Description Equivalence and Duality for Module Categories with Tilting and Cotilting for Rings (Cambridge Tracts in Mathematics) FB2

The course should teach the students the notions of Morita equivalence and duality, tilting and cotilting module. The students must be also able to apply creatively the learned theory, in order to establish equivalences or dualities between particular classes of modules.

Rings and Categories of Modules, Springer Verlag,   Rigid modules and Schur roots. Equivalence and duality for module categories. With tilting and cotilting for rings. Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, x+ pp.

Crawley-Boevey, W.: Exceptional sequences of representations of quivers. In: Representations of algebras (Ottawa, ON, ), Author: Christof Geiß, Bernard Leclerc, Jan Schröer.

Quasi-Frobenius Rings. By W. Nicholson and M. Yousif The Geometry of Total Curvature on Complete Open Surfaces. By Katsuhiro Shiohama, Takashi Shioya and Minoru Tanaka Approximation by Algebraic Numbers. By Yann Bugeaud Equivalence and Duality for Module Categories with Tilting and Cotilting for Rings.

By R. Colby Cited by:. A module T is called a partial cotilting module if T satis es the conditions (C1) and (C2). Dually we also de ne complements of partial cotilting modules. Also recall that a module T in mod is a tilting or a cotilting module if AddT and ProdT are replaced by addT and.

We relate the theory of envelopes and covers to tilting and cotilting theory, for (infinitely generated) modules over arbitrary rings.

Download Equivalence and Duality for Module Categories with Tilting and Cotilting for Rings (Cambridge Tracts in Mathematics) FB2

Our main result characterizes tilting torsion classes as the pretorsion classes providing special preenvelopes for all modules. A dual characterization is proved for cotilting torsion-free classes using the new notion of a cofinendo module.

We also construct.This book is intended to provide a reasonably self-contained account of a major portion of the general theory of rings and modules suitable as a text for introductory and more advanced graduate courses.

We assume the famil iarity with rings usually acquired in standard undergraduate algebra courses. Our general approach is categorical rather than arithmetical.