Product Description
This book provides a comprehensive categorical theory ofclosure operators, with applications to topological and uniformspaces, groups, R-modules, fields and topological groups, aswell as partially ordered sets and graphs. In particular, closureoperators are used to give solutions to the epimorphism andco-well-poweredness problem in many concrete categories.
The material is illustrated with many examples and exercises, and openproblems are formulated which should stimulate further research.
Audience: This volume will be of interest to graduate studentsand professional researchers in many branches of mathematics andtheoretical computer science. Knowledge of algebra, topology, and thebasic notions of category theory is assumed. ... Read more

Product Description
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. ... Read more

Product Description
This text offers comprehensive coverage of group cohomology, from introductory material through the most recent developments in the field. The primary motivation for this book is the interaction of group cohomology with representation theory, especially the geometry of support varieties over cohomology rings. The appendices, comprising computer calculations of the mod-2 cohomology rings of the groups whose orders divide 64, provide information useful for further developments in the field. A unique feature of this text is that it includes the concepts that are the subject of the calculations and are the source of some of the motivating conjectures for the computations. The programs for computing the cohomology rings were executed in the MAGMA computer algebra language. The text is a valuable resource for researchers in group cohomology and related disciplines. In addition, the book could be used as the text for an advanced graduate class or a graduate seminar. ... Read more

Product Description
The already broad range of applications of ring theory has been enhanced in the eighties by the increasing interest in algebraic structures of considerable complexity, the so-called class of quantum groups. One of the fundamental properties of quantum groups is that they are modelled by associative coordinate rings possessing a canonical basis, which allows for the use of algorithmic structures based on Groebner bases to study them. This book develops these methods in a self-contained way, concentrating on an in-depth study of the notion of a vast class of non-commutative rings (encompassing most quantum groups), the so-called Poincaré-Birkhoff-Witt rings. We include algorithms which treat essential aspects like ideals and (bi)modules, the calculation of homological dimension and of the Gelfand-Kirillov dimension, the Hilbert-Samuel polynomial, primality tests for prime ideals, etc. ... Read more

Product Description
This edited volume gives an overview of the concepts andmethods used in current research in quantum logic, viewed both as anoperational physical theory and in terms of purely mathematicalstructure. Far from being simply a collation of research papers, itconsists of 11 specially commissioned essays that, taken together,provide both an introduction to quantum logic and an overview ofcurrent research in this subject. The contributors, who include someof the most distinguished names in the field, discuss topics rangingfrom the algebraic theory of orthomodular lattices to recentapplications of category-theoretic methods and connections withtheoretical computer science. Moreover, an historical overview of thefield and an extensive citation and concept index are included.
Audience: This volume, dedicated to D.J. Foulis in honour of hisseminal contributions to quantum logic, should interest puremathematicians, theoretical physicists, computer scientists andphilosophers of science. ... Read more

Product Description
This book contains an introduction to the recently developedspectral theory of associative rings and Abelian categories, and itsapplications to the study of irreducible representations of classes ofalgebras which play an important part in modern mathematical physics.
Audience: A self-contained volume for researchers and graduatestudents interested in new geometric ideas in algebra, and in thespectral theory of noncommutative rings, currently invadingmathematical physics. Valuable reading for mathematicians working onrepresentation theory, quantum groups and related topics,noncommutative algebra, algebraic geometry, and algebraic K-theory.
... Read more

Product Description

The algebra of primary cohomology operations computed by the well-known Steenrod algebra is one of the most powerful tools of algebraic topology. This book computes the algebra of secondary cohomology operations which enriches the structure of the Steenrod algebra in a new and unexpected way.

The book solves a long-standing problem on the algebra of secondary cohomology operations by developing a new algebraic theory of such operations. The results have strong impact on the Adams spectral sequence and hence on the computation of homotopy groups of spheres.

Product Description
This book, the first printing of which was published as volume 38 of the Encyclopaedia of Mathematical Sciences, presents a modern approach to homological algebra, based on the systematic use of the terminology and ideas of derived categories and derived functors. The book contains applications of homological algebra to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of modules over rings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin explain all the main ideas of the theory of derived categories. Both authors are well-known researchers and the second, Manin, is famous for his work in algebraic geometry and mathematical physics. The book is an excellent reference for graduate students and researchers in mathematics and also for physicists who use methods from algebraic geometry and algebraic topology. ... Read more

Customer Reviews (1)

Homology Category
This is an excellent book for whom are familiar with the previous homological algebra techniques (as Cartan and Eilenberg's Homological algebra as first generation, Hilton and Stammbach's A course in Homological algebra as second generation, this book serves as third generation of derived category. This is not written for graduate students who are trying to learn this material as the book is in part of encyclopaedia.

The book explains well why, how and what the derived functors and derived category are introduced. Many concrete exmaples in algebra and geometry are listed in Chapter 3, in particular, Chapter 6 (Mixed Hodge structure), chapter 7(perverse sheaf) and chapter 8(D-modules) are served very good samples on the Derived categories and their equivalence which interprete many deep and fundamental theorems in these areas from category point of view.

Chapter 4(Derived categories and derived functors) and Chapter 5 (Triangulated categories) are the core of the homology category theory. The authors used the localization of homotopy category to construct the universal derived category which is familiar construction for category people under the name of fraction category. In fact many results can be obtained from calculus of fractions (see chapter 19 of the book by H. Schubert "Categories").

It would be better to include the rational homotopy model of Quillen and Sullivan's results and many other more explicit examples from infinitesimally topological calculations. There are few typos and undefined notations. Overall this is a very good reference book indeed. ... Read more

Product Description
Threading Homology through Algebratakes homological themes (Koszul complexes and their variations, resolutions in general) and shows how these affect the perception of certain problems in selected parts of algebra, as well as their success in solving a number of them.The text deals with regular local rings, depth-sensitive complexes, finite free resolutions, letter-place algebra, Schur and Weyl modules, Weyl-Schur complexes and determinantal ideals.Aimed at graduates and academics in mathematics, the book provides an overview of the developments that have taken place in these areas as well as an insight into some of the open problems which exist. ... Read more

Product Description
Let $R$ be a polynomial ring over an algebraically closed field and let $A$ be a standard graded Cohen-Macaulay quotient of $R$. The authors state that $A$ is a level algebra if the last module in the minimal free resolution of $A$ (as $R$-module) is of the form $R(-s)^a$, where $s$ and $a$ are positive integers. When $a=1$ these are also known as Gorenstein algebras.The basic question addressed in this paper is: What can be the Hilbert Function of a level algebra? The authors consider the question in several particular cases, e.g., when $A$ is an Artinian algebra, or when $A$ is the homogeneous coordinate ring of a reduced set of points, or when $A$ satisfies the Weak Lefschetz Property.The authors give new methods for showing that certain functions are NOT possible as the Hilbert function of a level algebra and also give new methods to construct level algebras.In a (rather long) appendix, the authors apply their results to give complete lists of all possible Hilbert functions in the case that the codimension of $A = 3$, $s$ is small and $a$ takes on certain fixed values. ... Read more

Product Description
This volume contains 43 research papers based on results presented at the Eighth International Conference on Representations of Algebras (ICRA VIII) held in Geiranger, Norway, in 1996. The papers, written by experts in the field, cover the most recent developments in the representation theory of artin algebras and related topics. The papers cover: representation of tame, biserial, cellular, factorial hereditary, Hopf, Koszul, non-polynomial growth, preprojective, Temperley-Lieb, tilted and quasitilted algebras.Other topics include: tilting/cotilting modules and generalizations as $*$-modules, exceptional sequences of modules and vector bundles, homological conjectures, Hochschild cohomology, cyclic homology, homologically finite subcategories, representations of posets, regular modules, vector space categories, triangulated categories, moduli spaces of representations of quivers, postprojective (and preprojective) partitions, stable and derived equivalences, and pure-injective, infinite dimensional, and endofinite representations. A general background in noncommutative algebra including rings, modules, and homological algebra is required.Following are the features: it presents a unique source for the developments in the representation theory of finite dimensional and artin algebras and related topics; and, a wide variety of important papers by leading researchers in the field, with references to earlier developments in the field. ... Read more

Product Description
Over 6000 reviews, which appeared in MR between 1940 and 1967, are included in this two-part volume. These reviews, with an author index, are arranged under 290 subject headings, and appear in their entirety. ... Read more

Product Description
This book provides a self-contained systematic treatment of the subject of relative homological algebra. It is designed for graduate students as well as researchers and specialists. It contains twelve chapters with abundant supply of important results with complete proofs covering material that is essential to understanding topics in algebra, algebraic geometry, and algebraic topology. The text also contains results that are in book form for the first time and thus provides essential reading for researchers and specialists. At the end of each section of each chapter, there are exercises that provide practice problems for students as well as additional important results for specialists.

The book can be used as a text for graduate students and as a handbook for researchers and specialists. The material in the first three chapters constitute notes from lectures of the authors at their respective universities and is suitable for an introductory course in module and ring theory. The following chapters are suitable for a course in relative homological algebra and its applications to commutative and non-commutative algebra. The last three chapters give applications to ring theory. These chapters together with Chapter 7 contain recent achievements including a solution to the flat cover conjecture. ... Read more

Customer Reviews (1)

A very disapointing book
This is NOT a textbook. It is a printout of the wikipedia-items on homological algebra, but without the diagrams and the mathematical expressions. In my opinion it is worthless.
... Read more

Product Description
This digital document is an article from Science and Its Times, brought to you by Gale®, a part of Cengage Learning, a world leader in e-research and educational publishing for libraries, schools and businesses.The length of the article is 1570 words.The article is delivered in HTML format and is available in your Amazon.com Digital Locker immediately after purchase.You can view it with any web browser.The histories of science, technology, and mathematics merge with the study of humanities and social science in this interdisciplinary reference work. Essays on people, theories, discoveries, and concepts are combined with overviews, bibliographies of primary documents, and chronological elements to offer students a fascinating way to understand the impact of science on the course of human history and how science affects everyday life. Entries represent people and developments throughout the world, from about 2000 B.C. through the end of the twentieth century. ... Read more