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# Resolution of Curve and Surface Singularities in Characteristic Zero

Resolution of Curve and Surface Singularities in Characteristic Zero by K. Kiyek , J. L. Vicente
English | PDF | 2004 | 506 Pages | ISBN : 1402020287 | 50.3 MB

The Curves The Point of View of Max Noether Probably the oldest references to the problem of resolution of singularities are found in Max Noether's works on plane curves [cf. , ]. And probably the origin of the problem was to have a formula to compute the genus of a plane curve. The genus is the most useful birational invariant of a curve in classical projective geometry. It was long known that, for a plane curve of degree n having l m ordinary singular points with respective multiplicities ri, i E {1, . . . , m}, the genus p of the curve is given by the formula = (n - l)(n - 2) _ ~ "r. (r. _ 1) P 2 2 L. . ,. •• . Of course, the problem now arises: how to compute the genus of a plane curve having some non-ordinary singularities. This leads to the natural question: can we birationally transform any (singular) plane curve into another one having only ordinary singularities? The answer is positive. Let us give a flavor (without proofs) 2 on how Noether did it • To solve the problem, it is enough to consider a special kind of Cremona trans­ formations, namely quadratic transformations of the projective plane. Let ~ be a linear system of conics with three non-collinear base points r = {Ao, AI, A }, 2 and take a projective frame of the type {Ao, AI, A ; U}.

# Groups, Matrices, and Vector Spaces: A Group Theoretic Approach to Linear Algebra (Repost)

Groups, Matrices, and Vector Spaces: A Group Theoretic Approach to Linear Algebra by James B. Carrell
English | PDF | 2017 | 414 Pages | ISBN : 0387794271 | 4.3 MB

This unique text provides a geometric approach to group theory and linear algebra, bringing to light the interesting ways in which these subjects interact. Requiring few prerequisites beyond understanding the notion of a proof, the text aims to give students a strong foundation in both geometry and algebra. Starting with preliminaries (relations, elementary combinatorics, and induction), the book then proceeds to the core topics: the elements of the theory of groups and fields (Lagrange's Theorem, cosets, the complex numbers and the prime fields), matrix theory and matrix groups, determinants, vector spaces, linear mappings, eigentheory and diagonalization, Jordan decomposition and normal form, normal matrices, and quadratic forms. The final two chapters consist of a more intensive look at group theory, emphasizing orbit stabilizer methods, and an introduction to linear algebraic groups, which enriches the notion of a matrix group.

# Commutative Algebra: Noetherian and Non-Noetherian Perspectives

Commutative Algebra: Noetherian and Non-Noetherian Perspectives by Marco Fontana, Salah-Eddine Kabbaj, Bruce Olberding, Irena Swanson
English | PDF(True) | 2011 | 490 Pages | ISBN : 1441969896 | 4.5 MB

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.

# The Use of Ultraproducts in Commutative Algebra (Repost)

The Use of Ultraproducts in Commutative Algebra by Hans Schoutens
English | PDF | 2010 | 215 Pages | ISBN : 3642133673 | 2.6 MB

In spite of some recent applications of ultraproducts in algebra, they remain largely unknown to commutative algebraists, in part because they do not preserve basic properties such as Noetherianity. This work wants to make a strong case against these prejudices. More precisely, it studies ultraproducts of Noetherian local rings from a purely algebraic perspective, as well as how they can be used to transfer results between the positive and zero characteristics, to derive uniform bounds, to define tight closure in characteristic zero, and to prove asymptotic versions of homological conjectures in mixed characteristic. Some of these results are obtained using variants called chromatic products, which are often even Noetherian. This book, neither assuming nor using any logical formalism, is intended for algebraists and geometers, in the hope of popularizing ultraproducts and their applications in algebra.

# Commutative Algebra: Volume II

Commutative Algebra: Volume II by Oscar Zariski , Pierre Samuel
English | PDF | 1960 | 425 Pages | ISBN : 3662277530 | 41.9 MB

This second volume of our treatise on commutative algebra deals largely with three basic topics, which go beyond the more or less classical material of volume I and are on the whole of a more advanced nature and a more recent vintage. These topics are: (a) valuation theory; (b) theory of polynomial and power series rings (including generalizations to graded rings and modules); © local algebra. Because most of these topics have either their source or their best motivation in algebraic geom­ etry, the algebro-geometric connections and applications of the purely algebraic material are constantly stressed and abundantly scattered through­ out the exposition.

# Combinatorial Aspects of Commutative Algebra and Algebraic Geometry: The Abel Symposium 2009 (Repost)

Combinatorial Aspects of Commutative Algebra and Algebraic Geometry: The Abel Symposium 2009 by Gunnar Fløystad, Trygve Johnsen, Andreas Leopold Knutsen
English | PDF | 2011 | 186 Pages | ISBN : 3642194915 | 2.6 MB

The Abel Symposium 2009 "Combinatorial aspects of Commutative Algebra and Algebraic Geometry", held at Voss, Norway, featured talks by leading researchers in the field.

# Approximation Theorems in Commutative Algebra: Classical and Categorical Methods

Approximation Theorems in Commutative Algebra: Classical and Categorical Methods by J. Alajbegović , J. Močkoř
English | PDF | 1992 | 339 Pages | ISBN : 0792319486 | 43.3 MB

Various types of approximation theorems are frequently used in general commutative algebra, and they have been found to be useful tools in valuation theory, the theory of Abelian lattice ordered groups, multiplicative ideal theory, etc.
Part 1 of this volume is devoted to the investigation of approximation theorems from a classical point of view. The chapters of this part deal with fields and rings, partly ordered groups, and with multirings and d-groups.

# Ideals, Varieties, and Algorithms

Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra by David A. Cox , John Little , Donal O’Shea
English | PDF(True) | 2015 | 653 Pages | ISBN : 3319167200 | 9.6 MB

This text covers topics in algebraic geometry and commutative algebra with a strong perspective toward practical and computational aspects. The first four chapters form the core of the book. A comprehensive chart in the Preface illustrates a variety of ways to proceed with the material once these chapters are covered. In addition to the fundamentals of algebraic geometry—the elimination theorem, the extension theorem, the closure theorem and the Nullstellensatz—this new edition incorporates several substantial changes, all of which are listed in the Preface. The largest revision incorporates a new Chapter (ten), which presents some of the essentials of progress made over the last decades in computing Gröbner bases. The book also includes current computer algebra material in Appendix C and updated independent projects (Appendix D).

# Universal Algebra

Universal Algebra by George Grätzer
English | PDF | 2008, Second Edition with updates, 1979 Second Edition | 601 Pages | ISBN : 0387774866 | 53.2 MB

Universal Algebra heralded as ". . . the standard reference in a field notorious for the lack of standardization . . .", has become the most authoritative, consistently relied on text in a field with applications in other branches of algebra and other fields such as combinatorics, geometry, and computer science.

# Classically Semisimple Rings: A Perspective Through Modules and Categories

Classically Semisimple Rings: A Perspective Through Modules and Categories by Martin Mathieu
English | PDF,EPUB | 2022 (2023 Edition) | 159 Pages | ISBN : 303114208X | 8.3 MB

Classically Semisimple Rings is a textbook on rings, modules and categories, aimed at advanced undergraduate and beginning graduate students.

# Numerical Semigroups and Applications (Repost)

Numerical Semigroups and Applications by Abdallah Assi, Pedro A. García-Sánchez
English | PDF | 2016 | 113 Pages | ISBN : 3319823256 | 1.7 MB

This work presents applications of numerical semigroups in Algebraic Geometry, Number Theory, and Coding Theory. Background on numerical semigroups is presented in the first two chapters, which introduce basic notation and fundamental concepts and irreducible numerical semigroups.

# C^\infinity - Differentiable Spaces (Repost)

C^\infinity - Differentiable Spaces by Juan A. Navarro González, Juan B. Sancho de Salas
English | PDF | 2003 | 191 Pages | ISBN : 354020072X | 2.8 MB

The volume develops the foundations of differential geometry so as to include finite-dimensional spaces with singularities and nilpotent functions, at the same level as is standard in the elementary theory of schemes and analytic spaces. The theory of differentiable spaces is developed to the point of providing a handy tool including arbitrary base changes (hence fibred products, intersections and fibres of morphisms), infinitesimal neighbourhoods, sheaves of relative differentials, quotients by actions of compact Lie groups and a theory of sheaves of Fréchet modules paralleling the useful theory of quasi-coherent sheaves on schemes. These notes fit naturally in the theory of C^\infinity-rings and C^\infinity-schemes, as well as in the framework of Spallek’s C^\infinity-standard differentiable spaces, and they require a certain familiarity with commutative algebra, sheaf theory, rings of differentiable functions and Fréchet spaces.

# Modular Invariant Theory

Modular Invariant Theory by H.E.A. Eddy Campbell, David L. Wehlau
English | PDF(True) | 2011 | 233 Pages | ISBN : 3642174035 | 2.75 MB

This book covers the modular invariant theory of finite groups, the case when the characteristic of the field divides the order of the group. It explains a theory that is more complicated than the study of the classical non-modular case, and it describes many open questions.

# Rings, Monoids and Module Theory: AUS-ICMS 2020, Sharjah, United Arab Emirates, February 6–9

Rings, Monoids and Module Theory: AUS-ICMS 2020, Sharjah, United Arab Emirates, February 6–9 by Ayman Badawi, Jim Coykendall
English | EPUB | 2022 | 317 Pages | ISBN : 9811684219 | 28 MB

This book contains select papers on rings, monoids and module theory which are presented at the 3rd International Conference on Mathematics and Statistics (AUS-ICMS 2020) held at the American University of Sharjah, United Arab Emirates, from 6–9 February 2020. This conference was held in honour of the work of the distinguished algebraist Daniel D. Anderson.

# Algebra, Revised Third Edition

Algebra, Revised Third Edition by Serge Lang
English | PDF | 2002 | 923 Pages | ISBN : 038795385X | 66.1 MB

This book is intended as a basic text for a one-year course in Algebra at the graduate level, or as a useful reference for mathematicians and professionals who use higher-level algebra. It successfully addresses the basic concepts of algebra. For the revised third edition, the author has added exercises and made numerous corrections to the text.