组合交换代数

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

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.