|
|
|
 |
|
国外优秀数学教材选评
最新更新：September 30, 2009
本书内容为原作者版权所有，未经协议授权，禁止下载使用
主 编 杨劲根
副主编 楼红卫 李振钱 郝群
编写人员（按汉语拼音为序）
陈超群 陈猛 东瑜昕 高威 郝群 刘东弟 吕志 童裕孙 王巨平 王泽军 徐晓津 杨劲根 应坚刚 张锦豪 张永前 周子翔 朱胜林
1. 序言
2. 非数学专业的数学教材
3. 数学分析和泛函分析
4. 单复变函数
5. 多复变函数
6. 代数
7.数论
8.代数几何
9.拓扑与微分几何
10.偏微分方程
11.概率论
12.计算数学
13.其他
14.附录
6 代数
数学系本科生课程中，线性代数是最重要的，一般要学一年，其次是抽象代数。一般这些代数知识还不够，在研究生阶段又要学一些更加专门的代数，如群表示、李代数、交换代数、同调代数、环论等。
数学系使用的线性代数教材和非数学专业的教材有很大的不同，前者注重定理的证明，习题中证明题的比重很大，而后者注重计算，习题也以计算题为主，因此非数学专业的学生如果使用数学专业的线性代数教材有可能会不适应。
在此，我们介绍了几本比较基本的优秀代数教材，大多是本科生或低年级研究生使用的。
书名：Linear Algebra, 2nd ed.
作者：K. Hoffmann and R. Kunze
出版商：Prentice Hall, Inc. (1971)
页数：407
适用范围：大学数学系本科低年级教材
预备知识：微积分
习题数量：600 多道习题
习题难度：各种难度都有
推荐强度：9
使用学校：
Central Michigan University，University of North Dakota，Indian Institute of Technology, Bombay，University of Pittsburgh，University of Texas，Johns Hopkins University，West Virginia University，University of Houston，Simon Fraser University，Washington University in St. Louis，University of Notre Dame，University of Wisconsin-Madison，Cornell University，University of South Carolina，University of Rhode Island，University of Missouri，University of Maryland，Stony Brook University，University of Michigan，Purdue University，University of Kansas，United Arab Emirates University，Rice University，Kenyon College，Temple University，Louisiana State University，Sonoma State University，North Carolina State University，University of Iowa
书评： 这是一本线性代数方面久负盛名的教材，在麻省理工学院作为大三课程的教材使用多年，第一版于 1961年出版，对此后世界各国编写的各种线性代数教材有很大影响。比方说，北大和复旦的线性代数教材的内容和编排次序与此书非常接近。
虽然作者在序言中声称为了照顾数学基础不是特别扎实的工科学生把很多地方写的浅一些，但是据我看这本教材只适合数学系或者对线性代数基础要求很高的专业用。
本书内容完整，线性代数的所有内容都有了，甚至多重线性代数的若干内容如外积也包含了。从第一章开始就定下一个基调：本书讲解的线性代数是在任意域上的。在叙述了域的定义后，先给读者一颗定心丸：但是几乎所有例子和习题是在数域上的。接下来马上又警告大家：奇怪的域是
有的，有限多个 1 相加可以等于 0. 这样的安排虽然对学生的要求比较高，我觉得还是利大于弊，可以一步到位了。另一个类似的情形出现在第五章讲行列式时一上来就定义任意交换环上的行列式。我国的教材大多把行列式定义为一个数，然而在例题和习题中很多行列式中含变元，使人觉得不很严格。提起行列式， Strang 教授在 MIT 的视频课程给我深刻印象，他并不先讲行列式的定义，而是先将它需要满足的几个关键性的性质，最后推出只存在唯一的方式来定义行列式满足这些关键性质。本书基本上也是按照这样的方式处理的，虽然费力一些，但把它的本质讲的比较透彻，这比急功近利地急急忙忙讲行列式的计算技巧深刻。更进一步，接下去作者马上讲交换环上模的概念以及交错型和外积，虽然显得有些过火，但这样的安排还是合情合理的，因为这些内容和行列式的关系太密切了。也许作为课堂教学这些内容只能跳过，否则一个学年讲不完这本书。
后半部分的内容是线性变换的各种标准形、内积空间以及它上的线性算子，比较常规。大量习题和这些习题的覆盖度大也是本书的一个特色。（杨劲根）
国外评论摘选
i) I got this book for my Linear Algebra class about four years ago. This is a great book if you are getting a degree in mathematics. It won't help if you are just trying to get by the class and don't like math. It is not very practical but if you are looking for a real math book on Linear Algebra this is it. It contains a wealth of theorems that only a math lover would appreciate. If you really want to learn about Linear Algebra from a rigorous mathematical point of view this is it. This book taught me so much.
ii) This was the textbook they used to use at MIT in the past few decades. Virtually, however, nobody uses this book in a regular undergraduate course anymore. Instead of developing the ideas in the familiar context of the real numbers, Hoffman and Kunze give a more abstract (and general) discussion. For example, the theorems about determinants work in all commutative rings. The rigorousness and the wealth of information are overwhelming for most undergraduates to handle. You will not learn anything if you just glance through the pages. Every line requires deep thought. Down-to-earth applications are not included. So I do not recommend this book for engineers.
书名：Lectures on Linear Algebra
作者：Gelfand
出版商：INTERSCIENCE
PUBLISHERS. INC. （60 年代）
页数：185
适用范围：大学数学系本科低年级教材
预备知识：微积分
习题数量：少
习题难度：比较大
推荐强度：8.5
书评：前苏联有一些数学大师写过一些给大学本科用的数学教材，就象我国华罗庚先生写过《高等数学引论》一样。作为泛函分析祖师爷级别的人物 Gelfand 写的线性代数讲义也不失为一本极好的教材。
本书是学院式的，结构和叙述十分严格和简洁，具有 Bourbaki 的风格，但又不象 Bourbaki 那样地追求一般性，所以不要求读者有很高的起点。不象一般的线性代数教材从解线性方程组或行列式开始，本书从 n 维内积空间作为第一章，甚至包括复内积空间，可谓开门见山。第二章讨论线性变换，特别是正交变换和酉变换。第三章是若当标准型，第四章是多重线性代数。
本书原文是俄文的，在 60 年代有多种文字的译本，包括中文本，由于是繁体字，现在已打入冷宫了。我在大学刚毕业不久（1971年）阅读此书得益非浅，感觉对线性代数乃至泛函分析有更深的认识，这是我把这本老书向读者介绍的主要原因。本人认为此书层次较高，适合于已经有一些初等线性代数知识的人读，它不一定对提高解题能力有利， 但对提高数学观点是绝对有益的。（杨劲根）
国外评论摘选
1) The professor who recommended this book made the comment that every time you re-read it, you notice something else that you missed the last time you read it. This is absolutely true.
I must say, the first time I picked up this book, I did not like it. The notation was not what I was used to, and the book dives right in, assuming a lot of background (matrices, determinants, etc.) but covering material which many people find boring (bases, etc.). However, when you read deeper, there's a lot here. Once you get past the ugly notation, the proofs are extraordinarily clear. And in spite of the books small size, there is a remarkable amount of motivation and discussion.
Like the other reviewer said, this is not a book to learn linear algebra from for the first time: this is an advanced book that is useful for graduate students who have already had a linear algebra course and who want to learn more topics, or understand topics on a deeper level.
This is an excellent book; the bottom line is that it's so cheap that there's no excuse NOT to buy it.
2) This is the best treatment of linear algebra that has been published. It starts with n-dimensional linear spaces and ends with an introduction to tensors. An excellent description of dual spaces is concisely presented. NO INDEX!
3) Lucid and clear notation , complete explanations . This books was first published in 1937 but until now it remains best text book in the field .
4) This is a good book if all you need is a condensed reference on theorems and proofs and it assumes that you go for practice (and instruction) elsewhere. If you are trying to actually learn linear algebra (especially on your own and especially if you want to learn how to solve practical problems) get one of Gilbert Strang's books and watch his video lectures at MIT web site. Another thing that I dislike about the Gelfand's book is that it puts too much emphasis on index notation - instead of matrix notation which is natural for linear algebra, almost all formulas and theorems are presented at very low level using expressions consisting of variables with multiple indices. Naturally it gets very messy and hard to follow at times. This doesn't present any more information than equivalent matrix notation but introduces unnecessary complexity and makes things that are really easy to understand very confusing.
书名：Linear Algebra Gems
作者： David Carlson, Charles R. Johnson,David C. Lay, A. Duane Porter
出版商： The Mathematical Association of America (2000左右)
页数：328
适用范围：大学数学系本科低年级参考读物
预备知识：微积分、线性代数
习题数量：123 题
习题难度：大
推荐强度：8.5
书评： 这本书不是线性代数的教材，而是兴趣浓厚的学生或教线性代数的老师的参考读物，同类 的书并不多见。美国最大的数学组织是美国数学会（AMS）, 其次就是美国数学协会（MAA）, 它 的主要目标是推动数学教学，尤其是大学本科的数学教学，和 AMS 一样，它也有不少出版物， 其中最主要的是美国数学月刊，简称 Monthly, 是一份历史悠久并且享有盛名的数学教育刊物， 上面的文章质量高于我国的数学通报，另有一个刊物 College Mathematics Journal，我国数学界不太熟悉。 本书是从多年的 Monthly 和 College Mathematics Journal 中选出几十篇与线性代数有关 的短文，又约稿请人写了若干文章，总共74篇按内容进行分类而构成的。大部分文章是教学心得和 若干有名的定理（比如若当标准型）和习题的进一步探讨。各篇文章互相独立，每篇文章一般在一个或半个小时内读完， 非常适合于充当大学生课外读物，特别是对大学生数学竞赛很有帮助。
内容共分十部分如下
PART 1 - PARTITIONED MATRIX MULTIPLICATION
PART 2 - DETERMINANTS
PART 3 - EIGENANALYSIS
PART 4 - GEOMETRY
PART 5 - MATRIX FORMS
PART 6 - POLYNOMIALS AND MATRICES
PART 7 - LINEAR SYSTEMS INVERSES AND RANK
PART 8 - APPLICATIONS
PART 9 - OTHER TOPICS
PART 10- PROBLEMS
象第 1，2，5，6，7 部分一看就知道有不少有技巧性的内容。第十部分是习题，大部分是竞赛级别的题。 （杨劲根）
书名：Algebra
作者： Michael Artin
出版商： Prentice Hall (1991), 机械工业出版社影印
页数：618
适用范围：大学数学系本科基础数学一学年的教材
预备知识：微积分和线性代数
习题数量：大
习题难度： 各种难度都有
推荐强度：9.8
书评： 本书是美国大数学家美国科学院院士 Michael Artin 的力作，从70年代早期开始就作为麻省理工学院数学系高年级 本科生教材，是一本极具特色的优秀教材，深受使用者欢迎。 与传统的抽象代数教材不同，本书以数学中的重要实例为主线索，引导出抽象的概念，对读者以启发为主，又不缺乏数学的严格性。虽然教材的主要内容是基本的代数结构，但字里行间不乏现代数学的烙印。代数数论、代数几何、表示论中的一些基本思想 也时时涌现，如整二次型的原理和应用、二次域的理想类、不定方程、紧群表示等。 全书分14章，从矩阵运算引入群概念直到最后一章伽罗华理论一气呵成，不使人感觉600多页篇幅的冗长。 本书的习题是作者20多年积累而得，很多是作者独创的习题，例如有一道2x2魔方的问题是70年代3x3魔方游戏刚问世时作者 编制的群论习题。大约有四分之一的习题有一定难度。 本人80年代在 MIT 攻读研究生期间为此课程作过多次助教，主讲人为作者本人或其他资深教授，每次大约有三十人修课，主要 学生是基础数学各专业的学生，也有一些计算机专业的本科生及研究生选修的。学生反映此课程质量很高，但比较难。
本书比较适合我国综合性大学数学系抽象代数课程的外文教材，尤其适合一学年。对于半年的抽象代数课程，则可选用部分章节。 程度较好的数学系本科生可选用此书作为抽象代数的课外读物。 （杨劲根）
国外评论摘选
1) Pretty much any introductory abstract algebra book on the market does a perfectly competent job of introducing the basic definitions and proving the basic theorems that any math student has to know. Artin's book is no exception, and I find his writing style to be very appropriate for this purpose. What sets this book apart is its treatment of topics beyond the basics--things like matrix groups and group representations. I suppose many introductory books shy away from much of the material on matrix groups in Artin's book because it involves a little analysis (and likewise for the section on Riemann surfaces in the chapter on field theory). However, Artin correctly realizes that a reasonably mathematically mature student--even one who doesn't know much analysis--will be able to profit from and enjoy the relatively informal treatments he gives these slightly more advanced topics. Of course these topics can also be found in graduate-level texts, but I for one would much rather be introduced to them via an example-based approach such as that in Artin than through the diagram-chasing obscurantism in more advanced books. I happened upon this book a little late--in fact, only after I'd taken a semester of graduate-level algebra and already felt like analysis was the path I wanted to take--but I'm beginning to think I would have been more keen on going into algebra if I'd first learned it from a book like this one.
2) I bought this book for a class that I ended up dropping. In the beginning, I hated this book. I found Herstein's "topics in algebra" much better, and more to the point. It was only when I was getting bored with Herstein that I bothered to pick this up again. I was pleasantly surprised. A lot of the material flowed very smoothly - exactly as if Artin was teaching the material to you. It must however be noted that people tend to love or hate this book. This is predominantly due to the author's writing style. Given how expensive this book is, you might perhaps want to peruse it somewhere before deciding to buy it. But if you do, you'll get a solid exposition on most of the introductory topics in algebra as well as some insight on groups and symmetry, lie groups, representation theory, galois theory and quadratic number fields. And a whole lot of intuition as well, for the more regular topics. Give this book a chance - it's worth the effort and money.
3) As an undergraduate I learned, or tried to learn, algebra from this book. Artin's pedagogical methods just didn't work for me. Although his idea of teaching through concrete, geometric examples sounds great in principle, in practice it's not so successful. It is very hard to see the forest for the trees, since Artin is so chatty and discursive. When he is discussing examples, he sometimes puts specialized results on par with more general theorems, which may be misleading. Many proofs are only sketched, and occasionally theorems are stated after their proofs, necessitating a rereading of the preceding paragraphs in order to grasp the points of the proof. The chapters on representation theory (Ch. 9) and arithmetic of quadratic number fields (Ch. 11) are nonstandard topics and interesting in themselves, but again, the level of detail tends to obscure, rather than enlighten.
The one saving grace of the book is the excellent problem sets at the end of each chapter. In doing them you will learn the algebra that the main body of the text attempts to impart.
书名：Codes and Curves
作者： Judy Walker
出版商： American Mathematical Society (2000)
页数：66
适用范围：大学数学系本科高年级参考书
预备知识：抽象代数
习题数量：小
习题难度： 容易
推荐强度：8.5
书评： 这本小册子是1999年美国数学会在 Princeton 组织的暑期学校的一门课程的讲稿，是代数几何码的入门读物。代数几何码是新发现的 一种纠错码，目前仍有大量问题在研究。本书前一半对纠错码的基本知识和若干经典的纠错码作了扼要的介绍，重点是 Reed-Solomon 码，因为代数几何码是它的推广。然后，作者不加证明地清楚地叙述了有限域上平面代数曲线的基本知识， 最后介绍了代数几何码以及好的代数几何码的构造方法。
本书的一个显著特点是提供了六个供本科生研究的课题。本人曾指导复旦大学数学系毕业班的六名学生报告这本书，并围绕 六个课题查阅文献资料，写作毕业论文，取得很好的效果。 （杨劲根）
国外评论摘选
1) The book gives an overview of algebraic coding theory. The first chapter introduces error correcting codes, the Hamming distance, Reed-Solomon codes, and concludes with a brief exposition of cyclic codes. The second chapter discusses some upper bounds on the minimum distance of a code such as the Singleton and Plotkin bounds.
The second theme of this book are algebraic curves. Chapter 3 contains the basic definitions and some examples of algebraic curves. The concept of a nonsingular curve is explained in Chapter 4. This chapter also contains a half page explanation of the genus of a curve. The Riemann-Roch theorem is finally covered in Chapter 5.
The two themes come together in Chapters 6 and 7. These chapters discuss the basic principles of algebraic geometry codes.
This little book gives the reader a first taste of an intriguing field. The most surprising part is how much is covered in so few pages [the main text without appendices has 44 pages]. The explanations are always accessible for undergraduate students of mathematics, computer science, or electrical engineering. The prerequisites are some knowledge of abstract algebra, but most material is reviewed in the appendices.
It is a lovely little book that is written in a lively style. The book nicely complements the typical college courses on coding theory. If you want to get an idea what algebraic geometric codes are and you want a quick answer, then this is the book for you.
2) There is a free version of the book available on the website of the University of Nebraska-Lincoln.
目录：
Chapter 1. Introduction to Coding Theory
1.1. Overview
1.2. Cyclic Codes
Chapter 2. Bounds on Codes
2.1. Bounds
2.2. Asymptotic Bounds
Chapter 3. Algebraic Curves
3.1. Algebraically Closed Fields
3.2. Curves and the Projective Plane
Chapter 4. Nonsingularity and the Genus
4.1. Nonsingularity
4.2. Genus
Chapter 5. Points, Functions, and Divisors on Curves
Chapter 6. Algebraic Geometry Codes Chapter
7. Good Codes from Algebraic Geometry
Appendix A. Abstract Algebra Review
A.1. Groups
A.2. Rings, Fields, Ideals, and Factor Rings
A.3. Vector Spaces
A.4. Homomorphisms and Isomorphisms
Appendix B. Finite Fields
B.l. Background and Terminology
B.2. Classification of Finite Fields
B.3. Optional Exercises
Appendix C. Projects
C.1. Dual Codes and Parity Check Matrices
C.2. BCH Codes
C.3. Hamming Codes
C.4. Golay Codes
C.5. MDS Codes
C.6. Nonlinear Codes
书名：Introduction to Commutative Algebra
作者： Michael Atiyah & I.G.MacDonald
出版商： Addison-Wesley Publishing Company (1991)
页数：126
适用范围：大学数学系本科基础数学高年级或研究生低年级教材
预备知识：抽象代数和点集拓扑
习题数量：大
习题难度： 较大
推荐强度：9
书评： 英国皇家科学院院士 Michael Atiyah 是当代大数学家，曾或菲尔滋奖。本书是交换代数的入门书籍，是一本优秀教材， 特别适合于代数几何、代数数论和其他代数专业的研究生使用。本书的篇幅虽小，内容却很丰富，包含了交换代数的核心内容。 学过一学期抽象代数的人可以顺利学习本书前九章，学习第十和第十一章需要点集拓扑的基本知识。正文中的定理的证明简明易懂，有很多重要的定理安排在习题中，所以要掌握此书内容必须化工夫做每一章后的大部分习题。
本书以诺特交换环和有限生成模作为重点，这正是代数几何和代数数论中出现最多的代数结构。 作者在序言中说到域论没有 涉及，这可以从别的优秀教材（如 Nagata 的“域论”）中得到补充。
国外很多名校的数学教授将此书作为交换代数教材的首选。我国引进此书也很早，它很受师生的欢迎。 （杨劲根）
国外评论摘选
1) Some people believe that, for getting into algebraic geometry (by this I mean Grothendieck-like AG, with schemes and all that), one needs a monolithic training in commutative algebra (something like both volumes of Zariski-Samuel, for example). I disagree. This little book seems to be specially suited to those who want to learn AG. It's a bit too brisk, specially at the beginning - if you don't already have an acquaintance with the basics of groups, rings and ideals, you may run into trouble - but very illuminating. Masterful choice of topics, great exercises (as a matter of fact, about half the topics of the book, and more specifically the ones that are directly related to AG, are treated in the exercises, some of them quite challenging) - like one said before, it looks like a "chapter 0" of Hartshorne's book on AG. The authors consciously estabilish relations between the commutative algebra and the modern foundations of AG over and over along the way, illuminating both topics. For the algebra itself, it also gets on well with Rotman's "Galois Theory" and MacDonald's out-of-print introduction to AG, "Algebraic Geometry - Introduction to Schemes", besides being the perfect preamble in commutative algebra to the books of Mumford and Hartshorne. A gem.
2)The strongest aspects of Atiyah & MacDonald's book are its brevity, accessibility to undergraduates, and subtle introduction of more advanced material.
Audience: I think an undergraduate with a solid understanding of material from a first course in abstract algebra (i.e., the chapter on rings--the modules chapter would help, but isn't necessary--from M. Artin's book 'Algebra' is more than sufficient) and some basic point-set topology from an intro real analysis course (or ch1-4 of Munkres) would be sufficient for fully appreciating the material. I think having experience in PS Topology is important for understanding parts of this book well; doing the exercises is possible if you learn it "on the fly," but I hadn't seen Urysohn's Lemma before, and even that caused me some "intuition" hangups; to fully appreciate the material, I would recommend doing a healthy number of problems in topology first.
Material: The material uses concepts from homological algebra, though in a disguised form; students with experience in category theory will find offhanded comments that recast some of the material in that language, but CT is absolutely not essential to understand the material well. It also provides exercises that lead naturally into topics from Algebraic Geometry and Algebraic Number Theory quite readily; a nice set of problems in CH1 walk a student through construction of the Zariski topology, prime spectrum, etc., and some functional properties of morphisms between spectra. Algebraic Number Theory starts showing up after chapter 4 in greater detail, and would lead comfortably into Lang's GTM on ALNT by CH9 (though I only read a bit of Lang, the first chapter felt natural).
The "details left to the reader" are usually reasonably tackled with the tools made available so far, and the book is short enough that one can cover a lot of ideas in a reasonable amount of time; the commentary made by the authors is brief, to the point, and never redundant as far as I can recall, so I consider this a highly efficient book (but not too efficient, it's self contained enough and not uncompromisingly terse).
Exercises: They are quite good, I think. Very few of them follow from "symbol-pushing" or "robotic theorem proving," and usually require some constructive argument. The exercises are mostly chosen to introduce more advanced material, and do a good job in that regard. The longer chapters have 25-30 exercises, and shorter chapters (a few pages) have maybe 10, so there are plenty of problems to do.
Hazards: The material on modules is brisk, the propositions in the first three sections on modules are mostly left without proof; however, the proofs follow from their analogues for rings, and aren't that hard, just be sure to actually do them because they are mentioned only briefly. Also, the book is not typo-free, but this only caused me one major hangup during the semester. After Chapter 3, the proofs are mostly complete, with a spattering of "left to the reader" exercises, which I usually found helpful.
Companion Material: I think Lang's 'Algebra' GTM would make a nice reference for the material on Homological Algebra and other miscellaneous things that come up in the proofs; I remember once a proof in the book required the notion of the adjoint of a matrix over a ring, and so I had to look it up in Lang, and also the basic category theory covered in CH1 of Lang would at least introduce (though in a very rapid way) the "abstract nonsense" mentioned offhandedly here and there. If you have a lot of money, or access to a good library, 'Categories for the Working Mathematician' is a slower and more thorough introduction to that language, and I would recommend at least having a look, though this isn't really central to the material from Commutative Algebra.
3) This is how mathematics texts SHOULD be written. As in technical writing, the smaller text is the better written text. Everything is clean and direct, with clairity obviously a prime consideration. One never gets mired down. The proofs are always as close to a "THE BOOK" proof as possible, with illuminating examples, and plenty of excercises, many with outlines for solution, which makes the book ideal for self study. This book is a revelation. If I had to take only one math text with me to a desert island, this would be the one.
4) This is a difficult book for undergraduates, even ones who have already had some abstract algebra. Many refer to the book's style as "terse", meaning that there is little explanation, few examples, and proofs are very condensed.
书名：HOPF ALGEBRAS
作者： MOSS E. SWEEDLER
出版商： W. A. BENJAMIN, INC. (1969)
页数：336
适用范围：大学数学系本科、数学专业研究生
预备知识：代数、环模基础理论
习题数量：小
习题难度： 容易
推荐强度：9
书评： 1941年，德国数学家H. Hopf在研究代数拓扑时引入了Hopf代数的概念。真正引 起人们对这类代数结构普遍关注的是1965年J.W. Milnor 和J.C. Moore的有关分 次Hopf代数的文章；到上世纪80年代末，量子群概念的出现及其在Knot不变量理 论中的应用将Hopf代数的研究推向一个新的高潮。如今，Hopf代数理论正在诸如代 数群、李代数、表示论、组合论以及量子力学等学科的研究中发挥着重要的作用。M.E. Sweedler所著的\textquotedblleft Hopf Algebra\textquotedblright是历史上第 一本系统介绍这方面Hopf代数知识的书籍。
这本书是从Sweedler给研究生的系列讲座内容中整理出来的，介绍的对象主要是非分 次的Hopf代数。域$k$上一个增广（augmented）代数$H$，若带有一个余结合的 （coassociative）和余单位的（counitary）代数映射$\Delta \colon H\rightarrow H\otimes H$，则称$H$是一个双代数（bialgebra），Hopf代数是指带 有antipode的双代数。本书开始先引入sigma-符号；而后一步步将上世纪70年代 以来有关Hopf代数的最新结果，其中大部分是作者与其合作者当时取得的进展，呈现 给读者；最后一章以证明域上由所有有限维交换、余交换的Hopf代数组成的范畴是交 换的（abelian）范畴作为结束。整本书的内容简洁，易懂，且自我包含，是一部很好 的关于Hopf代数知识的入门教材。它的不尽完美之处是没有列出任何参考文献 。
本书共有16章，有些章节非常短。前4章，给出了余代数、余模、及Hopf代数的 初步介绍，其中包括从模中构建余模的有理模构造方法及余代数的基本定理---该定 理阐明了在一个余代数中，任意有限个元素均包含于一个有限维子余代数中，故而任一 余代数都是有限维余代数的直接极限。第5章讨论了积分（integral）， \textquotedblleft积分\textquotedblright这一名称是因它很像紧群上关于 Haar测度的积分运算而得名。这一章的重要结果是证明任一有限维Hopf代数必存在 一维的积分空间；作为一个推论，本章中导出了群环的著名Maschke定理。
第6章对一个代数引入并讨论了它的对偶余代数。第7章到第9章主要介绍了度量 、smash积和外积等概念，为第13章余交换点（pointed）Hopf代数的结构定理的 证明作了前期的准备工作。
第10章的主要内容是Hopf代数作用的Galois理论；第11章对本原元进行了重点 讨论并引入了分次双代数的概念，在这一章以及后面的第14章中余代数的基本定理起 到了很大的作用。第12章考虑了shuffle代数和相关既约的点余代数万有映射性质 ，此外还讨论了divided power。
第13章中证明了著名的结论：任一既约的点余交换Hopf代数一定同构于Lie代数 或限制Lie代数的包络代数；据此导出了余交换Hopf代数的结构定理。剩下的两章 讨论了仿射群和由交换、余交换Hopf代数构成的Abelian范畴。 （朱胜林）
7 数论
数论是数学中历史悠久但又有生命力的分支，也很有趣味。数论有分初等数论、代数数论、解析数论等。初等数论主要用初等的方法讨论整数的性质，如同余方程、不定方程、二次剩余等。
代数数论是讨论代数数的分支，要使用很深的代数工具。近年来，代数数论和代数几何合起来形成了一门称为算术几何的新分支，是非常艰深的。解析数论则是用数学分析和复变函数论来研究数论的问题，当今数学中第一号未解决问题黎曼猜想就属于解析数论的范围。
大学本科阶段学习一些数论是有用的，特别对学习抽象代数有很大帮助。研究生阶段一般只有数论专业的学生才学数论。
我们在这里介绍的数论教材大多是供数学专业本科生使用的。
书名：Elementary Methods in Number Theory
作者： Melvyn B.Nathanson
出版商： Springer-Verlag
页数：509
适用范围：大学数学系本科基础数学学生、数学专业研究生
预备知识：微积分
习题数量：大
习题难度： 中等，多数习题很容易
推荐强度：9.5
书评： 本书是Springer-Verlag出版的研究生系列教材中的一本，编号第195，2000年出版。全书分 为三个部分，第一部分介绍了初等数论的基本内容，整除性，同余，原根，Gauss二次互反律，有限 交换群上的Fourier分析，以及abc猜想的一个简单介绍。第二部分讨论了一些算术函数的性质， 给出了素数定理的初等证明。第三部分介绍了加法数论中的三个问题，即Waring问题， 正整数表为整数的平方和的问题，以及分拆函数的渐进估计的问题。 本书的一个特点是给出了许多深刻的数论定理的初等证明， 比如，Selberg的素数定理的初等证明，Linnik关于Waring问题初等证明， 一个整数表示为偶数个整数的平方和的个数的Liouville方法，以及Erdos关于分拆函数的渐进估计的结果。 事实上，本书的所有的证明都只使用了初等的方法，不涉及解析方法以及其他的高等方法， 因此本书也是一本很好的大学生数论教材。本书第一部分和第二部分作为大学生一个学期的课程是合适的。 （王巨平）
国外评论摘选
1) Every serious student of number theory should have this classic book on their shelf. Even though only "elementary" calculus and abstract algebra are used, a certain mathematical maturity is required. I feel the book is strongest in the area of elementary --not necessarily easy though -- analytic number theory (Hardy was a world class expert in analytic number theory). An elementary, but difficult proof of the Prime number Theorem using Selberg's Theorem is thoroughly covered in chapter 22.
While modern results in the area of algorithmic number theory are not presented nor is a systematic presentation of number theory given (it is not a textbook), it contains a flavor, inspiration and feel that is completely unique. It covers more disparate topics in number theory than any other n.t. book I know of. The fundamental results in classical, algebraic, additive, geometric, and analytic number theory are all covered. A beautifully written book.
Other recommended books on number theory in increasing order of difficulty:
1) Elementary Number Theory, By David Burton, Third Edition. Covers classical number theory. Suitable for an upper level undergraduate course. Primarily intended as a textbook for a one semester number theory course. No abstract algebra required for this book. Not a gem of a book like Davenport's The Higher Arithmetic, but a great book to seriously start learning number theory.
2) The Queen of Mathematics, by Jay Goldman. A historically motivated guide to number theory. A very clearly written book that covers number theory at a graduate or advanced undergraduate level. Covers much of the material in Gauss's Disquisitiones, but without all the detail. The book covers elementary number theory, binary quadratic forms, cyclotomy, Gaussian integers, quadratic fields, ideals, algebraic curves, rational points on elliptic curves, geometry of numbers, and introduces p-adic numbers. Only a slight bit of analytic number theory is covered. The best book in my opinion to start learning algebraic number theory. Wonderfully fills the otherwise troublesome gap between undergraduate and graduate level number theory.
Full of historical information hard to find elsewhere, very well researched. To cover all the material in this book would likely take two semesters, though most of the important material could be covered in one semester. Requires a background in abstract algebra (undergraduate level), and a little advanced calculus. Some complex analysis for sections 19.7 and 19.8 would be helpful, but not at all a requirement. The author recommends Harold Davenport's The Higher Arithmetic, as a companion volume for the first 12 chapters; according to Goldman it is a gem of a book.
3) Additive Number Theory, by Melvyn Nathanson. Graduate level text in additive number theory, covers the classical bases. This book is the first comprehensive treatment of the subject in 40 years. Some highlights: 1) Chen's theorem that every sufficiently large even integer is the sum of a prime and a number that is either prime or the product of two primes. 2) Brun's sieve for upper bound on the number of twin primes. 3) Vinogradov's simplification of the Hardy, Littlewood, and Ramanujan's circle method.
2) My initial reaction through the first chapters was one of embarrassment at my lack of understanding. I could not believe a book, hailed by so many as a standard and essential resource, could be so much out of my reach. Then, amid the last page or so of chapter 1 I had an epiphany. The book, from that point on, was completely clear and logical while retaining an extraordinary amount of breadth in coverage. Add my staunch support and recommendation to the long list of kudos that this book has accrued. There are, to my knowledge, no better books for the beginning student of number theory. If you have any interest whatsoever in the theory of numbers, this book is essential.
书名：A course in arithmetic
作者： J.-P. Serre
出版商： Springer Verlag (1973) ISBN 0-387-90041-1
页数：113
适用范围：大学数学系本科基础数学高年级或研究生低年级教材
预备知识：抽象代数,复分析
习题数量：很少
习题难度： 较大
推荐强度：10
书评： 法国大数学家，菲尔滋奖和阿贝尔奖获得者 Serre 写过不少短小精悍的小册子，大部分从他亲自所讲授的课程的讲稿整理而成。 本书是他非常有代表性的本科生高年级的数论教材，曾在西方评为某年度世界最佳数学教材。
本书并不是数论的系统教程，作者选择数论中三个重要专题扼要叙述了它们的内容和方法，这三个专题是：二次型、素数的 Dirichlet 定理 和模形式。读者可以化较少的时间学到一些近代数论的知识。最令读者欣赏的是定理的证明将大数学家的技巧展现得淋漓尽致，阅读中不禁 拍案叫绝。非定型幺模偶整格的分类定理非常漂亮，但其完整的证明在很多代数教科书中很难找到，本人所知道的就是本书以及 Milnor 和 Husemoeller 写的 Symmetric bilinear form 一书中的证明。这两本书都是70年代出版的，经过这两位菲尔滋奖得主之手的证明已经很难再 作改进，因此后人写的书大多只是引用而不再重写了。
具备抽象代数的知识就可以读懂前半本书，后一半需要复分析的准备知识。由于叙述简洁，习题数量少，作为教材使用会有一定困难。 作为自学的参考书对读者的数学素养也有较高的要求。 （杨劲根）
国外评论摘选
1) The book is divided into two parts -- algebraic and analytic. I've only worked through the analytic part. Anything by Serre is worth its weight in gold and this book is no exception; everything Serre covers is of the utmost importance. But Serre's style is extremely condensed and spare, and he makes no concessions to the reader in terms of motivation or examples. I can't digest more than half a page of Serre a day; however if one wants to understand the structure of a theory, Serre is ideal. I worked through "A Course in Arithmetic" over a decade back. As I recall I covered Riemann's zeta function and the Prime Number Theorem, the proof of Dirichlet's theorem on primes in arithmetical progressions using group characters in the context of arithmetical functions, and some of the basic theory of modular functions. All of this material is also covered in Apostol's two books on analytic number theory ("Introduction to Analytic Number Theory", and "Dirichlet Series and Modular Functions in Number Theory"); Apostol goes further than Serre in the analytic part -- which is only to be expected since he is devoting two whole texts to the subject.
2) Serre's work could best be summarized in one word - Elegance. The book comprises of two distinct parts. The first one is the 'algebraic' part. Serre's goal in this section is to give a complete classification of the quadratic forms over the rationals. As preliminaries to reaching this goal, he introduces the reader to quadratic reciprocity, $p-$adic fields and the Hilbert Symbol. After these three, he spends the next chapter detailing the properties of quadratic forms over ${\mathbb Q$ and ${\mathbb Q_p$ (the $p-$adic field). The reason to work over ${\mathbb Q_p$ is the Hasse-Minkowski Theorem (which says that if you have a quadratic form, it has solutions in Q if and only if it has solutions in ${\mathbb Q_p$). Using Hensels Lemma, checking for solutions in ${\mathbb Q_p$ is (almost) as easy as checking for solutions in Z/pZ. After doing that, he spends yet another chapter talking about the quadratic forms over the integers. (Note: the classification goal is already achieved in previous chapter). The second half of the book is the 'analytic' one. The first chapter in this section gives a complete proof of Dirichlet's theorem while the second one studies the properties of modular forms (these are good!) Due to the extreme elegance, the book is sometimes hard to read. This might sound like a paradox, but it's not and I'll explain why. The book takes some effort to read because it's terse and it often takes a while to figure out why something is 'obvious'. However, once you see it all, you'll realize that a great mind was guiding you through the pursuit. The choice of topics is just right to achieve the goals that the author sets out for himself. Also, I'd rather think for myself and read a smaller book than be given a huge fat tome where the author details his own thought process. This book was my first foray into number theory and I absolutely enjoyed it. If you're considering reading it, I wish you joy in your pursuits.
书名：Introduction to Analytic Number Theory
作者： Tom Apostol
出版商： Springer Verlag (1976) ISBN 0-387-90163-9
页数：328
适用范围：大学数学系本科数论教材
预备知识：微积分,复分析
习题数量：大
习题难度：一般
推荐强度：9
书评： 这是一本非常受欢迎的数论入门教材，写得极其清楚仔细而又不烦琐。虽然书名是解析数论，事实上也包括了初等数论。 由于书的自封性能好，习题又经过精心挑选，适合于大学低年级的数论教材。
本书由于其良好声誉而多次再版，被选入 Springer 的 UTM 系列。同一作者的微积分教材（见本书的另一篇书评）也有好口碑。 （杨劲根）
国外评论摘选
1) I think that there will be little harm if the title of the book is changed to 'Introduction to elementary number theory' instead. The author presumes that the reader has not any knowledge of number theory. As a result, materials like congruence equation, primitive roots, and quadratic reciprocity are included. Of course as the title indicates, the book focusses more on the analytic aspect. The first 2 chapters are on arithmetic functions, asymptotic formulas for averaging sums, using elementary methods like Euler-Maclaurin formula .This lay down the foundation for further discussion in later chapters, where complex analysis is involved in the investigation. Then the author explain congruence in chapter 4 and 5. Chapter 6 introduce the important concept of character. Since the purpose of this chapter is to prepare for the proof of Dirichlet's theorem and introduction of Gauss sums, the character theory is developed just to the point which is all that's needed. ( i.e. the orthogonal relation). Chapter 7 culminates on the elementary proof on Dirichlet's theorem on primes in arithmetic progression. The proof still uses $L-$function of course, but the estimates, like the non-vanishing of $L(1)$ , are completely elementary and is based only on the first 2 chapters. The author then introduce primitve roots to further the theory of Dirichlet characters. Gauss sums can then be introduced. 2 proofs of quadratic reciprocity using Gauss sums are offered. The complete analytic proof, using contour integration to evaluate explicitly the quadratic Gauss sums, is a marvellous illustration of how truth about integers can be obtained by crossing into the complex domains. The book then turns in to the analyic aspect. General Dirichlet series, followed by the Riemann zeta function, L function ,are introduced. It's shown that the $L-$functions have meremorphic continuation to the whole complex plane by establishing the functional equation $L(s)=$ elementary factor $* L(1-s).$ The reader should be familiar with residue calculus to read this part. Chapter 13 may be a high point of this book, where the Prime Number Theorem is proved. Arguably, it's the Prime Number Theorem which stimulate much of the theory of complex analysis and analyic number theory. As Riemann first pointed out, the Prime Number Theorem can be proved by expressing the prime counting function as a contour integral of the Riemann zeta function, then estimate the various contours. The proof given in this book , although not exactly that envisaged by Riemann , is a variant that run quite smoothly. As is well known , a key point is that one can move the contour to the line $Re(s)=1,$ and to do this one have to verify that $\zeta(s)$ does not vanish on $Re(s)=1.$ The proof , due to de la vale-Poussin, is a clever application of a trigonometric identity. Unfortunately, the method does not allow one penetrate into the region $0<1,$ where the distribution of zeroes in this region contain the information about the flunctuation of $\Pi(x)$ around $x/\log x.$ The famous Riemann Hypothesis states that the only zeroes in this region lis on the line $Re(s)=1/2.$ After more than 100 years, although the Riemann Hpothesis has natural generalisation to number fields, neither of these RH is proven, which indicates the difficulties of this problem. Recently some new directions, related to quantum statistical mechanics, has been connected with this old problem. If the RH is proven, then the set of prime numbers , although looks completely random locally ( like the occurences of twin primes), is governed by clear-cut laws on the large after all. The last Chapter is of quite differnt flavour, the so-called additive number theory. Here the author only focusses on the simplest partition function ---the unrestricted partition. However interesting phenomeon occur already at this level. The first result is Euler's pentagonal number theorem, which leads to a simple recursion formula for the partition function p(n). 3 proofs are given. The most beautiful one is no doubt a combinatorial proof due to Franklin. The third proof is through establishing the Jacobi triple product identity, which leads to lots of identites besides Euler's pentagonal number theorem. Jacobi's original proof uses his theory of theta functions, but it turns out that power series manipulaion is all that's needed. The book ends with an indication of deeper aspect of partition theory--- Ramanujan's remarkable congrence and identities ( the simplest one being $p(5m+4)= 0 \pmod{5$ ). To prove these mysterious identites, the "natural"way is to plow through the theory of modular functions, which Ramanujan had left lots more theorem ( unfortunately most without proof). However an elementary proof of one these identites is outlined in the exercises. This book is well written, with enough exercises to balance the main text. Not bad for just an 'introduction'.
2) This book has excellent exercises at the end of each chapter. The exercises are interesting and challenging and supplement the main text by showing additional consequences and alternate approaches. The book covers a mixture of elementary and analytic number theory, and assumes no prior knowledge of number theory. Analytic ideas are introduced early, wherever they are appropriate. The exposition is very clear and complete. Some novel features include: three chapters on arithmetic functions and their averages (including a simple Tauberian theorem due to Shapiro); Polya's inequality for character sums; and an evaluation of Gaussian sums (by contour integration), used in one proof of quadratic reciprocity.
8 代数几何
代数几何是核心数学的重要分支，内容比较高深，不太容易入门。由于它所用的知识比较多，学习的周期相对比较长。一般本科生阶段不设代数几何课程。
代数几何对抽象代数、复分析、拓扑等都有较高的要求，特别交换代数和同调代数是它的不可缺少的工具。我们在这里介绍一些目前在国际上使用最多的一批基础性的教材供有兴趣和有志向的读者参考。
书名：An Invitation to Algebraic Geometry
作者：K.Smith etc.
出版商：Springer-Verlag, New York, 2000. ISBN 0-387-98980-3
页数：155
适用范围：基础数学本科高年级或非代数几何专业研究生低年级
预备知识：线性代数，群，环，域扩张，Galois
理论的基础知识，最基本的点集拓扑
习题数量：大
习题难度：容易
推荐强度：8.5
书评：
本书由作者 1996 年的为非代数几何专业的数学研究生开设的 20 小时代数几何课的讲义 整理修改而成，是一本非常好的代数几何入门书。
本书从最基本的交换代数代数几何概念开始，用简明的方式引入仿射簇和射影簇。重点的内容是一些经典的例子： Veronese 映射、计数几何、Segr\'{e}
嵌入、 Grassman 簇等。最后介绍代数几何中的一些重大问题如奇点解消、射影簇分类、典范映射等。
本书把代数几何讲解的具体易懂，不拘泥于细节，一些关键定理给出清晰的解释而不是详细的证明，如 Hilbert 零点定理、 B\'{e}zout 定理、 Bertini
定理等，有一部分内容的简单证明放入习题。有些其他教材不提及的问题也作了简短介绍，如 Gauss 映射用来 2 页和 5 道习题。书中不时插入对一些重要问题研究的历史和现状，颇有 Wikipedia
的风格。在三个合适的地方叙述三个未解决的难题： Jabobian 猜想、空间曲线的完全交问题、 Iitaka 猜想。
本书篇幅不大，适合初学者在较短的时间内对代数几何的特点有初步的了解，为进一步的深入学习作准备。（杨劲根）
国外评论摘选
1）This book has a great deal to recommend it:
a. It is a genuinely entry-level book that begins with the definition of a prime ideal and the Nullstellensatz.
b. The style of explanation is clearly geared to noninsiders. In addition to giving examples of algebraic varieties, some "nonexamples" are given that might have occurred to, say, an analyst as reasonable objects to study but that do not qualify as varieties.
c. The illustrations are frequent, relevant, and well executed.
d. The authors go out of their way to help the reader develop geometric intuition and to relate it to the accompanying algebraic description. For example, in the careful treatment of the geometry of a family of hyperbolas in chapter 6, the geometry of the general hyperplane section is beautifully illustrated, and the reader's geometric intuition is stimulated into action.
e. Many of the constructions covered are classical-the Grassmannian, the Veronese, the Gauss mapping, the secant variety of a variety-yet the book almost seamlessly connects this with more modern material, such as resolution of singularities and vector bundles.
f. There is a consistent policy throughout the book of tying in elementary algebraic geometry to recent developments by current
leaders such as Kollar, Kontsevich,Mori, Lazarsfeld, and de Jong, so that readers come away with a clear conception of where this is all going and what the next steps might be if a particular topic sparks their interest. Overall, readers will find this book easy to get into and enjoyable to read. Outsiders to the subject will feel that they are hiking up a gently sloping trail, at the end of which they reach a number of pleasant viewing spots from which they. can see rather far in a number of different directions. Students contemplating algebraic geometry as a field of specialization will also find this an attractive and instructive place to start. ( by Mark Green, {\em The American Mathematical Monthly,} Vol. {\bf 109}, 675-678(2002))
2）This could be your only book on algebraic geometry if you just want a sound idea of what algebraic geometry can do. If you actually want to know the field, and you do not already have a lot of expert friends telling you about it, then the advanced books will go much more easily with this expert around. It is a terrific guide to the key ideas--what they mean, how they work, how they look. The only book like this one in brevity and scope is Reid UNDERGRADUATE ALGEBRAIC GEOMETRY--with its highly informed, highly polemical, final chapter on the state of the art. Both are very good. This one is more advanced. Beyond what Reid covers, Smith sketches Hilbert polynomials, Hironaka's (and very briefly even De Jong's) approach to removing singularities, and ample line bundles. You do need a bit of topology and analysis to follow it. Smith has very many fewer concrete examples than Reid. They are beautifully chosen classics, like Veronese maps and Segre maps, so they teach a lot. And the more you know to start with, the more you will see in each.
The book does geometry over the complex numbers. It is good old conservative material, with terrific graphics of curves and
surfaces. The proofs and partial proofs are very clear, intuitive and to the point. But, in fact, just because the proofs are so clear and to the point they usually work in a much broader setting. Long stretches of the book apply just as well over any field or any algebraically complete field. This generality is only mentioned a few times, in passing, but is there if you want it. Smith describes schemes very briefly, and mentions them at each point where they naturally arise. You will not know what schemes "are" at the end of this book. You will know some things they DO. She has no time for fights between "concretely complex" and "abstractly scheming" approaches--for her it is all geometry.
3）For people just starting on Algebraic Geometry, Robin Hartshorne's book, is very daunting--but it is the ULTIMATE book for
professional and advanced readers. But for starters, Karen Smith's "An Invitation to Algebraic Geometry" is simply a SPLENDID way to start working on the basic ideas. The author has some stunning graphs and pictures to help understand material. I loved the book the minute I opened it.
书名：Introduction to Commutative Algebra and Algebraic Geometry
作者： Ernst Kunz
出版商： Birkhauser Boston, (1985) ISBN 3-7643-3065-1
页数：238
适用范围：基础数学本科高年级或研究生低年级
预备知识：线性代数，群，环，域扩张，Galois 理论的基础知识，最基本的点集拓扑
习题数量：大
习题难度： 大部分中等，少量难题
推荐强度：9
书评： 作为交换代数的入门书，它不如 Atiyah-McDonald 的有名，也不如 Eisenbud 的大， 但是我认为对于有志学习代数几何的大学生来讲，这是最好的入门书。此书与大学基础课程 的衔接非常紧密，不管是自学还是用此书当教材都比较轻松，如果再认真做习题则效果更好。 交换代数的内容甚广，作者完全按经典代数几何的需要选择交换代数的内容，重点是多变量 多项式环的商环及它上的有限生成模。除了基础性的材料外，也有少量研究性的题材。现在 代数几何最流行的研究生教材是 Hartshorne 的 Algebraic Geometry, 本书可以认为是 Hartshorne 的教程的前续课程。本书作者是著名的交换代数专家，原书用德文写作，后翻译成英文，美国 代数几何大师 David Mumford 写了序言，称此书是美国学代数几何学生久等的一本书，它填补 了一个空白。（杨劲根）
书名：Basic Algebraic Geometry (Second, Revised and Expanded Edition)
作者： Shafarevich
出版商： Springer-Verlag (1988) ISBN 3-540-54812-2
页数：上册 303 下册 269
适用范围：基础数学研究生
预备知识：近世代数、复分析、点集拓扑
习题数量：大
习题难度：较难
推荐强度：9
书评： 本书是俄罗斯的数学大师 Shafarevich 的力作，由英国著名代数几何学家 Miles Reid 翻译成英文。本书内容非常丰富且不枯燥，叙述和证明清晰，比较容易读，是非常收欢迎的一本 代数几何书，国内外不少院校开设代数几何课曾将此书选为研究生教材。全书分三大部分， 第一部分是射影簇，内容包含经典代数几何，一直讲到代数曲面的分类和奇点，其中不乏其它 教科书中不多见的内容。这一部分占了整个上册，作为一个学期的课程内容够多的。 第二部分是概形理论，用现代的语言来刻画代数簇，最后讲到 Hilbert 概形。本部分内容比较简要，基本上讲清概形和层论的威力。第三部分是复代数流形的拓扑和几何，很多内容如代数簇的拓扑 分类和 uniformization 在其它代数几何教科书很难找到。总之，本书基本上讲述了代数几何的所有方法。 习题非常丰富，大部分的习题很有意思，可以看出是作者和他的助手们多年积累而编成的。 还有一个显著的特点是本书不需要交换代数的预备知识，当然学过交换代数在看此书更加轻松。 （杨劲根）
国外评论摘选
1) I have been a student of AG for the past six years and I have come to the conclusion that Shafarevich is a great place to start. Having said this, one must have the necessary background in algebra and topology. I disagree with the other reviewer about doing this after Hartshorne--start here then do Hartshorne!!!
书名：Algebraic Geometry
作者: Robin Hartshorne
出版商：Springer-Verlag
页数：495
适用范围：基础数学研究生
预备知识：近世代数、交换代数、同调代数、复分析、基础拓扑
习题数量：多
习题难度： 又难又繁
推荐强度：8.8
书评： 本书是现代代数几何的标准教科书，适合代数几何专业的研究生使用。从代数几何的发展历史来看， 60年代由 Grothendieck 提出的以概形为基础的新理论完成了代数几何的一次新的革命，至今代数几何仍以 Grothendieck 的理论为基础，他和 Dieudonne 合写的庞大的 EGA (Elements de Geometrie Algebrique) 可堪为代数几何的圣经。但是由于规模太大，EGA 无法当作教科书使用。事实上在 EGA 之前，Grothendieck 在日本的东北数学杂志上一篇同调代数的长文也是代数几何的奠基性的文献之一。此后，出现了两本有 很大影响的书，其一是 Matsumura 的 Commutative Algebra, 这本看上去象是研究笔记的专著把 Grothendieck 的 EGA 中的交换代数部分和部分同调代数整理出来加以详细证明。另一本就是 Hartshorne 的 Algebraic Geometry, 此书用两章约230页的篇幅介绍 Grothendieck 的概形理论。作者能完成此举得益于两点：第一，所有和交换代数 有关的内容都引用 Matsumura 的有关章节。第二，作者牺牲一般性而大大简化了很多大定理的证明，具体来讲， 在大部分章节作者把概形限制为诺特概形，把态射限制为有限型的态射。这样的简化对代数几何的主流方向的研究 来说影响不大。从某种意义来看，Hartshorne 的书是 Grothendieck 的 EGA 的浓缩简化版。这本书中的很多英文 名词现在已经获得代数几何界的普遍认可。
本书中定理的证明比较简洁，认真的读者需要补充不少细节，从这点来看，此书的浓度比较大，要读懂此书大部分得化一年以上的 时间，对于没有学过交换代数或学的不多的读者，最好先读 Kunz 的 Introduction to Commutative Algebra and Algebraic Geometry 一书。会法文的读者可以参考 Grothendieck 的 EGA 学习，因为有些定理的证明在 EGA 中写的更为详细。 习题也是本书的一大难点，不少研究生抱怨这本书中的习题太难做。网上有些地方甚至有 Hartshorne 习题部分解答下载。 （杨劲根）
国外评论摘选
1) This book is one of the most used in graduate courses in algebraic geometry and one that causes most beginning students the most trouble. But it is a subject that is now a "must-learn" for those interested in its many applications, such as cryptography, coding theory, physics, computer graphics, and engineering. That algebraic geometry has so many applications is quite amazing, since it was not too long ago that it was thought of as a highly abstract, esoteric topic. That being said, most of the books on the subject, including this one, are written from a very formal point of view. Those interested in applications will have to face up to this when attempting to learn the subject. To read this book productively one should gain a thorough knowledge of commutative algebra, a good start being Eisenbud's book on this subject. Also, it is important to dig into the original literature on algebraic geometry, with the goal of gaining insight into the constructions and problems involved. The author of this book does not make an attempt to motivate the subject with historical examples, and so such a perusal of the literature is mandatory for a deeper appreciation of algebraic geometry. The study of algebraic geometry is well worth the time however, since it is one that is marked by brilliant developments, and one that will no doubt find even more applications in this century. Varieties, both affine and projective, are introduced in chapter 1. The discussion is purely formal, with the examples given unfortunately in the exercises. The Zariski topology is introduced by first defining algebraic sets, which are zero sets of collections of polynomials. The algebraic sets are closed under intersection and under finite unions. Therefore their complements form a topology which is the Zariski topology. The properties of varieties are discussed, along with morphisms between them. "Functionals" on varieties, called regular functions in algebraic geometry, are introduced to define these morphisms. Rational and birational maps, so important in "classical" algebraic geometry are introduced here also. Blowing up is discussed as an example of a birational map. A very interesting way, due to Zariski, of defining a nonsingular variety intrinsically in terms of local rings is given. The more specialized case of nonsingular curves is treated, and the reader gets a small taste of elliptic curves in the exercises. A very condensed treatment of intersection theory in projective space is given. The discussion is primarily from an algebraic point of view. It would have been nice if the author would have given more motivation of why graded modules are necessary in the definition of intersection multiplicity.
The theory of schemes follows in chapter 2, and to that end sheaf theory is developed very quickly and with no motivation (such as could be obtained from a discussion of analytic continuation in complex analysis). Needless to say scheme theory is very abstract and requires much dedication on the reader's part to gain an in-depth understanding. I have found the best way to learn this material is via many examples: try to experiment and invent some of your own. The author's discussion on divisors in this chapter is fairly concrete however.
The reader is introduced to the cohomology of sheaves in chapter 3, and the reader should review a book on homological algebra before taking on this chapter. Derived functors are used to construct sheaf cohomology which is then applied to a Noetherian affine scheme, and shown to be the same as the Cech cohomology for Noetherian separated schemes. A very detailed discussion is given of the Serre duality theorem.
Things get much more concrete in the next chapter on curves. After a short proof o the Riemann-Roch theorem, the author studies morphisms of curves via Hurwitz's theorem. The author then treats embeddings in projective space, and shows that any curve can be embedded in P(3), and that any curve can be mapped birationally into P(2) if one allows nodes as singularities in the image. And then the author treats the most fascinating objects in all of mathematics: elliptic curves. Although short, the author does a fine job of introducing most important results.
This is followed in the next chapter by a discussion of algebraic surfaces in the last chapter of the book. The treatment is again much more concrete than the earlier chapters of the book, and the author details modern formulations of classical constructions in algebraic geometry. Ruled surfaces, and nonsingular cubic surfaces in P(3) are discussed, as well as intersection theory. A short overview of the classification of surfaces is given. The reader interested in more of the details of algebraic surfaces should consult some of the early works on the subject, particularly ones dealing with Riemann surfaces. It was the study of algebraic functions of one variable that led to the introduction of Riemann surfaces, and the later to a consideration of algebraic functions of two variables. A perusal of the works of some of the Italian geometers could also be of benefit as it will give a greater appreciation of the methods of modern algebraic geometry to put their results on a rigorous foundation.
2) This is THE book to use if you're interested in learning algebraic geometry via the language of schemes. Certainly, this is a difficult book; even more so because many important results are left as exercises. But reading through this book and completing all the exercises will give you most of the background you need to get into the cutting edge of AG. This is exactly how my advisor prepares his students, and how his advisor prepared him, and it seems to work. Some helpful suggestions from my experience with this book: 1) if you want more concrete examples of schemes, take a look at Eisenbud and Harris, The Geometry of Schemes; 2) if you prefer a more analytic approach (via Riemann surfaces), Griffiths and Harris is worth checking out, though it lacks exercises.
书名：Principles of Algebraic Geometry
作者： Phillip Griffiths, Joseph Harris
出版商： John Wiley & Sons, Inc., 1978; ISBN 0-471-32792-1
页数：813
适用范围：基础数学研究生
预备知识：近世代数、复分析、点集拓扑
习题数量：无
推荐强度：8
书评： 代数几何原理的作者之一 P. Griffiths 是美国科学院院士，国际著名的数学家，他是 普林斯顿高等研究院教授。本书从比较解析的方面介绍代数几何，它给人的印象是代数几何也很具体， 这是一本代数几何的入门引导课本。
本书前两章处理复流形理论的一些结果和技巧，同时强调它们在射影代数簇上的应用； 第2章开始介绍黎曼面和代数曲线的理论；第3章中介绍了流、陈类、Riemann-Roch 公式等基本工具，然后在第4章中介绍代数曲面理论；本书最后介绍 Quadric Line Complex.
本书的选材十分合适，内容基本自我包容，给读者以直观和易懂的感觉， 但笔者认为该书在排版过程中过于仓促，书中（尤其第0章）累出打印错误和符号冲突。 如果有谁能从现在的版本重新修订并浓缩成一本稍薄的代数几何入门书，那将是代数几何爱好者的福音。（陈猛）
国外评论摘选
1) Once thought to be highly esoteric and useless by those interested in applications, algebraic geometry has literally taken the world by storm. Indeed, coding theory, cryptography, steganography, computer graphics, control theory, and artificial intelligence are just a few of the areas that are now making heavy use of algebraic geometry. This book would probably be one the most useful one for those interested in applications, for it is an overview of algebraic geometry from the complex analytic point of view, and complex analysis is a subject that most engineers and scientists have had to learn at some point in their careers. But one must not think that this book is entirely concrete in its content. There are many places where the authors discuss concepts that are very abstract, particularly the discussion of sheaf theory, and this might make its reading difficult. The complex analytic point of view however is the best way of learning the material from a practical point of view, and mastery of this book will pave the way for indulging oneself in its many applications.
Algebraic geometry is an exciting subject, but one must master some background material before beginning a study of it. This is done in the initial part of the book (Part 0), wherein the reader will find an overview of harmonic analysis (potential theory) and Kahler geometry in the context of compact complex manifolds. Readers first encountering Kahler geometry should just view it as a generalization of Euclidean geometry in a complex setting. Indeed, the so-called Kahler condition is nothing other than an approximation of the Euclidean metric to order 2 at each point.
The authors choose to introduce algebraic varieties in a projective space setting in chapter 1, i.e. they are the set of complex zeros of homogeneous polynomials in projective space. The absence of a global holomorphic function for a compact complex manifold motivates a study of meromorphic functions and divisors. Divisors are introduced as formal sums of irreducible analytic hypersurfaces, but they are related to the defining functions for these hypersurfaces also, via the poles and zeros of meromorphic functions. For the mathematical purist, a "sheafified" version of divisors is also outlined. Divisors and line bundles are basically "linear" tools used to investigate complex varieties through their representation as complex submanifolds of projective space. In addition, various approaches are used to study codimension-one subvarieties, such as the results of Kodaira and Spencer. Although the famous Kodaira vanishing theorem is clothed in the language of Cech cohomology, this cohomology is represented by harmonic forms, thus making its understanding more accessible. The authors also show explicitly to what extent an algebraic variety can be thought of as a compact complex manifold via the Kodaira embedding theorem. Projective space of course is not the most complicated of constructions, as readers familiar with the theory of vector bundles will know. Grassmannians are an example of this, and they are introduced and discussed in the book as generalizations of projective space. And, just as in the ordinary theory of vector bundles, the authors show how to use Grassmannians to act as universal bundles for holomorphic vector bundles.
The presence of meromorphic functions will alert the astute reader as to the role of Riemann surfaces in the study of complex algebraic varieties. Indeed, in chapter 2, the authors cast many classical complex analytic results to modern ones, and they prove the famous Riemann-Roch theorem, which essentially counts the number of meromorphic functions on a Riemann surface of genus g. The theory of Abelian varieties is outlined, and the reader gets a taste of "Italian" algebraic geometry but done in the rigorous setting of Plucker formulas and coordinates.
Chapter 3 is a summary of some of the other methodologies and techniques used to study general analytic varieties, the first of these being the theory of currents, i.e differential forms with distribution coefficients. It is perhaps not surprising to see this applied here, given that it can handle both the smooth and piecewise smooth chains simultaneously. The currents are associated to analytic varieties and allow a definition of their intersection numbers and a proof that they are positive. The all-important Chern classes are introduced here, and it is shown that the Chern classes of a holomorphic vector bundle over an algebraic variety are fundamental classes of algebraic cycles. Most importantly the authors introduce spectral sequences, a topic that is usually formidable for newcomers to algebraic geometry.
The study of surfaces is studied in chapter 4, with the differences between its study and the theory of curves (Riemann surfaces) emphasized. The reader gets a first crack at the notion of a rational map, and the birational classification of surfaces is shown. Intuitively, one expects that the classification of surfaces would be easy if it were not for "singular points", and this is born out in the use of blowing up singularities in this chapter. Rational surfaces are characterized using Noether's lemma, and a rather detailed discussion is given of surfaces that are not rational, giving the reader more examples of rigorous "Italian" geometry.
2) If you are a graduate student in mathematics or related fields and you are interested in learning algebraic geometry in the Griffiths-Harris way, then I suggest before buying this book to have a good background in the following: 1. Complex Analysis 2. Differential Geometry and calculus on manifolds 3. Homology-Cohomology Theory 4. Undergraduate Algebraic Geometry
Do not expect chapter 0, "Foundational Material", to be the place where you are supposed to build your "foundation". You can try the books of Michael Spivak, David A. Cox, Fangyang Zheng, among other books for foundational material but not chapter 0.
However, if you have most of the above-mentioned foundational material, then this book is good in presenting complex manifolds for example in chapter 0 section 2 and also in presenting (complex) holomorphic vector bundles, as well as many other things.
So, in summary, I would say a good book but not for students trying to learn the basics in algebraic geometry.
书名：The Red Book of Varieties and Schemes
作者： David Mumford
出版商： Springer-Verlag (1994) ISBN 3-540-50497-4
页数：309
适用范围：基础数学研究生
预备知识：近世代数、复分析、点集拓扑
习题数量：很少
习题难度： 较难
推荐强度：8.6
书评： 代数几何学家 Mumford 是美国科学院院士，菲尔滋奖获得者，在哈佛大学任教多年。 本书实际上是 60 年代他在哈佛的代数几何课程的讲义。即使到80年代有了 Hartshorne, Shafarevich 写的优秀教科书，很多初学者仍然喜欢 Mumford 的老讲义,油印本在研究生中广为流传。Springer Verlag 经专家的推荐便将这些讲义原封不动地出版了，由于原来的油印的封面是红色的，故此书就被亲切地取名为 red book. 比较可惜的是这本书只有三章，由于种种原因作者未能把原来写讲义的庞大计划执行到底。
代数几何是抽象概念非常多的一门数学分支，初学者需要化很大的精力来理解、消化和记住一大堆基本概念。 Mumford 的讲义用朴实无华的方式解释代数几何这些概念的来龙去脉，一点不落俗套。从目录上看，这些内容和 其它同类的书差不多，事实上具体的论述还是很不一样的，讲义行文的非正式的风格也使枯燥的数学变的生动。 作者自嘲地称这本书里一个定理也没有，这多少有些夸张， 很多应该叫做定理的结论在这位大师面前大概不能称为定理。 （杨劲根）
国外评论摘选
1）In a nutshell, reading this book is like reading the mind of a great mathematician as he thinks about a great new idea. Anyone interested in schemes should read it. But a review needs more detail: The RED BOOK is a concise, brilliant survey of schemes, by one of the first mathematicians to learn of them from Grothendieck. He gives wonderfully intuitive pictures of schemes, especially of "arithmetic schemes" where number theory appears as geometry. The geometry shines through it all: as in differentials, and etale maps, and how unique factorization relates to non-singularity. There is a bravura discussion of Zariski's Main Theorem (the algebraic property of being "normal" implies that a variety has only one branch at each point) comparing forms of it from older algebraic geometry, topology, power series, and schemes. Mumford cites proofs of these but does not give them. In fact, this theorem was one of the first things Mumford could use, to get Zariski to respect schemes.
Many accomplished algebraic geometers say this book got them started. But you probably cannot learn to work in the subject from this book alone--you either have to work with people who work with it, or use some other books besides (maybe both). The other book would probably be Hartshorne ALGEBRAIC GEOMETRY, which is far more detailed, has far more examples, goes very much farther into cohomology--and is very much longer and denser (though also clearly written).
2）There is a problem in getting going with alg. geo. To learn the geometry you need commutative algebra and to contextualize commutative algebra you need algebraic geometry. Mumford is an excellent book to get going without the need for the heavy prereqs of the more classic books like Hartshorne or Griffiths-Harris. A really good read. This is not however a terrific reference text, you'll need something else as a reference. Its much to expository and their is no index.
书名：Compact Complex Surfaces, 2nd edition
作者：W.P. Barth, K. Hulek, C.A.M. Peters, A. Van de Ven
出版商：Springer-Verlag (2003) ISBN 3-540-00832-2
页数：436
适用范围：代数几何、复几何、微分几何方向研究生
预备知识：多复变函数论、代数几何、复微分几何
习题数量：无
推荐强度：9
书评： "紧复曲面"专著第一版于1984年出版，自其出版以来因其选材精致，重点突出而广受青睐， 内行称其为"BPV"（第一版的三位作者的姓氏的开头字母）。第二版增加了不少新的研究成果， 但笔者认为第二版的组织过于仓促，反而给人画蛇添足的感觉，尽管如此， 这仍不失为一本优秀的专业工具书。
本书第一章列出了必需的预备知识，虽然没有证明，但笔者认为该内容十分恰当。 第二章中，作者分别介绍了曲面上的曲线、Riemann-Roch定理、相交理论；第三章中介绍了曲面的奇点、 纤维化方法和稳定纤维化的周期映射。然后从第四章开始，本书着重讲述曲面的一般性质、 特殊曲面的分类和一般型曲面的典范分类。本书的最后一章中主要介绍曲面的拓扑和微分结构。 笔者认为，本书对于一般型曲面的分类内容略显陈旧。
总的来说，这是一本介绍代数曲面理论的极好工具书。（陈猛）
|
|
|
