eisenbud交换代数 solutions

近世代数第9讲置换群(pormutation group)本讲的教学目的和要求:置换群是一种特殊的变换群。

换句话说，置换群就是有限集上的变换群。

由于是定义在有限集上，故每个置换的表现形式，固有特点都是可揣测的。

这一讲主要要求：1、弄清置换与双射的等同关系。

2、掌握置换—轮换—对换之间的联系和置换的奇偶性。

3、置换的分解以及将轮换表成对换之积的基本方法要把握。

4、对称群与交错群的结构以及有限群的cayley定理需要理解。

本讲的重点与难点：对于置换以及置换群需要侧重注意的是：对称群和交错群的结构和置换的分解定理（定理2）。

注意：由有限群的cayley定理可知：如把所有置换群研究清楚了。

就等于把所有有限群都研究清楚了，但经验告诉我们，研究置换群并不比研究抽象群容易。

所以，一般研究抽象群用的还是直接的方法。

并且也不能一下子把所有群都不得找出来。

因为问题太复杂了。

人们的方法是将群分成若干类（即附加一定条件）；譬如有限群；无限群；变换群；非变换群等等。

对每个群类进行研究以设法回答上述三个问题。

可惜 ， 人们能弄清的群当今只有少数几类（后面的循环群就是完全解决了的一类群）大多数还在等待人们去解决。

变换群是一类应用非常广泛的群，它的具有代表性的特征—置换群，是现今所研究的一切抽象群的来源，是抽象代数创始人E.Galais(1811-1832)在证明次数大于四的一元代数方程不可能用根号求解时引进的。

一． 置换群的基本概念定义1.任一集合A 到自身的映射都叫做A 的一个变换，如果A 是有限集且变换是一一变换（双射），那么这个变换为A 的一个置换。

有限集合A 的若干个置换若作成群，就叫做置换群。

含有n 个元素的有限群A 的全体置换作成的群，叫做n 次对称群。

通常记为n S .明示：由定义1知道，置换群就是一种特殊的变换群（即有限集合上的变换群）而n 次对称群n S 也就是有限集合A 的完全变换群。

现以{}321 , , a a a A =为例，设π：A →A 是A 的一一变换。

Z[x]的素理想与Krull维数Advances in Applied Mathematics 应⽤数学进展, 2017, 6(8), 942-945Published Online November 2017 in Hans. /doc/045644411.html/journal/aam https:///doc/045644411.html/10.12677/aam.2017.68113Prime Ideals and Krull Dimension of []xRongzheng JiaoSchool of Mathematics Science, Yangzhou University, Yangzhou JiangsuReceived: Nov. 4th, 2017; accepted: Nov. 17th, 2017; published: Nov. 23rd, 2017AbstractUsing elementary method, we get all the prime ideals of integral domain []x , which give an ex-plicit proof of a result in Mumford’s red book. We get the Krull dimension 2 of []x by directcomputation as a by-product.KeywordsIntegral Domain, Prime Ideal, Maximal Ideal, Krull Dimension, Euclid Domain[]x 的素理想与Krull 维数焦荣政扬州⼤学数学科学学院，江苏扬州收稿⽇期：2017年11⽉4⽇；录⽤⽇期：2017年11⽉17⽇；发布⽇期：2017年11⽉23⽇摘要本⽂⽤初等⽅法考虑⼀元多项式环[]x 上的素理想、极⼤理想。

学数学的必看GTM经典著作下载三202 Introduction to Topological Manifolds,John M.Lee(拓扑流形入门)镜像下载(4874KB，英文版，DJVU格式，支持关键词检索，点击打开下载页面，支持迅雷、快车下载)203 The Symmetric Group,Bruce E.Sagan 204 Galois Theory,Jean-Pierre Escofier 205 Rational Homotopy Theory,Yves Félix,Stephen Halperin,Jean-Claude Thomas(有理同伦论)镜像下载(5220KB，英文版，DJVU格式，支持关键词检索，点击打开下载页面，支持迅雷、快车下载)有理同伦论是由Sullivan创立的。

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曾任JAC的主编，现在是组合学期刊(JC)电子版的主编。

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KressGTM182《Ordinary Differential Equations》Wolfgang WalterGTM183《An introduction to Banach Spaces》Robert E.MegginsonGTM184《Modern Graph Theory》Béla Bollobás（现代图论）GTM185《Using Algebraic Geomety》David A.Cox, John Little, Donal O’Shea（应用代数几何）GTM186《Fourier Analysis on Number Fields》Dinakar Ramakrishnan, Robert J.Valenza GTM187《Moduli of Curves》Joe Harris, Ian Morrison（曲线模）GTM188《Lectures on the Hyperreals：An Introduction to Nonstandard Analysis》Robert GoldblattGTM189《Lectures on Modules and Rings》m（模和环讲义）GTM190《Problems in Algebraic Number Theory》M.Ram Murty, Jody Esmonde（代数数论中的问题）GTM191《Fundamentals of Differential Geometry》Serge Lang（微分几何基础）GTM192《Elements of Functional Analysis》Francis Hirsch, Gilles LacombeGTM193《Advanced Topics in Computational Number Theory》Henri CohenGTM194《One-Parameter Semigroups for Linear Evolution Equations》Klaus-Jochen Engel, Rainer Nagel（线性发展方程的单参数半群）GTM195《Elementary Methods in Number Theory》Melvyn B.Nathanson（数论中的基本方法）GTM196《Basic Homological Algebra》M.Scott OsborneGTM197《The Geometry of Schemes》David Eisenbud, Joe HarrisGTM198《A Course in p-adic Analysis》Alain M.RobertGTM199《Theory of Bergman Spaces》Hakan Hedenmalm, Boris Korenblum, Kehe Zhu（Bergman空间理论）GTM200《An Introduction to Riemann-Finsler Geometry》D.Bao, S.-S.Chern, Z.Shen GTM201《Diophantine Geometry An Introduction》Marc Hindry, Joseph H.Silverman GTM202《Introduction to Topological Manifolds》John M.LeeGTM203《The Symmetric Group》Bruce E.SaganGTM204《Galois Theory》Jean-Pierre EscofierGTM205《Rational Homotopy Theory》Yves Félix, Stephen Halperin, Jean-Claude Thomas（有理同伦论）GTM206《Problems in Analytic Number Theory》M.Ram MurtyGTM207《Algebraic Graph Theory》Chris Godsil, Gordon Royle（代数图论）GTM208《Analysis for Applied Mathematics》Ward CheneyGTM209《A Short Course on Spectral Theory》William Arveson（谱理论简明教程）GTM210《Number Theory in Function Fields》Michael RosenGTM211《Algebra》Serge Lang（代数）GTM212《Lectures on Discrete Geometry》Jiri Matousek（离散几何讲义）GTM213《From Holomorphic Functions to Complex Manifolds》Klaus Fritzsche, Hans Grauert（从正则函数到复流形）GTM214《Partial Differential Equations》Jüergen Jost（偏微分方程）GTM215《Algebraic Functions and Projective Curves》David M.Goldschmidt（代数函数和投影曲线）GTM216《Matrices：Theory and Applications》Denis Serre（矩阵：理论及应用）GTM217《Model Theory An Introduction》David Marker（模型论引论）GTM218《Introduction to Smooth Manifolds》John M.Lee（光滑流形引论）GTM219《The Arithmetic of Hyperbolic 3-Manifolds》Colin Maclachlan, Alan W.Reid GTM220《Smooth Manifolds and Observables》Jet Nestruev（光滑流形和直观）GTM221《Convex Polytopes》Branko GrüenbaumGTM222《Lie Groups, Lie Algebras, and Representations》Brian C.Hall（李群、李代数和表示）GTM223《Fourier Analysis and its Applications》Anders Vretblad（傅立叶分析及其应用）GTM224《Metric Structures in Differential Geometry》Gerard Walschap（微分几何中的度量结构）GTM225《Lie Groups》Daniel Bump（李群）GTM226《Spaces of Holomorphic Functions in the Unit Ball》Kehe Zhu（单位球内的全纯函数空间）GTM227《Combinatorial Commutative Algebra》Ezra Miller, Bernd Sturmfels（组合交换代数）GTM228《A First Course in Modular Forms》Fred Diamond, Jerry Shurman（模形式初级教程）GTM229《The Geometry of Syzygies》David Eisenbud（合冲几何）GTM230《An Introduction to Markov Processes》Daniel W.Stroock（马尔可夫过程引论）GTM231《Combinatorics of Coxeter Groups》Anders Bjröner, Francesco Brenti（Coxeter 群的组合学）GTM232《An Introduction to Number Theory》Graham Everest, Thomas Ward（数论入门）GTM233《Topics in Banach Space Theory》Fenando Albiac, Nigel J.Kalton（Banach空间理论选题）GTM234《Analysis and Probability：Wavelets, Signals, Fractals》Palle E.T.Jorgensen（分析与概率）GTM235《Compact Lie Groups》Mark R.Sepanski（紧致李群）GTM236《Bounded Analytic Functions》John B.Garnett（有界解析函数）GTM237《An Introduction to Operators on the Hardy-Hilbert Space》Rubén A.Martínez-Avendano, Peter Rosenthal（哈代-希尔伯特空间算子引论）GTM238《A Course in Enumeration》Martin Aigner（枚举教程）GTM239《Number Theory：VolumeⅠTools and Diophantine Equations》Henri Cohen GTM240《Number Theory：VolumeⅡAnalytic and Modern Tools》Henri Cohen GTM241《The Arithmetic of Dynamical Systems》Joseph H.SilvermanGTM242《Abstract Algebra》Pierre Antoine Grillet（抽象代数）GTM243《Topological Methods in Group Theory》Ross GeogheganGTM244《Graph Theory》J.A.Bondy, U.S.R.MurtyGTM245《Complex Analysis：In the Spirit of Lipman Bers》Jane P.Gilman, Irwin Kra, Rubi E.RodriguezGTM246《A Course in Commutative Banach Algebras》Eberhard KaniuthGTM247《Braid Groups》Christian Kassel, Vladimir TuraevGTM248《Buildings Theory and Applications》Peter Abramenko, Kenneth S.Brown GTM249《Classical Fourier Analysis》Loukas Grafakos（经典傅里叶分析）GTM250《Modern Fourier Analysis》Loukas Grafakos（现代傅里叶分析）GTM251《The Finite Simple Groups》Robert A.WilsonGTM252《Distributions and Operators》Gerd GrubbGTM253《Elementary Functional Analysis》Barbara D.MacCluerGTM254《Algebraic Function Fields and Codes》Henning StichtenothGTM255《Symmetry Representations and Invariants》Roe Goodman, Nolan R.Wallach GTM256《A Course in Commutative Algebra》Kemper GregorGTM257《Deformation Theory》Robin HartshorneGTM258《Foundation of Optimization》Osman GülerGTM259《Ergodic Theory：with a view towards Number Theory》Manfred Einsiedler, Thomas WardGTM260《Monomial Ideals》Jurgen Herzog, Takayuki HibiGTM261《Probability and Stochastics》Erhan CinlarGTM262《Essentials of Integration Theory for Analysis》Daniel W.StroockGTM263《Analysis on Fock Spaces》Kehe ZhuGTM264《Functional Analysis, Calculus of Variations and Optimal Control》Francis ClarkeGTM265《Unbounded Self-adjoint Operatorson Hilbert Space》Konrad Schmüdgen GTM266《Calculus Without Derivatives》Jean-Paul PenotGTM267《Quantum Theory for Mathematicians》Brian C.HallGTM268《Geometric Analysis of the Bergman Kernel and Metric》Steven G.Krantz GTM269《Locally Convex Spaces》M.Scott Osborne。

美国数学本科生、研究生基础课程参考书目在网上找书的时候恰好看到这个，看着觉得的确是经典书目大全，贴在这里供学弟学妹们参考：）其中所谓第几学年云云，各校要求不同，像我所在的学校，一般学生第一年选三到四门基础课（代数、分析、几何三大类中至少各挑一门），学年末进行qualifying笔试。

第二年开始选自己喜爱方向的高级课程，并通过qualifying口试。

第三年开始做research，并通过第二语言考试（法语或德语或俄语，一般人都选法语，因为代数几何经典大作都是法语的）. 而Princeton 就没有基础课，只有seminar类型的课。

第一学年几何与拓扑：1、James R. Munkres, Topology：较新的拓扑学的教材适用于本科高年级或研究生一级；2、Basic Topology by Armstrong：本科生拓扑学教材；3、Kelley, General Topology：一般拓扑学的经典教材，不过观点较老；4、Willard, General Topology：一般拓扑学新的经典教材；5、Glen Bredon, Topology and geometry：研究生一年级的拓扑、几何教材；6、Introduction to Topological Manifolds by John M. Lee：研究生一年级的拓扑、几何教材，是一本新书；7、from calculus to cohomology by Madsen：很好的本科生代数拓扑、微分流形教材。

代数：1、Abstract Algebra Dummit：最好的本科代数学参考书，标准的研究生一年级代数材；2、Algebra Lang：标准的研究生一、二年级代数教材，难度很高，适合作参考书；3、Algebra Hungerford：标准的研究生一年级代数教材，适合作参考书；4、Algebra M,Artin：标准的本科生代数教材；5、Advanced Modern Algebra by Rotman：较新的研究生代数教材，很全面；6、Algebra：a graduate course by Isaacs：较新的研究生代数教材；7、Basic algebra Vol I&II by Jacobson：经典的代数学全面参考书，适合研究生参考。

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eisenbud交换代数Eisenbud交换代数是现代代数表面理论的一个重要分支。

它是由David Eisenbud在20世纪80年代中期提出的，以研究代数曲线在仿射流形上的理想和矢量丛的几何性质为出发点。

Eisenbud交换代数的主要研究对象是拟凸或凸流形上的交换代数，这些代数在某种意义下是仿射代数簇上坐标环的广义推广。

Eisenbud交换代数的研究方法主要是通过引入使非交换性能有一定控制的辅助结构，以及利用几何工具和代数工具相互之间的相互联系，来研究非交换环的性质和结构。

最初的几何工具就是亚纯函数和亚纯函数理论，在此基础上，Eisenbud发展了一系列有力工具和概念，如奇异连续函数和微分流形的剪切结构，从而使得Eisenbud交换代数具有很强的几何性质和结构性质。

Eisenbud交换代数的一个重要方向是研究非交换代数的表示理论。

表示理论是研究抽象代数结构的一种重要方法，它将代数结构转化为线性代数结构，从而为代数结构的研究提供了新的视角和工具。

Eisenbud交换代数通过引入表示理论的方法来研究非交换代数的结构和性质，这为研究非交换代数的代数性质和几何性质提供了一种新的途径和思路。

另一个重要的研究方向是Eisenbud交换代数在代数几何中的应用。

代数几何是研究代数对象的几何性质和代数性质的一门学科，它是数学的一个重要分支。

Eisenbud交换代数在代数几何中的应用是通过发展Eisenbud环和Eisenbud模等一系列新的代数工具，来研究代数几何中的关键问题，如切空间的几何性质、剪切环的结构、局部稳定性和局部代数性质等。

此外，Eisenbud交换代数还在代数拓扑学、代数几何流形、非交换系统的拓扑性质和结构、代数K理论等领域有着广泛的应用和影响。

总的来说，Eisenbud交换代数作为现代代数表面理论的一个重要分支，通过引入辅助结构和利用几何工具和代数工具的相互联系，为代数问题的研究提供了一种新的方法和思路，开辟了代数学和几何学的新研究领域，对于代数学和数学的发展具有重要的意义。

数学基础学习阶段◆分析学微积分学教程（1、2、3册）菲赫金哥尔茨数学分析教程（上、下册）史济怀Principles of Mathematical Analysis (Third Edition) Walter Rudin实变函数江泽坚实变函数论周民强复分析导论（上、下册）沙巴特泛函分析讲义（上、下）张恭庆Real and Complex Analysis(Third Edition) Walter RudinFuctional Analysis(Second Edition) Walter Rudin◆代数学高等代数（北京大学数学与力学系）前代数小组代数学引论（聂灵沼、丁石孙）Algebra HungerfordAlgebra Lang美国大学数学参考书目录:第一学年几何与拓扑：1、James R. Munkres, Topology：较新的拓扑学的教材适用于本科高年级或研究生一年级；2、Basic Topology by Armstrong：本科生拓扑学教材；3、Kelley, General Topology：一般拓扑学的经典教材，不过观点较老；4、Willard, General Topology：一般拓扑学新的经典教材；5、Glen Bredon, Topology and geometry：研究生一年级的拓扑、几何教材；6、Introduction to Topological Manifolds by John M. Lee：研究生一年级的拓扑、几何教材，是一本新书；7、From calculus to cohomology by Madsen：很好的本科生代数拓扑、微分流形教材。

代数：1、Abstract Algebra Dummit：最好的本科代数学参考书，标准的研究生一年级代数教材；2、Algebra Lang：标准的研究生一、二年级代数教材，难度很高，适合作参考书GTM；3、Algebra Hungerford：标准的研究生一年级代数教材，适合作参考书GTM；4、Algebra M,Artin：标准的本科生代数教材；5、Advanced Modern Algebra by Rotman：较新的研究生代数教材，很全面；6、Algebra：a graduate course by Isaacs：较新的研究生代数教材；7、Basic algebra Vol I&II by Jacobson：经典的代数学全面参考书，适合研究生参考。

eisenbud交换代数【原创实用版】目录1.交换代数的概念2.Eisenbud 交换代数的定义3.Eisenbud 交换代数的性质4.Eisenbud 交换代数的应用正文1.交换代数的概念交换代数是代数学的一个重要分支，主要研究交换环（即具有加法交换律的环）的性质及其表示。

交换代数起源于 19 世纪末 20 世纪初，当时数学家们对线性代数和代数几何的研究日益深入，交换代数作为这两个领域的交叉学科应运而生。

交换代数有着广泛的应用，例如在数学的各个领域，以及物理学、计算机科学等领域。

2.Eisenbud 交换代数的定义Eisenbud 交换代数是一种特殊的交换代数，由美国数学家 Eisenbud 于 20 世纪 50 年代提出。

Eisenbud 交换代数是在给定向量空间上的线性变换的基础上，通过引入一种新的乘法运算定义而成的。

具体地，设 V 是一个向量空间，L(V) 是 V 上的线性变换组成的集合，那么 Eisenbud 交换代数可以表示为：A = span{L(V)}，其中 span 表示线性组合。

3.Eisenbud 交换代数的性质Eisenbud 交换代数具有以下几个重要性质：（1）结合律：对于 A 中的任意元素 a 和 b，有 a * b = b * a。

（2）交换律：对于 A 中的任意元素 a 和 b，有 a * b = b * a。

（3）分配律：对于 A 中的任意元素 a、b 和 c，有 a * (b + c) = a * b + a * c。

（4）单位元：存在一个元素 e，对于 A 中的任意元素 a，满足 a * e = e * a = a。

（5）乘法对加法的分配律：对于 A 中的任意元素 a、b 和 c，有 a * (b + c) = a * b + a * c。

4.Eisenbud 交换代数的应用Eisenbud 交换代数在数学领域有着广泛的应用，例如在代数几何、线性代数、微积分等分支中都有重要的应用。

美国大学本科数学专业的必修课及教材（Required courses and materials for undergraduate mathematics in the United States）What are the required courses and teaching materials for undergraduate math majors in the United States? Questioner: 2008-10-18 22:01 _ blue shadow mourning drunk, | reward points: 100 | Views: 2387I'm going to America next year undergraduate mathematics, professional choice, now want to own a preview, hope to understand the required curriculum in the United States about 40 of university mathematics and the use of the teaching materials are (what is the most important about mathematics curriculum) hope that good hearted people tell me, thank you...2008-10-31 19:10 satisfied answer American undergraduate mathematics, Graduate basic curriculum reference bookFirst academic yearGeometry and topology:1, James, R., Munkres, Topology: the new topology of teaching materials for undergraduate senior or graduate freshmen;2, Basic, Topology, by, Armstrong: Undergraduate topology textbooks;3, Kelley, General, Topology: the classic textbook of general topology, but the older point of view;4, Willard, General, Topology: new classical textbook of general topology;5, Glen, Bredon, Topology, and, geometry: Graduate freshmen topology and geometry textbooks;6, Introduction, to, Topological,, Manifolds, by, John, M., Lee: Graduate freshman topology and geometry textbook, is a new book;7, From, calculus, to, cohomology, by, Madsen: good undergraduate algebra topology and differential manifold teaching materials.Algebra：1, Abstract, Algebra, Dummit: the best undergraduate algebra reference books, standard graduate students, first-year algebra textbooks;2, Algebra Lang: standard graduate student, grade one or two algebra teaching material, very difficult, suitable for reference books;3, Algebra, Hungerford: standard graduate freshmen algebra textbooks, suitable for reference books;4, Algebra, M, Artin: standard undergraduate algebra textbooks;5, Advanced, Modern, Algebra, by, Rotman: newer graduatealgebra textbooks, very comprehensive;6, Algebra:a, graduate, course, by, Isaacs: newer graduate algebra textbooks;7, Basic, algebra, Vol, I&II, by, Jacobson: classical algebra comprehensive reference book, suitable for graduate reference.Analysis basis:1, Walter, Rudin, Principles, of, mathematical, analysis: standard reference books for undergraduate mathematical analysis;2, Walter, Rudin, Real, and, complex, analysis: standard graduate freshmen analysis textbooks;3, Lars, V., Ahlfors, Complex, analysis: senior undergraduate and graduate students in the first grade classic analysis of teaching materials;4, Functions, of, One, Complex, Variable, I, J.B.Conway: graduate level single variable complex analysis classic;5, Lang, Complex, analysis: graduate level single variable complex analysis reference book;6, Complex, Analysis, by, Elias, M., Stein: newer graduate level single variable complex analysis textbooks;7, Lang, Real, and, Functional, analysis: graduate levelanalytical reference books;8, Royden, Real, analysis: standard graduate students in the first year of the actual analysis of teaching materials;9, Folland, Real, analysis: standard graduate students in the first year of the actual analysis of teaching materials.Second academic yearAlgebra：1, Commutative, ring, theory, by, H., Matsumura: newer graduate exchange algebra standard textbook;2, Commutative, Algebra, I&II, by, Oscar, Zariski, Pierre, Samuel: classical commutative algebra reference book;3, An, introduction, to, Commutative, Algebra, by, Atiyah: standard introductory textbook on commutative algebra;4, An, introduction, to, homological, algebra, by, Weibel: newer graduate students' two year coherence algebra textbooks;5, A, Course, in, Homological, Algebra, by, P.J.Hilton, U.Stammbach: classical and comprehensive homological algebra reference book;6, Homological, Algebra, by, Cartan: classical homological algebra reference books;7, Methods, of, Homological, Algebra, by, Sergei, I., Gelfand, Yuri, I., Manin: advanced and classical homological algebra reference books;8, Homology, by, Saunders, Mac, Lane: an introduction to the classical homological algebra system;9, Commutative, Algebra, with, a,, view, toward, Algebraic, Geometry, by, Eisenbud: Advanced Algebra geometry, commutative algebra reference book, the latest exchange algebra comprehensive reference.Algebraic topology:1, Algebraic, Topology, A., Hatcher: the latest graduate algebra topology standard textbook;2, Spaniers, "Algebraic, Topology": classical algebraic topology reference book;3, Differential, forms, in, algebraic, topology, by, Raoul, Bott, and, Tu, Loring, W.: Graduate algebra topology standard textbook;4, Massey, A, basic, course, in, Algebraic, topology: classical graduate algebra topology textbooks;5, Fulton, Algebraic, topology:a, first, course: very good algebra algebra reference for freshmen and graduate students in their freshman year;6, Glen, Bredon, Topology, and, geometry: standard graduate algebra topology textbooks, there is a considerable space to talk about smooth manifolds;7, Algebraic, Topology, Homology, and, Homotopy: advanced and classical algebraic topological reference books;8, A, Concise, Course, in,, Algebraic, Topology, by, J.P.May: Graduate algebra topology introductory materials, covering a wide range;9, Elements, of, Homotopy, Theory, by, G.W., Whitehead: advanced and classical algebraic topological reference books.Real analysis and functional analysis:1, Royden, Real, analysis: standard graduate analysis textbooks;2, Walter, Rudin, Real, and, complex, analysis: standard graduate analysis textbooks;3, Halmos, "Measure Theory": Classic graduate analysis of teaching materials, suitable for reference books;4, Walter, Rudin, Functional, analysis: standard graduate functional analysis textbooks;5, Conway, A, course, of, Functional, analysis: standard graduate functional analysis textbooks; 6, Folland, Real,analysis: standard graduate student analysis of teaching materials;7, Functional, Analysis, by, Lax: advanced graduate functional analysis textbooks;8, Functional, Analysis, by, Yoshida: advanced graduate functional analysis reference book;9, Measure, Theory, Donald, L., Cohn: classical measurement theory reference book.Differential topology Li Qun and Lie algebra1, Hirsch, Differential, topology: standard graduate differential topology textbooks, quite difficult;2, Lang, Differential, and, Riemannian, manifolds: graduate reference books of Differential Manifolds, higher difficulty;3, Warner, Foundations, of, Differentiable, manifolds, and, Lie, groups: standard graduate Differential Manifolds teaching materials, there is considerable space about Li Qun;4, Representation, theory:, a, first, course, by, W., Fulton, and, J., Harris: Li Qun and its representation standards;5, Lie, groups, and, algebraic, groups, by, A., L., Onishchik, E., B., Vinberg: Li Qun's reference book;6, Lectures, on, Lie, Groups, W.Y.Hsiang: Li Qun's referencebook;7, Introduction, to, Smooth, Manifolds, by, John, M., Lee: the newer standard textbook on smooth manifolds;8, Lie, Groups, Lie, Algebras, and, Their, Representation, by, V.S., Varadarajan: the most important reference book of Li Qun and Li algebra;9, Humphreys,李代数及其表示理论，介绍SpringerVerlag，gtm9：标准的李代数入门教材。

eisenbud交换代数摘要：I.引言A.交换代数的概念B.Eisenbud 交换代数的背景与意义II.Eisenbud 交换代数的定义与性质A.定义B.性质C.举例说明III.Eisenbud 交换代数与其他交换代数的联系与区别A.与经典交换代数的联系与区别B.与其他交换代数的联系与区别IV.Eisenbud 交换代数的应用A.在代数几何中的应用B.在数论中的应用C.在其他领域的应用V.Eisenbud 交换代数的局限性与未来展望A.局限性B.未来展望VI.结论A.对Eisenbud 交换代数的评价B.对未来研究的启示正文：交换代数是代数学中的一个重要分支，主要研究具有交换性的结合律和分配律的代数结构。

Eisenbud 交换代数是这一领域的一个重要成果，它的定义、性质及其与其他交换代数的关系等问题一直是数学家们关注的焦点。

Eisenbud 交换代数是由数学家David Eisenbud 在20 世纪80 年代提出的，它的主要思想是将经典的交换代数进行扩展，以更好地刻画代数几何中的一些现象。

具体来说，Eisenbud 交换代数是一种具有某些局部性质的交换代数，这种局部性质使得它能够更好地处理代数几何中的局部问题。

与经典交换代数相比，Eisenbud 交换代数具有更强的表达能力。

例如，它可以更好地描述代数几何中的奇点现象。

然而，Eisenbud 交换代数也有其局限性，比如在处理一些全局性质问题时，它的表达能力可能不如经典交换代数。

在应用方面，Eisenbud 交换代数在代数几何、数论等领域都取得了重要的成果。

例如，它在研究代数曲面的奇点、代数簇的有限生成性质等问题时发挥了关键作用。

同时，Eisenbud 交换代数也为这些领域的研究提供了新的研究方法和思路。

总之，Eisenbud 交换代数是交换代数学的一个重要成果，它在刻画代数几何中的局部性质方面具有独特的优势。

尽管它还存在一些局限性，但无疑为代数学的研究提供了一个有力的工具。

Heisenberg群上二阶左不变LPDO的局部可解性—辛变换
技巧
佚名
【期刊名称】《兰州大学学报（自然科学版）》
【年(卷),期】1992(000)001
【摘要】无
【总页数】1页(P1)
【正文语种】中文
【中图分类】O1
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