Constructive Commutative Algebra

Author : Ihsen Yengui
Publisher : Springer
Page : 277 pages
File Size : 43,9 Mb
Release : 2015-12-11
Category : Mathematics
ISBN : 9783319194943

Get Book

Constructive Commutative Algebra by Ihsen Yengui Pdf

The main goal of this book is to find the constructive content hidden in abstract proofs of concrete theorems in Commutative Algebra, especially in well-known theorems concerning projective modules over polynomial rings (mainly the Quillen-Suslin theorem) and syzygies of multivariate polynomials with coefficients in a valuation ring. Simple and constructive proofs of some results in the theory of projective modules over polynomial rings are also given, and light is cast upon recent progress on the Hermite ring and Gröbner ring conjectures. New conjectures on unimodular completion arising from our constructive approach to the unimodular completion problem are presented. Constructive algebra can be understood as a first preprocessing step for computer algebra that leads to the discovery of general algorithms, even if they are sometimes not efficient. From a logical point of view, the dynamical evaluation gives a constructive substitute for two highly nonconstructive tools of abstract algebra: the Law of Excluded Middle and Zorn's Lemma. For instance, these tools are required in order to construct the complete prime factorization of an ideal in a Dedekind ring, whereas the dynamical method reveals the computational content of this construction. These lecture notes follow this dynamical philosophy.

Author : Henri Lombardi,Claude Quitté
Publisher : Springer
Page : 996 pages
File Size : 50,8 Mb
Release : 2015-07-22
Category : Mathematics
ISBN : 9789401799447

Get Book

Commutative Algebra: Constructive Methods by Henri Lombardi,Claude Quitté Pdf

Translated from the popular French edition, this book offers a detailed introduction to various basic concepts, methods, principles, and results of commutative algebra. It takes a constructive viewpoint in commutative algebra and studies algorithmic approaches alongside several abstract classical theories. Indeed, it revisits these traditional topics with a new and simplifying manner, making the subject both accessible and innovative. The algorithmic aspects of such naturally abstract topics as Galois theory, Dedekind rings, Prüfer rings, finitely generated projective modules, dimension theory of commutative rings, and others in the current treatise, are all analysed in the spirit of the great developers of constructive algebra in the nineteenth century. This updated and revised edition contains over 350 well-arranged exercises, together with their helpful hints for solution. A basic knowledge of linear algebra, group theory, elementary number theory as well as the fundamentals of ring and module theory is required. Commutative Algebra: Constructive Methods will be useful for graduate students, and also researchers, instructors and theoretical computer scientists.

Author : David Eisenbud
Publisher : Springer Science & Business Media
Page : 784 pages
File Size : 41,7 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781461253501

Get Book

Commutative Algebra by David Eisenbud Pdf

This is a comprehensive review of commutative algebra, from localization and primary decomposition through dimension theory, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. The book gives a concise treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Many exercises included.

Author : David Eisenbud
Publisher : Unknown
Page : 808 pages
File Size : 46,9 Mb
Release : 2014-09-01
Category : Electronic
ISBN : 1461253519

Get Book

Commutative Algebra by David Eisenbud Pdf

Author : Ray Mines,Fred Richman,Wim Ruitenburg
Publisher : Springer Science & Business Media
Page : 355 pages
File Size : 48,8 Mb
Release : 2012-09-10
Category : Mathematics
ISBN : 9781441986405

Get Book

A Course in Constructive Algebra by Ray Mines,Fred Richman,Wim Ruitenburg Pdf

The constructive approach to mathematics has enjoyed a renaissance, caused in large part by the appearance of Errett Bishop's book Foundations of constr"uctiue analysis in 1967, and by the subtle influences of the proliferation of powerful computers. Bishop demonstrated that pure mathematics can be developed from a constructive point of view while maintaining a continuity with classical terminology and spirit; much more of classical mathematics was preserved than had been thought possible, and no classically false theorems resulted, as had been the case in other constructive schools such as intuitionism and Russian constructivism. The computers created a widespread awareness of the intuitive notion of an effecti ve procedure, and of computation in principle, in addi tion to stimulating the study of constructive algebra for actual implementation, and from the point of view of recursive function theory. In analysis, constructive problems arise instantly because we must start with the real numbers, and there is no finite procedure for deciding whether two given real numbers are equal or not (the real numbers are not discrete) . The main thrust of constructive mathematics was in the direction of analysis, although several mathematicians, including Kronecker and van der waerden, made important contributions to construc tive algebra. Heyting, working in intuitionistic algebra, concentrated on issues raised by considering algebraic structures over the real numbers, and so developed a handmaiden'of analysis rather than a theory of discrete algebraic structures.

Author : Krister Forsman
Publisher : Unknown
Page : 192 pages
File Size : 49,5 Mb
Release : 1991
Category : Commutative algebra
ISBN : 9178708273

Get Book

Constructive Commutative Algebra in Nonlinear Control Theory by Krister Forsman Pdf

Author : Wolmer Vasconcelos
Publisher : Springer Science & Business Media
Page : 432 pages
File Size : 42,7 Mb
Release : 2004-05-18
Category : Mathematics
ISBN : 3540213112

Get Book

Computational Methods in Commutative Algebra and Algebraic Geometry by Wolmer Vasconcelos Pdf

This ACM volume deals with tackling problems that can be represented by data structures which are essentially matrices with polynomial entries, mediated by the disciplines of commutative algebra and algebraic geometry. The discoveries stem from an interdisciplinary branch of research which has been growing steadily over the past decade. The author covers a wide range, from showing how to obtain deep heuristics in a computation of a ring, a module or a morphism, to developing means of solving nonlinear systems of equations - highlighting the use of advanced techniques to bring down the cost of computation. Although intended for advanced students and researchers with interests both in algebra and computation, many parts may be read by anyone with a basic abstract algebra course.

Author : Holger Brenner,Jürgen Herzog,Jurgen Herzog,Orlando E. Villamayor
Publisher : American Mathematical Soc.
Page : 190 pages
File Size : 41,9 Mb
Release : 2008
Category : Mathematics
ISBN : 9780821844342

Get Book

Three Lectures on Commutative Algebra by Holger Brenner,Jürgen Herzog,Jurgen Herzog,Orlando E. Villamayor Pdf

This book provides careful and detailed introductions to some of the latest advances in three significant areas of rapid development in commutative algebra and its applications. The book is based on courses at the Winter School on Commutative Algebra and Applications held in Barcelona: Tight closure and vector bundles, by H. Brenner; Combinatorics and commutative algebra, by J. Herzog; and Constructive desingularization, by O. Villamayor. The exposition is aimed at graduate students who have some experience with basic commutative algebra or algebraic geometry but may also serve as an introduction to these modern approaches for mathematicians already familiar with commutative algebra. This book is published in cooperation with Real Sociedad Matematica Espanola.

Author : Huishi Li
Publisher : World Scientific
Page : 198 pages
File Size : 41,5 Mb
Release : 2004
Category : Mathematics
ISBN : 9812389512

Get Book

An Introduction to Commutative Algebra by Huishi Li Pdf

- Contains many examples and problems (with hints) - Provides a good introduction for beginners in algebraic number theory and algebraic geometry

Author : Michael F. Atiyah,I.G. MacDonald
Publisher : CRC Press
Page : 140 pages
File Size : 40,8 Mb
Release : 2018-03-09
Category : Mathematics
ISBN : 9780429973260

Get Book

Introduction To Commutative Algebra by Michael F. Atiyah,I.G. MacDonald Pdf

First Published in 2018. This book grew out of a course of lectures given to third year undergraduates at Oxford University and it has the modest aim of producing a rapid introduction to the subject. It is designed to be read by students who have had a first elementary course in general algebra. On the other hand, it is not intended as a substitute for the more voluminous tracts such as Zariski-Samuel or Bourbaki. We have concentrated on certain central topics, and large areas, such as field theory, are not touched. In content we cover rather more ground than Northcott and our treatment is substantially different in that, following the modern trend, we put more emphasis on modules and localization.

Author : Andrea Ferretti
Publisher : American Mathematical Society
Page : 394 pages
File Size : 44,7 Mb
Release : 2023-09-26
Category : Mathematics
ISBN : 9781470471279

Get Book

Commutative Algebra by Andrea Ferretti Pdf

This book provides an introduction to classical methods in commutative algebra and their applications to number theory, algebraic geometry, and computational algebra. The use of number theory as a motivating theme throughout the book provides a rich and interesting context for the material covered. In addition, many results are reinterpreted from a geometric perspective, providing further insight and motivation for the study of commutative algebra. The content covers the classical theory of Noetherian rings, including primary decomposition and dimension theory, topological methods such as completions, computational techniques, local methods and multiplicity theory, as well as some topics of a more arithmetic nature, including the theory of Dedekind rings, lattice embeddings, and Witt vectors. Homological methods appear in the author's sequel, Homological Methods in Commutative Algebra. Overall, this book is an excellent resource for advanced undergraduates and beginning graduate students in algebra or number theory. It is also suitable for students in neighboring fields such as algebraic geometry who wish to develop a strong foundation in commutative algebra. Some parts of the book may be useful to supplement undergraduate courses in number theory, computational algebra or algebraic geometry. The clear and detailed presentation, the inclusion of computational techniques and arithmetic topics, and the numerous exercises make it a valuable addition to any library.

Author : O. Zariski,P. Samuel
Publisher : Springer Science & Business Media
Page : 433 pages
File Size : 44,9 Mb
Release : 1976-03-29
Category : Mathematics
ISBN : 9780387901718

Get Book

Commutative Algebra II by O. Zariski,P. Samuel Pdf

From the Preface: "topics are: (a) valuation theory; (b) theory of polynomial and power series rings (including generalizations to graded rings and modules); (c) local algebra... the algebro-geometric connections and applications of the purely algebraic material are constantly stressed and abundantly scattered throughout the exposition. Thus, this volume can be used in part as an introduction to some basic concepts and the arithmetic foundations of algebraic geometry."

Author : Nguyen Tu CUONG,Le Tuan HOA,Ngo Viet TRUNG
Publisher : Springer
Page : 258 pages
File Size : 55,5 Mb
Release : 2018-08-02
Category : Mathematics
ISBN : 9783319755656

Get Book

Commutative Algebra and its Interactions to Algebraic Geometry by Nguyen Tu CUONG,Le Tuan HOA,Ngo Viet TRUNG Pdf

This book presents four lectures on recent research in commutative algebra and its applications to algebraic geometry. Aimed at researchers and graduate students with an advanced background in algebra, these lectures were given during the Commutative Algebra program held at the Vietnam Institute of Advanced Study in Mathematics in the winter semester 2013 -2014. The first lecture is on Weyl algebras (certain rings of differential operators) and their D-modules, relating non-commutative and commutative algebra to algebraic geometry and analysis in a very appealing way. The second lecture concerns local systems, their homological origin, and applications to the classification of Artinian Gorenstein rings and the computation of their invariants. The third lecture is on the representation type of projective varieties and the classification of arithmetically Cohen -Macaulay bundles and Ulrich bundles. Related topics such as moduli spaces of sheaves, liaison theory, minimal resolutions, and Hilbert schemes of points are also covered. The last lecture addresses a classical problem: how many equations are needed to define an algebraic variety set-theoretically? It systematically covers (and improves) recent results for the case of toric varieties.

Author : Dilip P. Patil,Uwe Storch
Publisher : World Scientific
Page : 220 pages
File Size : 43,5 Mb
Release : 2010
Category : Mathematics
ISBN : 9789814304573

Get Book

Introduction to Algebraic Geometry and Commutative Algebra by Dilip P. Patil,Uwe Storch Pdf

Along the lines developed by Grothendieck, this book delves into the rich interplay between algebraic geometry and commutative algebra. With concise yet clear definitions and synopses a selection is made from the wealth of meterial in the disciplines including the Riemann-Roch theorem for arbitrary projective curves."--pub. desc.

Author : Luchezar L. Avramov,Mark Green,Craig Huneke,Karen E. Smith,Bernd Sturmfels
Publisher : Cambridge University Press
Page : 7 pages
File Size : 54,5 Mb
Release : 2004-12-13
Category : Mathematics
ISBN : 9780521831956

Get Book

Trends in Commutative Algebra by Luchezar L. Avramov,Mark Green,Craig Huneke,Karen E. Smith,Bernd Sturmfels Pdf

This book describes the interaction of commutative algebra with other areas of mathematics, including algebraic geometry, group cohomology, and combinatorics.