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Homological Algebra

Homological Algebra Author Marco Grandis
ISBN-10 9789814425933
Release 2013-01-11
Pages 356
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We propose here a study of ‘semiexact’ and ‘homological' categories as a basis for a generalised homological algebra. Our aim is to extend the homological notions to deeply non-abelian situations, where satellites and spectral sequences can still be studied. This is a sequel of a book on ‘Homological Algebra, The interplay of homology with distributive lattices and orthodox semigroups’, published by the same Editor, but can be read independently of the latter. The previous book develops homological algebra in p-exact categories, i.e. exact categories in the sense of Puppe and Mitchell — a moderate generalisation of abelian categories that is nevertheless crucial for a theory of ‘coherence’ and ‘universal models’ of (even abelian) homological algebra. The main motivation of the present, much wider extension is that the exact sequences or spectral sequences produced by unstable homotopy theory cannot be dealt with in the previous framework. According to the present definitions, a semiexact category is a category equipped with an ideal of ‘null’ morphisms and provided with kernels and cokernels with respect to this ideal. A homological category satisfies some further conditions that allow the construction of subquotients and induced morphisms, in particular the homology of a chain complex or the spectral sequence of an exact couple. Extending abelian categories, and also the p-exact ones, these notions include the usual domains of homology and homotopy theories, e.g. the category of ‘pairs’ of topological spaces or groups; they also include their codomains, since the sequences of homotopy ‘objects’ for a pair of pointed spaces or a fibration can be viewed as exact sequences in a homological category, whose objects are actions of groups on pointed sets. Homological Algebra: The Interplay of Homology with Distributive Lattices and Orthodox Semigroups Contents:IntroductionSemiexact categoriesHomological CategoriesSubquotients, Homology and Exact CouplesSatellitesUniversal ConstructionsApplications to Algebraic TopologyHomological Theories and Biuniversal ModelsAppendix A. Some Points of Category Theory Readership: Graduate students, professors and researchers in pure mathematics, in particular category theory and algebraic topology. Keywords:Non Abelian Homological Algebra;Spectral Sequences;Distributive Lattices;Orthodox Semigroups;Categories of RelationsReviews: “The range of applications and examples is considerable and many are outside the reach of more standard forms of homological algebra, but the methods used here also give insight as to 'why' the classical theory works and how its results can be interpreted.” Zentralblatt MATH

Homological Algebra

Homological Algebra Author Marco Grandis
ISBN-10 9789814407069
Release 2012
Pages 369
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In this book we want to explore aspects of coherence in homological algebra, that already appear in the classical situation of abelian groups or abelian categories. Lattices of subobjects are shown to play an important role in the study of homological systems, from simple chain complexes to all the structures that give rise to spectral sequences. A parallel role is played by semigroups of endorelations. These links rest on the fact that many such systems, but not all of them, live in distributive sublattices of the modular lattices of subobjects of the system. The property of distributivity allows one to work with induced morphisms in an automatically consistent way, as we prove in a 'Coherence Theorem for homological algebra'. (On the contrary, a 'non-distributive' homological structure like the bifiltered chain complex can easily lead to inconsistency, if one explores the interaction of its two spectral sequences farther than it is normally done.) The same property of distributivity also permits representations of homological structures by means of sets and lattices of subsets, yielding a precise foundation for the heuristic tool of Zeeman diagrams as universal models of spectral sequences. We thus establish an effective method of working with spectral sequences, called 'crossword chasing', that can often replace the usual complicated algebraic tools and be of much help to readers that want to apply spectral sequences in any field.

Category Theory and Applications

Category Theory and Applications Author Marco Grandis
ISBN-10 9789813231085
Release 2018-01-16
Pages 304
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Category Theory now permeates most of Mathematics, large parts of theoretical Computer Science and parts of theoretical Physics. Its unifying power brings together different branches, and leads to a deeper understanding of their roots. This book is addressed to students and researchers of these fields and can be used as a text for a first course in Category Theory. It covers its basic tools, like universal properties, limits, adjoint functors and monads. These are presented in a concrete way, starting from examples and exercises taken from elementary Algebra, Lattice Theory and Topology, then developing the theory together with new exercises and applications. Applications of Category Theory form a vast and differentiated domain. This book wants to present the basic applications and a choice of more advanced ones, based on the interests of the author. References are given for applications in many other fields. Contents: Introduction Categories, Functors and Natural Transformations Limits and Colimits Adjunctions and Monads Applications in Algebra Applications in Topology and Algebraic Topology Applications in Homological Algebra Hints at Higher Dimensional Category Theory References Indices Readership: Graduate students and researchers of mathematics, computer science, physics. Keywords: Category TheoryReview: Key Features: The main notions of Category Theory are presented in a concrete way, starting from examples taken from the elementary part of well-known disciplines: Algebra, Lattice Theory and Topology The theory is developed presenting other examples and some 300 exercises; the latter are endowed with a solution, or a partial solution, or adequate hints Three chapters and some extra sections are devoted to applications

A Singular Introduction to Commutative Algebra

A Singular Introduction to Commutative Algebra Author Gert-Martin Greuel
ISBN-10 9783540735427
Release 2007-09-23
Pages 689
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This substantially enlarged second edition aims to lead a further stage in the computational revolution in commutative algebra. This is the first handbook/tutorial to extensively deal with SINGULAR. Among the book’s most distinctive features is a new, completely unified treatment of the global and local theories. Another feature of the book is its breadth of coverage of theoretical topics in the portions of commutative algebra closest to algebraic geometry, with algorithmic treatments of almost every topic.

A Course in Commutative Algebra

A Course in Commutative Algebra Author Gregor Kemper
ISBN-10 3642035450
Release 2010-12-02
Pages 248
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This textbook offers a thorough, modern introduction into commutative algebra. It is intented mainly to serve as a guide for a course of one or two semesters, or for self-study. The carefully selected subject matter concentrates on the concepts and results at the center of the field. The book maintains a constant view on the natural geometric context, enabling the reader to gain a deeper understanding of the material. Although it emphasizes theory, three chapters are devoted to computational aspects. Many illustrative examples and exercises enrich the text.

Introduction to Commutative Algebra and Algebraic Geometry

Introduction to Commutative Algebra and Algebraic Geometry Author Ernst Kunz
ISBN-10 0817630651
Release 1985
Pages 238
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Introduction to Commutative Algebra and Algebraic Geometry has been writing in one form or another for most of life. You can find so many inspiration from Introduction to Commutative Algebra and Algebraic Geometry also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Introduction to Commutative Algebra and Algebraic Geometry book for free.

Contributions to Algebra

Contributions to Algebra Author Hyman Bass
ISBN-10 9781483268064
Release 2014-05-10
Pages 446
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Contributions to Algebra: A Collection of Papers Dedicated to Ellis Kolchin provides information pertinent to commutative algebra, linear algebraic group theory, and differential algebra. This book covers a variety of topics, including complex analysis, logic, K-theory, stochastic matrices, and differential geometry. Organized into 29 chapters, this book begins with an overview of the influence that Ellis Kolchin's work on the Galois theory of differential fields has had on the development of differential equations. This text then discusses the background model theoretic work in differential algebra and discusses the notion of model completions. Other chapters consider some properties of differential closures and some immediate consequences and include extensive notes with proofs. This book discusses as well the problems in finite group theory in finding the complex finite projective groups of a given degree. The final chapter deals with the finite forms of quasi-simple algebraic groups. This book is a valuable resource for students.

A Course in Homological Algebra

A Course in Homological Algebra Author P.J. Hilton
ISBN-10 9781468499360
Release 2013-03-09
Pages 340
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In this chapter we are largely influenced in our choice of material by the demands of the rest of the book. However, we take the view that this is an opportunity for the student to grasp basic categorical notions which permeate so much of mathematics today, including, of course, algebraic topology, so that we do not allow ourselves to be rigidly restricted by our immediate objectives. A reader totally unfamiliar with category theory may find it easiest to restrict his first reading of Chapter II to Sections 1 to 6; large parts of the book are understandable with the material presented in these sections. Another reader, who had already met many examples of categorical formulations and concepts might, in fact, prefer to look at Chapter II before reading Chapter I. Of course the reader thoroughly familiar with category theory could, in principal, omit Chapter II, except perhaps to familiarize himself with the notations employed. In Chapter III we begin the proper study of homological algebra by looking in particular at the group ExtA(A, B), where A and Bare A-modules. It is shown how this group can be calculated by means of a projective presentation of A, or an injective presentation of B; and how it may also be identified with the group of equivalence classes of extensions of the quotient module A by the submodule B.

Operator Algebras Operator Theory and Applications

Operator Algebras  Operator Theory and Applications Author J. J. Grobler
ISBN-10 3034601743
Release 2009-12-24
Pages 294
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Operator Algebras Operator Theory and Applications has been writing in one form or another for most of life. You can find so many inspiration from Operator Algebras Operator Theory and Applications also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Operator Algebras Operator Theory and Applications book for free.

Mal cev Protomodular Homological and Semi Abelian Categories

Mal cev  Protomodular  Homological and Semi Abelian Categories Author Francis Borceux
ISBN-10 1402019610
Release 2004-02-29
Pages 479
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The purpose of the book is to take stock of the situation concerning Algebra via Category Theory in the last fifteen years, where the new and synthetic notions of Mal'cev, protomodular, homological and semi-abelian categories emerged. These notions force attention on the fibration of points and allow a unified treatment of the main algebraic: homological lemmas, Noether isomorphisms, commutator theory. The book gives full importance to examples and makes strong connections with Universal Algebra. One of its aims is to allow appreciating how productive the essential categorical constraint is: knowing an object, not from inside via its elements, but from outside via its relations with its environment. The book is intended to be a powerful tool in the hands of researchers in category theory, homology theory and universal algebra, as well as a textbook for graduate courses on these topics.

A Non Hausdorff Completion

A Non Hausdorff Completion Author Saul Lubkin
ISBN-10 9789814667401
Release 2015-05-28
Pages 352
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This book introduces entirely new invariants never considered before, in homological algebra and commutative (and even non-commutative) algebra. The C-completion C(M), and higher C-completions, Cn(M), are defined for an arbitrary left module M over a topological ring A. Spectral sequences are defined that use these invariants. Given a left module over a topological ring A, under mild conditions the usual Hausdorff completion: M^ can be recovered from the C-completion C(M), by taking the quotient module by the closure of {0}. The new invariants and tools in this book are expected to be used in the study of p-adic cohomology in algebraic geometry; and also in the study of p-adic Banach spaces — by replacing the cumbersome "complete tensor product" of p-adic Banach spaces, with the more sophisticated "C-complete tensor product", discussed in this book. It is also not unlikely that the further study of these new invariants may well develop into a new branch of abstract mathematics - connected with commutative algebra, homological algebra, and algebraic topology.

Set theoretic methods in homological algebra and Abelian groups

Set theoretic methods in homological algebra and Abelian groups Author Paul C. Eklof
ISBN-10 UOM:39015017308928
Release 1980
Pages 117
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Set theoretic methods in homological algebra and Abelian groups has been writing in one form or another for most of life. You can find so many inspiration from Set theoretic methods in homological algebra and Abelian groups also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Set theoretic methods in homological algebra and Abelian groups book for free.

Commutative Algebra

Commutative Algebra Author Marco Fontana
ISBN-10 144196990X
Release 2010-09-29
Pages 492
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Commutative algebra is a rapidly growing subject that is developing in many different directions. This volume presents several of the most recent results from various areas related to both Noetherian and non-Noetherian commutative algebra. This volume contains a collection of invited survey articles by some of the leading experts in the field. The authors of these chapters have been carefully selected for their important contributions to an area of commutative-algebraic research. Some topics presented in the volume include: generalizations of cyclic modules, zero divisor graphs, class semigroups, forcing algebras, syzygy bundles, tight closure, Gorenstein dimensions, tensor products of algebras over fields, as well as many others. This book is intended for researchers and graduate students interested in studying the many topics related to commutative algebra.

Extended Abstracts Spring 2015

Extended Abstracts Spring 2015 Author Dolors Herbera
ISBN-10 9783319454412
Release 2016-12-14
Pages 192
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This book includes 33 expanded abstracts of selected talks given at the two workshops "Homological Bonds Between Commutative Algebra and Representation Theory" and "Brave New Algebra: Opening Perspectives," and the conference "Opening Perspectives in Algebra, Representations, and Topology," held at the Centre de Recerca Matemàtica (CRM) in Barcelona between January and June 2015. These activities were part of the one-semester intensive research program "Interactions Between Representation Theory, Algebraic Topology and Commutative Algebra (IRTATCA)." Most of the abstracts present preliminary versions of not-yet published results and cover a large number of topics (including commutative and non commutative algebra, algebraic topology, singularity theory, triangulated categories, representation theory) overlapping with homological methods. This comprehensive book is a valuable resource for the community of researchers interested in homological algebra in a broad sense, and those curious to learn the latest developments in the area. It appeals to established researchers as well as PhD and postdoctoral students who want to learn more about the latest advances in these highly active fields of research.

An Introduction to Homological Algebra

An Introduction to Homological Algebra Author Charles A. Weibel
ISBN-10 9781139643078
Release 1995-10-27
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The landscape of homological algebra has evolved over the last half-century into a fundamental tool for the working mathematician. This book provides a unified account of homological algebra as it exists today. The historical connection with topology, regular local rings, and semi-simple Lie algebras are also described. This book is suitable for second or third year graduate students. The first half of the book takes as its subject the canonical topics in homological algebra: derived functors, Tor and Ext, projective dimensions and spectral sequences. Homology of group and Lie algebras illustrate these topics. Intermingled are less canonical topics, such as the derived inverse limit functor lim1, local cohomology, Galois cohomology, and affine Lie algebras. The last part of the book covers less traditional topics that are a vital part of the modern homological toolkit: simplicial methods, Hochschild and cyclic homology, derived categories and total derived functors. By making these tools more accessible, the book helps to break down the technological barrier between experts and casual users of homological algebra.

Homological and Computational Methods in Commutative Algebra

Homological and Computational Methods in Commutative Algebra Author Aldo Conca
ISBN-10 9783319619439
Release 2017-11-16
Pages 256
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This volume collects contributions by leading experts in the area of commutative algebra related to the INdAM meeting “Homological and Computational Methods in Commutative Algebra” held in Cortona (Italy) from May 30 to June 3, 2016 . The conference and this volume are dedicated to Winfried Bruns on the occasion of his 70th birthday. In particular, the topics of this book strongly reflect the variety of Winfried Bruns’ research interests and his great impact on commutative algebra as well as its applications to related fields. The authors discuss recent and relevant developments in algebraic geometry, commutative algebra, computational algebra, discrete geometry and homological algebra. The book offers a unique resource, both for young and more experienced researchers seeking comprehensive overviews and extensive bibliographic references.

Nonabelian Algebraic Topology

Nonabelian Algebraic Topology Author Ronald Brown
ISBN-10 3037190833
Release 2011
Pages 668
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Nonabelian Algebraic Topology has been writing in one form or another for most of life. You can find so many inspiration from Nonabelian Algebraic Topology also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Nonabelian Algebraic Topology book for free.