| 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 regularity 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 |
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