组合交换代数

组合交换代数

(美) 米勒 (Miller,E.) , (美) 施图姆费尔斯 (Sturmfels,B.) , 著

出版社:世界图书出版公司北京公司

年代:2007

定价:49.0

书籍简介:

组合交换代数学是一个活跃的研究领域,它与纯数学与应用数学的其他一些领域的发展有着紧密的联系。本书的内容自成体系,重点介绍了多重网格多项式环的组合技巧,半群代数以及行列式环。全书共有十八章,广泛涉及谱理论,从单项式理想的同调不变量及其多面体分解,到运用群的作用来学习代数簇的简易工具,如 toric 簇,旗簇,箭图轨迹以及Hilbert概型。

书籍目录:

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

内容摘要:

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.

书籍规格:

书籍详细信息
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出版地北京出版单位世界图书出版公司北京公司
版次1版印次1
定价(元)49.0语种英文
尺寸14装帧平装
页数印数 1000

书籍信息归属:

组合交换代数是世界图书出版公司北京公司于2007.08出版的中图分类号为 O187.3 的主题关于 交换环-英文 的书籍。