Book Description
Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry. The author presents a comprehensive view of commutative algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text. One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Applications of the theory and even suggestions for computer algebra projects are included. This book will appeal to readers from beginners to advanced students of commutative algebra or algebraic geometry. To help beginners, the essential ideals from algebraic geometry are treated from scratch. Appendices on homological algebra, multilinear algebra and several other useful topics help to make the book relatively self- contained. Novel results and presentations are scattered throughout the text.
Customer Reviews:
Not for the beginner.......2007-02-27
Well, the strength of this book lies in where it takes you. There is so much material here that when finished, you'll be prepared for a lot. Personally I think it is too wordy (my preferance is Atiyah & MacDonald) and the typesetting overall isn't all that impressive, so read up or consult other texts before/during your first encounter. M.Reids book is a better place to start.
Good book of reference.......2006-03-18
I purchased this as a book of reference. When I want to know something about Commutative Algebra (while reading Hartshorne's Algebraic Geometrry), I like a standard book of reference. But it seems a good book to learn commutative algebra aswell.
very good, but should be read slowly.......2004-09-25
Some proofs are somewhat abstract to the beginner. Although you are forced to check them on the paper, I think it is very good for the study. Also, you need a professor to instruct you, because in math, any language could only express the part of the oringins. Anyway, algebraic geometry is the course that you have to have a good professor to help you, otherwise stop study this field. In one word, it is a very very good book, so read it slowly!!!!!!
Superb.......2001-09-04
If one is interested in taking on a thorough study of algebraic geometry, this book is a perfect starting point. The writing is excellent, and the student will find many exercises that illustrate and extend the results in each chapter. Readers are expected to have an undergraduate background in algebra, and maybe some analysis and elementary notions from differential geometry. Space does not permit a thorough review here, so just a brief summary of the places where the author has done an exceptional job of explaining or motivating a particular concept:
(1) The history of commutative algebra and its connection with algebraic geometry, for example the origin of the concept of an "ideal" of a ring as generalizing unique factorization.
(2) The discussion of the concept of localization, especially its origins in geometry. A zero dimensional ring (collection of "points") is a ring whose primes are all maximal, as expected.
(3) The theory of prime decomposition as a generalization of unique prime factorization. Primary decomposition is given a nice geometric interpretation in the book.
(4) Five different proofs of the Nullstellensatz discussed, giving the reader good insight on this important result.
(5) The geometric interpretation of an associated graded ring corresponding to the exceptional set in the blowup algebra.
(6) The notion of flatness of a module as a continuity of fibers and a test for this using the Tor functor.
(7) The characterization of Hensel's lemma as a version of Newton's method for solving equations. The geometric interpretation of the completion as representing the properties of a variety in neighborhoods smaller than Zariski open neighborhoods.
(8) The characterization of dimension using the Hilbert polynomial.
(9) The fiber dimension and the proof of its upper semicontinuity.
(10) The discussion of Grobner bases and flat families. Nice examples are given of a flat family connecting a finite set of ideals to their initial ideals.
(11) Computer algebra projects for the reader using the software packages CoCoA and Macaulay.
(12) The theory of differentials in algebraic geometry as a generalization of what is done in differential geometry.
(13) The discussion of how to construct complexes using tensor products and mapping cones in order to study the Koszul complex.
(14) The connection of the Koszul complex to the cotangent bundle of projective space.
(15) The geometric interpretation of the Cohen-Macauley property as a map to a regular variety.
The standard text.......2000-07-28
This is often referred to as the standard text on commutative algebra.
It is an exceptionally good book on a subject that is normally difficult to get a handle on. Eisenbud's readable book gives intuitive and motivated proofs of even very technical results in commutative algebra, often illustrated with instructive examples, such as the useful figures illustrating embedded primes. A very nice feature is that he gives proofs to all the results in commutative algebra used by Robin Hartshorne's popular "Algebraic Geometry," making them a nice pair of books to read together.
I found this to be useful as a reference as well as a text. Most sections are fairly self-contained and many important topics are included in depth. I almost always find that it is the best place to learn any of the material covered.
This book belongs on the shelf of anyone learning algebraic geometry, although it will spend plenty of time off the shelf as well.
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Local Cohomology: An Algebraic Introduction with Geometric Applications (Cambridge Studies in Advanced Mathematics)
M. P. Brodmann , and
R. Y. Sharp
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ASIN: 0521372860 |
Book Description
This book provides a careful and detailed algebraic introduction to Grothendieck's local cohomology theory, and illustrates many applications for the theory in commutative algebra and in the geometry of quasi-affine and quasi-projective varieties. Topics covered include Castelnuovo-Mumford regularity, the Fulton-Hansen connectedness theorem for projective varieties, and connections between local cohomology and both reductions of ideals and sheaf cohomology. It is designed for graduate students who have some experience of basic commutative algebra and homological algebra, and also for experts in commutative algebra and algebraic geometry.
Average customer rating:
- very comprehensive treatment on the subject
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Cyclic Homology (Grundlehren der mathematischen Wissenschaften)
Jean-Louis Loday
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A User's Guide to Spectral Sequences (Cambridge Studies in Advanced Mathematics)
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Counterexamples in Topology
ASIN: 3540630740 |
Book Description
From the reviews: "This is a very interesting book containing material for a comprehensive study of the cyclid homological theory of algebras, cyclic sets and S1-spaces. Lie algebras and algebraic K-theory and an introduction to Connes'work and recent results on the Novikov conjecture. The book requires a knowledge of homological algebra and Lie algebra theory as well as basic technics coming from algebraic topology. The bibliographic comments at the end of each chapter offer good suggestions for further reading and research. The book can be strongly recommended to anybody interested in noncommutative geometry, contemporary algebraic topology and related topics." European Mathematical Society Newsletter
In this second edition the authors have added a chapter 13 on MacLane (co)homology.
Customer Reviews:
very comprehensive treatment on the subject.......2004-02-21
I think that this is probably the only comprehensive book on the subject. There are several lecture notes published, but no other books have collected all up to date recent research results on the subject like this book did. Probably must-read classical book if you are interested in cyclic homology, or if you want to use it for you research. Very unfortunate fact is that this book is very rare, so that very hard to obtain. You may have lots of trouble buying this book.
Book Description
Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory. The algorithms to answer questions such as those posed above are an important part of algebraic geometry. This book bases its discussion of algorithms on a generalization of the division algorithm for polynomials in one variable that was only discovered in the 1960's. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century. This has changed in recent years, and new algorithms, coupled with the power of fast computers, have let to some interesting applications, for example in robotics and in geometric theorem proving. In preparing a new edition of
Ideals, Varieties and
Algorithms the authors present an improved proof of the Buchberger Criterion as well as a proof of Bezout's Theorem. Appendix C contains a new section on Axiom and an update about Maple , Mathematica and REDUCE.
Customer Reviews:
Symbolic computation.......2003-08-29
This book explains and illustrates the algorithms used by symbolic math packages such as Mathematica, Maple, CoCoA, MatLab, MuPAD,... to solve problems involving polynomials in many variables, and along the way teaches the elements of real algebraic geometry-- most mathematics texts concentrate on the complex-variable version. It is not just for undergraduates; electrical engineers, for instance, should see it. Lots of pictures!
Easiest introduction to Algebraic Geometry.......2003-04-23
This is the easiest introduction to algebraic geometry and commutative algebra, the authors had done a great job in writing a book that assume very little from the readers. To learn some algebraic geometry, you can either start with this book, or you can spend a year to read a lot of background materials in algebra and then go to a Graduate Text like Harris' book. Of course, if you want to be an expert in algebra, you eventually need a lot of background, what this book can help you is to offer you a quick start, much quicker than you would ever imagine.
Straightforward and lucidly written.......2002-04-09
Having just finished using this text in the course of an undergraduate seminar, I can attest to the fact that the authors' style is outstanding - they are able to synthesize an enormous amount of material in this volume and present it in a manner that is highly accessible to almost all students of mathematics. The presentation of important theorems (for example, Hilbert's Nullstellensatz and Basis Theorem) along with just the right amount of copncrete examples makes for a book of superb quality. All-around, I highly recommend this volume to anyone who has an interest in learning about Algebraic Geometry.
Good book.......2001-05-27
I don't have the second edition of this book but did read the first, and the authors do a fine job of introducing the reader to the computational side of algebraic geometry. I will forego a chapter by chapter review therefore, but no doubt the second edition (which I do not own) is as well-written as the first. I would recommend it to anyone interested in the many applications of algebraic geometry and to those who need to understand how to compute things in algebraic geometry. The good thing about this book is that it gives a concrete flavor to a highly abstract subject. Algebraic geometry, through its applications to coding theory, cryptography, and computer graphics, is fast becoming the subject to learn. It is no longer just an esoteric, high-brow subject but one that is taking on major importance in the information age. Even without applications though it is a fascinating subject, and readers will get a taste of this in this book.
The best book on the topic.......2001-01-26
I learned the basics of Groebner bases from this book and its the best introductory book on this topic. Authors have explained all concepts with the help of examples which makes it readable for people from other fields also. It also talks about applications of Groebner bases to other fields. The book gives lot of exercises which help in understanding the contents more. I recommend that if you wish to learn Algebraic Geometry and Groebner bases then this is the book to start with.
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Lie Algebras and Algebraic Groups (Springer Monographs in Mathematics)
Patrice Tauvel , and
Rupert W.T. Yu
Manufacturer: Springer
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Elliptic Curves
ASIN: 3540241701 |
Book Description
The theory of Lie algebras and algebraic groups has been an area of active research for the last 50 years. It intervenes in many different areas of mathematics: for example invariant theory, Poisson geometry, harmonic analysis, mathematical physics. The aim of this book is to assemble in a single volume the algebraic aspects of the theory, so as to present the foundations of the theory in characteristic zero. Detailed proofs are included and some recent results are discussed in the final chapters. All the prerequisites on commutative algebra and algebraic geometry are included.
Book Description
Combinatorial commutative algebra is an active area of research with thriving connections to other fields of pure and applied mathematics. This book provides a self-contained introduction to the subject, with an emphasis on combinatorial techniques for multigraded polynomial rings, semigroup algebras, and determinantal rings. The eighteen chapters cover a broad spectrum of topics, ranging from homological invariants of monomial ideals and their polyhedral resolutions, to hands-on tools for studying algebraic varieties with group actions, such as toric varieties, flag varieties, quiver loci, and Hilbert schemes. Over 100 figures, 250 exercises, and pointers to the literature make this book appealing to both graduate students and researchers.
Customer Reviews:
Coherent and Useful.......2006-07-25
This book does a good job at introducing some of the most interesting combinatorial problems in monomial resolutions. Although their chapter on toric varieties is somewhat weak, most of the other chapters (square-free monomial ideals, Borel-fixed ideals, staircases associated to monomial ideals in three variables, etc.) are very well written with challenging and revealing exercises at the end.
Book Description
Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated?
The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory.
The algorithms to answer questions such as those posed above are an important part of algebraic geometry. Although the algorithmic roots of algebraic geometry are old, it is only in the last forty years that computational methods have regained their earlier prominence. New algorithms, coupled with the power of fast computers, have led to both theoretical advances and interesting applications, for example in robotics and in geometric theorem proving.
In addition to enhancing the text of the second edition, with over 200 pages reflecting changes to enhance clarity and correctness, this third edition of Ideals, Varieties and Algorithms includes: A significantly updated section on Maple in Appendix C; Updated information on AXIOM, CoCoA, Macaulay 2, Magma, Mathematica and SINGULAR; A shorter proof of the Extension Theorem presented in Section 6 of Chapter 3.
From the 2nd edition: "I consider the book to be wonderful. ... The exposition is very clear, there are many helpful pictures, and there are a great many instructive exercises, some quite challenging ... offers the heart and soul of modern commutative and algebraic geometry." The American Mathematical Monthly
Book Description
This is an English translation of the now classic "Algèbre Locale - Multiplicités" originally published by Springer as LNM 11, in several editions since 1965. It gives a short account of the main theorems of commutative algebra, with emphasis on modules, homological methods and intersection multiplicities ("Tor-formula"). Many modifications to the original French text have been made by the author for this English edition: they make the text easier to read, without changing its intended informal character.
Customer Reviews:
Serre as usual.......2005-11-02
This book is crystal clear and reads like a bedtime story. I wish I had it when I took commutative algebra.
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Computations in Algebraic Geometry with Macaulay 2
Bernd Sturmfels
Manufacturer: Springer
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The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra (Graduate Texts in Mathematics)
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Combinatorial Commutative Algebra (Graduate Texts in Mathematics)
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The Geometry of Schemes
ASIN: 3540422307 |
Book Description
This book presents algorithmic tools for algebraic geometry and experimental applications of them. It also introduces a software system in which the tools have been implemented and with which the experiments can be carried out. Macaulay 2 is a computer algebra system devoted to supporting research in algebraic geometry, commutative algebra, and their applications. The reader of this book will encounter Macaulay 2 in the context of concrete applications and practical computations in algebraic geometry. The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems. These expositions will be valuable to both the users of other programs similar to Macaulay 2 (for example, Singular and CoCoA) and those who are not interested in explicit machine computations at all. The first part of the book is primarily concerned with introducing Macaulay2, whereas the second part emphasizes the mathematics.
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Free Resolutions in Commutative Algebra and Algebraic Geometry: Sundance Ninety (Research Notes in Mathematics, 2) (Research Notes in Mathematics)
David Eisenbud
Manufacturer: AK Peters
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Binding: Paperback
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ASIN: 0867202858 |
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The selected contributions in this volume originated at the Sundance conference, which was devoted to discussions of current work in the area of free resolutions. The papers include new research, not otherwise published, and expositions that develop current problems likely to influence future developments in the field.
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