商品簡介
The last decade has seen a number of exciting developments at the intersection of commutative algebra with combinatorics. New methods have evolved out of an influx of ideas from such diverse areas as polyhedral geometry, theoretical physics, representation theory, homological algebra, symplectic geometry, graph theory, integer programming, symbolic computation, and statistics. The purpose of this volume is to provide a selfcontained introduction to some of the resulting combinatorial techniques for dealing with polynomial rings, semigroup rings, and determinantal rings.Our exposition mainly concerns combinatorially defined ideals and their quotients, with a focus on numerical invariants and resolutions, especially under gradings more refined than the standard integer grading.自Von Neumann起，將公里化方法應用于有限維向量空間理論，使此理論得到了系統的發展。有限維向量空間理論已成為研究線性泛函分析的主要方法。本書通過更一般理論的方法來討論有限維向量空間中的線性變換，意在強調數學的很多領域中常見的幾何概念及其應用，並用清晰而通俗的表述告訴讀者關於積分方程以及Hilbert空間的一些定理的基本證明思想。本書是第二版，與前一版相比，除了一些局部內容的略微調整外，還增加了一些新的內容，例如：域的簡論，帶有內積的向量空間（特別是歐氏空間）,利用多重線性型理論給出的行列式定義，此外還有大量練習。這些習題是全書內容的重要補充，相信會對讀者的學習起到很大的幫助作用。
讀者對象：數學專業的本科生
目次：空間；變換；正交性；分析學；附錄，Hilbert 空間Paul R.Halmos 20世紀世界著名的數學家。1983年于伊利諾伊大學獲得數學博士學位。曾在許多大學任職，其中包括芝加哥大學，密歇根大學和印第安納大學。Halmos以他在許多數學領域的研究著稱於世，其中包括：泛函分析，遍歷理論，測度論，布爾代數理論等。除本書外，還著有《Naïve Set Theory》、《Measure Theory》、《Problems For Mathematicians Young And Old》、《I Want To Be A Mathematician》,這些著作早已成為經典名著，至今仍在印刷出版。他對數學的很多領域保持著濃厚的興趣，對於數學的主要進展十分關注，這顯示出他作為一位數學家的地位，遠超出狹窄的工匠之上。這位偉大的數學家及數學教育家于2006年10月在美國逝世，享年90歲。
目次
Preface
I Monomial Ideals
1 Squarefree monomial ideals
1.1 Equivalent descriptions
1.2 Hilbert series
1.3 Simplicial complexes and homology
1.4 Monomial matrices
1.5 Betti numbers
Exercises
Notes
2 Borel-fixed monomial ideals
2.1 Group actions
2.2 Generic initial ideals
2.3 The Eliahou-Kervaire resolution
2.4 Lex-segment ideals
Exercises
Notes
3 Three-dimensional staircases
3.1 Monomial ideals in two variables
3.2 An example with six monomials.
3.3 The Buchberger graph
3.4 Genericity and deformations...
3.5 The planar resolution algorithm.
Exercises
Notes
4 Cellular resolutions
4.1 Construction and exactness
4.2 Betti numbers and K-polynomials
4.3 Examples of cellular resolutions
4.4 The hull resolution
4.5 Subdividing the simplex
Exercises
Notes
5 Alexander duality
5.1 Simplicial Alexander duality
5.2 Generators versus irreducible components.
5.3 Duality for resolutions
5.4 Cohull resolutions and other applications
5.5 Projective dimension and regular!ty
Exercises
Notes
6 Generic monomial ideals
6.1 Taylor complexes and genericity
6.2 The Scarf complex
6.3 Genericity by deformation
6.4 Bounds on Betti numbers
6.5 Cogeneric monomial ideals
Exercises
Notes
II Toric Algebra
7 Semigroup rings
7.1 Semigroups and lattice ideals
7.2 Affine semigroups and polyhedral cones
7.3 Hilbert bases
7.4 Initial ideals of lattice ideals
Exercises
Notes
8 Multigraded polynomial rings
8.1 Multigradings
8.2 Hilbert series and K-polynomials
8.3 Multigraded Betti numbers
8.4 K-polynomials in nonpositive gradings
8.5 Multidegrees
Exercises
Notes
9 Syzygies of lattice ideals
9.1 Betti numbers
9.2 Laurent monomial modules
9.3 Free resolutions of lattice ideals
9.4 Genericity and the Scarf complex
Exercises
Notes
……
III Determinants
References
Glossary of notation
Index
