K-theory, Projective Modules, and Complete Intersections
Author: Benjamin Parker Nicholson
Publisher:
Published: 1998
Total Pages: 94
ISBN-13:
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Projective Modules and Complete Intersections
Author: Satya Mandal
Publisher: Springer
Published: 2006-11-14
Total Pages: 121
ISBN-13: 3540695982
DOWNLOAD EBOOKIn these notes on "Projective Modules and Complete Intersections" an account on the recent developments in research on this subject is presented. The author's preference for the technique of Patching isotopic isomorphisms due to Quillen, formalized by Plumsted, over the techniques of elementary matrices is evident here. The treatment of Basic Element theory here incorporates Plumstead's idea of the "generalized dimension functions." These notes are highly selfcontained and should be accessible to any graduate student in commutative algebra or algebraic geometry. They include fully self-contained presentations of the theorems of Ferrand-Szpiro, Cowsik-Nori and the techniques of Lindel.
Ideals and Reality
Author: Friedrich Ischebeck
Publisher: Springer Science & Business Media
Published: 2005-11-22
Total Pages: 339
ISBN-13: 3540263705
DOWNLOAD EBOOKBesides giving an introduction to Commutative Algebra - the theory of c- mutative rings - this book is devoted to the study of projective modules and the minimal number of generators of modules and ideals. The notion of a module over a ring R is a generalization of that of a vector space over a field k. The axioms are identical. But whereas every vector space possesses a basis, a module need not always have one. Modules possessing a basis are called free. So a finitely generated free R-module is of the form Rn for some n E IN, equipped with the usual operations. A module is called p- jective, iff it is a direct summand of a free one. Especially a finitely generated R-module P is projective iff there is an R-module Q with P @ Q S Rn for some n. Remarkably enough there do exist nonfree projective modules. Even there are nonfree P such that P @ Rm S Rn for some m and n. Modules P having the latter property are called stably free. On the other hand there are many rings, all of whose projective modules are free, e. g. local rings and principal ideal domains. (A commutative ring is called local iff it has exactly one maximal ideal. ) For two decades it was a challenging problem whether every projective module over the polynomial ring k[X1,. . .
Leavitt Path Algebras and Classical K-Theory
Author: A. A. Ambily
Publisher: Springer Nature
Published: 2020-01-17
Total Pages: 340
ISBN-13: 9811516111
DOWNLOAD EBOOKThe book offers a comprehensive introduction to Leavitt path algebras (LPAs) and graph C*-algebras. Highlighting their significant connection with classical K-theory—which plays an important role in mathematics and its related emerging fields—this book allows readers from diverse mathematical backgrounds to understand and appreciate these structures. The articles on LPAs are mostly of an expository nature and the ones dealing with K-theory provide new proofs and are accessible to interested students and beginners of the field. It is a useful resource for graduate students and researchers working in this field and related areas, such as C*-algebras and symbolic dynamics.
Polytopes, Rings, and K-Theory
Author: Winfried Bruns
Publisher: Springer Science & Business Media
Published: 2009-06-12
Total Pages: 461
ISBN-13: 0387763562
DOWNLOAD EBOOKThis book examines interactions of polyhedral discrete geometry and algebra. What makes this book unique is the presentation of several central results in all three areas of the exposition - from discrete geometry, to commutative algebra, and K-theory.
Categories and Modules with K-Theory in View
Author: A. J. Berrick
Publisher: Cambridge University Press
Published: 2000-05-25
Total Pages: 384
ISBN-13: 9780521632768
DOWNLOAD EBOOKThis book, first published in 2000, is a concise introduction at graduate level to ring theory, module theory and number theory.
Algebraic K-Theory
Author: Vasudevan Srinivas
Publisher: Springer Science & Business Media
Published: 2009-05-21
Total Pages: 357
ISBN-13: 0817647392
DOWNLOAD EBOOKAlgebraic K-Theory has become an increasingly active area of research. With its connections to algebra, algebraic geometry, topology, and number theory, it has implications for a wide variety of researchers and students in mathematics. This book is based on lectures given by the author at the Tata Institute in Bombay and elsewhere. This new edition includes an appendix on algebraic geometry that contains required definitions and results needed to understand the core of the book.
Algebraic K-Theory and Its Applications
Author: Jonathan Rosenberg
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 404
ISBN-13: 1461243149
DOWNLOAD EBOOKAlgebraic K-Theory is crucial in many areas of modern mathematics, especially algebraic topology, number theory, algebraic geometry, and operator theory. This text is designed to help graduate students in other areas learn the basics of K-Theory and get a feel for its many applications. Topics include algebraic topology, homological algebra, algebraic number theory, and an introduction to cyclic homology and its interrelationship with K-Theory.
Algebra, $K$-Theory, Groups, and Education
Author: Hyman Bass
Publisher: American Mathematical Soc.
Published: 1999
Total Pages: 250
ISBN-13: 0821810871
DOWNLOAD EBOOKThis volume includes expositions of key developments over the past four decades in commutative and non-commutative algebra, algebraic $K$-theory, infinite group theory, and applications of algebra to topology. Many of the articles are based on lectures given at a conference at Columbia University honoring the 65th birthday of Hyman Bass. Important topics related to Bass's mathematical interests are surveyed by leading experts in the field. Of particular note is a professional autobiography of Professor Bass, and an article by Deborah Ball on mathematical education. The range of subjects covered in the book offers a convenient single source for topics in the field.
Algebraic K-theory: The Homotopy Approach Of Quillen And An Approach From Commutative Algebra
Author: Satya Mandal
Publisher: World Scientific
Published: 2023-06-22
Total Pages: 680
ISBN-13: 9811269408
DOWNLOAD EBOOKIn this book the author takes a pedagogic approach to Algebraic K-theory. He tried to find the shortest route possible, with complete details, to arrive at the homotopy approach of Quillen [Q] to Algebraic K-theory, with a simple goal to produce a self-contained and comprehensive pedagogic document in Algebraic K-theory, that is accessible to upper level graduate students. That is precisely what this book faithfully executes and achieves.The contents of this book can be divided into three parts — (1) The main body (Chapters 2-8), (2) Epilogue Chapters (Chapters 9, 10, 11) and (3) the Background and preliminaries (Chapters A, B, C, 1). The main body deals with Quillen's definition of K-theory and the K-theory of schemes. Chapters 2, 3, 5, 6, and 7 provide expositions of the paper of Quillen [Q], and chapter 4 is on agreement of Classical K-theory and Quillen K-theory. Chapter 8 is an exposition of the work of Swan [Sw1] on K-theory of quadrics.The Epilogue chapters can be viewed as a natural progression of Quillen's work and methods. These represent significant benchmarks and include Waldhausen K-theory, Negative K-theory, Hermitian K-theory, 𝕂-theory spectra, Grothendieck-Witt theory spectra, Triangulated categories, Nori-Homotopy and its relationships with Chow-Witt obstructions for projective modules. In most cases, the proofs are improvisation of methods of Quillen [Q].The background, preliminaries and tools needed in chapters 2-11, are developed in chapters A on Category Theory and Exact Categories, B on Homotopy, C on CW Complexes, and 1 on Simplicial Sets.
