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A Singular Introduction to Commutative Algebra / by Gert-Martin Greuel, Gerhard Pfister
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: SpringerLink BücherPublisher: Berlin, Heidelberg : Springer-Verlag Berlin Heidelberg, 2007Description: Online-Ressource (XX, 690p. 49 illus, digital)ISBN:- 9783540735427
- 512.44
- 512
- 510
- 510
- QA150-272
- QA251.3
Contents:
Summary: Rings, Ideals and Standard Bases -- Modules -- Noether Normalization and Applications -- Primary Decomposition and Related Topics -- Hilbert Function and Dimension -- Complete Local Rings -- Homological Algebra.Summary: From the reviews: "…It is certainly no exaggeration to say that Greuel and Pfister's A Singular Introduction to Commutative Algebra aims to lead a further stage in the computational revolution in commutative algebra, in which computational methods and results become central to how the subject is taught and learned. […] Among the great strengths and most distinctive features of Greuel and Pfister's book is a new, completely unified treatment of the global and local theories. The realization that the two cases could be combined to this extent was decisive in the design of the Singular system, making it one of the most flexible and most efficient systems of its type. The authors present the first systematic development of this unified approach in a textbook here, and this aspect alone is almost worth the price of admission. Another distinctive feature of this book is the degree of integration of explicit computational examples into the flow of the text. Strictly mathematical components of the development (often quite terse and written in a formal "theorem-proof" style) are interspersed with parallel discussions of features of Singular and numerous Singular examples giving input commands, some extended programs in the Singular language, and output. […] Yet another strength of Greuel and Pfister's book is its breadth of coverage of theoretical topics in the portions of commutative algebra closest to algebraic geometry, with algorithmic treatments of almost every topic. A synopsis of the table of contents will make this clear. […] Greuel and Pfister have written a distinctive an highly useful book that should be in the library of every commutative algebrais and algebraic geometer, expert and novice alike. I hope that it achieves the educational impact it deserves." John B. Little, Monthly of The Mathematical Association of America, March 2004 "... The authors' most important new focus is the presentation of non-well orderings that allow them the computational approach for local commutative algebra. In fact the book provides an introduction to commutative algebra from a computational point of view. So it might be helpful for students and other interested readers (familiar with computers) to explore the beauties and difficulties of commutative algebra by computational experiences. In this respect the book is the one of the first samples of a new kind of textbooks in algebra." P.Schenzel, Zentralblatt für Mathematik 1023.13001, 2003.PPN: PPN: 1645737004Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
""Preface to the Second Edition""; ""Preface to the First Edition""; ""Contents""; ""1. Rings, Ideals and Standard Bases""; ""1.1 Rings, Polynomials and Ring Maps""; ""Exercises""; ""1.2 Monomial Orderings""; ""Exercises""; ""1.3 Ideals and Quotient Rings""; ""Exercises""; ""1.4 Local Rings and Localization""; ""Exercises""; ""1.5 Rings Associated to Monomial Orderings""; ""Exercises""; ""1.6 Normal Forms and Standard Bases""; ""Exercises""; ""1.7 The Standard Basis Algorithm""; ""Exercises""; ""1.8 Operations on Ideals and Their Computation""; ""Exercises""
""1.9 Non�Commutative G�Algebras""""2. Modules""; ""2.1 Modules, Submodules and Homomorphisms""; ""Exercises""; ""2.2 Graded Rings and Modules""; ""Exercises""; ""2.3 Standard Bases for Modules""; ""Exercises""; ""2.4 Exact Sequences and Free Resolutions""; ""Exercises""; ""2.5 Computing Resolutions and the Syzygy Theorem""; ""Exercises""; ""2.6 Modules over Principal Ideal Domains""; ""Exercises""; ""2.7 Tensor Product""; ""Exercises""; ""2.8 Operations on Modules and Their Computation""; ""Exercises""; ""3. Noether Normalization and Applications""; ""3.1 Finite and Integral Extensions""
""Exercises""""3.2 The Integral Closure""; ""Exercises""; ""3.3 Dimension""; ""Exercises""; ""3.4 Noether Normalization""; ""Exercises""; ""3.5 Applications""; ""Exercises""; ""3.6 An Algorithm to Compute the Normalization""; ""Exercises""; ""3.7 Procedures""; ""4. Primary Decomposition and Related Topics""; ""4.1 The Theory of Primary Decomposition""; ""Exercises""; ""4.2 Zero�dimensional Primary Decomposition""; ""Exercises""; ""4.3 Higher Dimensional Primary Decomposition""; ""Exercises""; ""4.4 The Equidimensional Part of an Ideal""; ""Exercises""; ""4.5 The Radical""; ""Exercises""
""4.6 Characteristic Sets""""Exercises""; ""4.7 Triangular Sets""; ""Exercises""; ""4.8 Procedures""; ""5. Hilbert Function and Dimension""; ""5.1 The Hilbert Function and the Hilbert Polynomial""; ""Exercises""; ""5.2 Computation of the Hilbert�Poincare Series""; ""Exercises""; ""5.3 Properties of the Hilbert Polynomial""; ""Exercises""; ""5.4 Filtrations and the Lemma of Artin�Rees""; ""Exercises""; ""5.5 The Hilbert�Samuel Function""; ""Exercises""; ""5.6 Characterization of the Dimension of Local Rings""; ""Exercises""; ""5.7 Singular Locus""; ""Exercises""
""6. Complete Local Rings""""6.1 Formal Power Series Rings""; ""Exercises""; ""6.2 Weierstraß Preparation Theorem""; ""Exercises""; ""6.3 Completions""; ""Exercises""; ""6.4 Standard Bases""; ""Exercises""; ""7. Homological Algebra""; ""7.1 Tor and Exactness""; ""Exercises""; ""7.2 Fitting Ideals""; ""Exercises""; ""7.3 Flatness""; ""Exercises""; ""7.4 Local Criteria for Flatness""; ""Exercises""; ""7.5 Flatness and Standard Bases""; ""Exercises""; ""7.6 Koszul Complex and Depth""; ""Exercises""; ""7.7 Cohen�Macaulay Rings""; ""Exercises""
""7.8 Further Characterization of Cohen�Macaulayness""
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