Serre's Problem on Projective Modules

Serre's Problem on Projective Modules
Author: T.Y. Lam
Publisher: Springer Science & Business Media
Total Pages: 412
Release: 2010-05-17
Genre: Mathematics
ISBN: 3540345752

An invaluable summary of research work done in the period from 1978 to the present

Ideals and Reality

Ideals and Reality
Author: Friedrich Ischebeck
Publisher: Springer Science & Business Media
Total Pages: 339
Release: 2005-11-22
Genre: Mathematics
ISBN: 3540263705

Besides 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,. . .

Projective Modules and Complete Intersections

Projective Modules and Complete Intersections
Author: Satya Mandal
Publisher: Springer
Total Pages: 121
Release: 2006-11-14
Genre: Mathematics
ISBN: 3540695982

In 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.

Serre's Conjecture

Serre's Conjecture
Author: T.Y. Lam
Publisher: Springer
Total Pages: 240
Release: 2006-11-15
Genre: Mathematics
ISBN: 3540359265

From the Preface: "I felt it would be useful for graduate students to see a detailed account of the sequence of mathematical developments which was inspired by the Conjecture, and which ultimately led to its full solution.... I offered a course on Serre's Conjecture to a small group of graduate students in January, 1977 [at the University of California, Berkeley] one year after its solution by Quillen and Suslin. My course was taught very much in the spirit of a mathematical 'guided tour'. Volunteering as the guide, I took upon myself the task of charting a route through all the beautiful mathematics surrounding the main problem to be treated; the 'guide' then leads his audience through the route, on to the destination, pointing out the beautiful sceneries and historical landmarks along the way."

Algebra, Mathematical Logic, Number Theory, Topology

Algebra, Mathematical Logic, Number Theory, Topology
Author: Ivan Matveevich Vinogradov
Publisher: American Mathematical Soc.
Total Pages: 284
Release: 1986
Genre: Algebra
ISBN: 9780821830963

Collection of papers on the current research in algebra, mathematical logic, number theory and topology.

Leavitt Path Algebras and Classical K-Theory

Leavitt Path Algebras and Classical K-Theory
Author: A. A. Ambily
Publisher: Springer Nature
Total Pages: 340
Release: 2020-01-17
Genre: Mathematics
ISBN: 9811516111

The 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.

Integral Closure of Ideals, Rings, and Modules

Integral Closure of Ideals, Rings, and Modules
Author: Craig Huneke
Publisher: Cambridge University Press
Total Pages: 446
Release: 2006-10-12
Genre: Mathematics
ISBN: 0521688604

Ideal for graduate students and researchers, this book presents a unified treatment of the central notions of integral closure.

Abelian l-Adic Representations and Elliptic Curves

Abelian l-Adic Representations and Elliptic Curves
Author: Jean-Pierre Serre
Publisher: CRC Press
Total Pages: 203
Release: 1997-11-15
Genre: Mathematics
ISBN: 1439863865

This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on the Taniyama-Weil conjecture and Fermat's Last Theorem. The initial chapters are devoted to the Abelian case (complex multiplication), where one

Ring Theory, Waterloo 1978

Ring Theory, Waterloo 1978
Author: D. Handelman
Publisher: Springer
Total Pages: 361
Release: 2007-01-05
Genre: Mathematics
ISBN: 3540350438

A ring theory conference took place at the University of Waterloo, 12-16 June 1978, and these are its proceedings. This conference was held as a part of the Summer Research Institute in Ring Theory, at Waterloo, sponsored by the Canadian Mathematical Society. In soliciting speakers, and contributors to the Proceedings, we attempted to represent those portions of ring theory which seemed to us interesting. There was thus considerable emphasis on lower K-theory and related topics, Artinian and Noetherian rings, as well as actions and representations of groups on rings. Regrettably, we could only obtain one paper in the mainstream of commutative ring theory, but we believe that the lack of quantity is more than made up for by the quality. We also took the liberty of including a survey of results in a field which we feel deserves more attention by ring theorists, C* algebras from an algebraic point of view.

Applicable Mathematics in the Golden Age

Applicable Mathematics in the Golden Age
Author: J. C. Misra
Publisher: Alpha Science Int'l Ltd.
Total Pages: 528
Release: 2003
Genre: Mathematics
ISBN: 9788173194870

This comprehensive volume introduces educational units dealing with the various emerging branches of Applicable Mathematics. It consists of chapters that deal with the major areas of pure mathematics and emphasises the cross-field application of the skills conveyed within.