Computations and Combinatorics in Commutative Algebra EACA School, Valladolid 2013

Featuring up-to-date coverage of three topics lying at the intersection of combinatorics and commutative algebra, namely Koszul algebras, primary decompositions and subdivision operations in simplicial complexes, this book has its focus on computations. "Computations and Combinatorics in Commut...

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Bibliographic Details
Main Authors Bigatti, Anna M, Gimenez, Philippe, Sáenz-de-Cabezón, Eduardo
Format eBook Book
LanguageEnglish
Published Cham Springer Nature 2017
Springer
Springer International Publishing AG
Springer International Publishing
Edition1
SeriesLecture Notes in Mathematics
Subjects
Algebra & number theory
Algorithms
Combinatorial analysis
Combinatorics
Commutative algebra
Commutative algebra-Congresses
Commutative Rings and Algebras
Mathematics
Mathematics and Statistics
Online AccessGet full text
ISBN3319513192
9783319513195
9783319513188
3319513184
ISSN0075-8434
1617-9692
DOI10.1007/978-3-319-51319-5

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Table of Contents:
  • 5.2.1 Barycentric Subdivision -- 5.2.2 Edgewise Subdivision -- 5.2.3 Interval Subdivision -- 6 First Relations to Algebra -- 7 Few Subdivisions, Real Rootedness, Koszul and Veronese Algebras, Charney-Davis Conjecture -- 7.1 Barycentric Subdivision -- 7.2 Edgewise Subdivision -- 7.3 Interval Subdivision -- 7.4 Subdivisions and the Charney-Davis Conjecture -- 8 Few Subdivisions, Lefschetz Properties and Real Rootedness -- 8.1 Barycentric Subdivision -- 8.2 Edgewise Subdivision -- 8.3 Interval Subdivision -- 8.4 Lefschetz Properties and Real Rootedness -- 9 Many Subdivisions, Limit Behaviour -- 9.1 The f- and h-Vectors -- 9.1.1 Barycentric Subdivision -- 9.1.2 Edgewise Subdivision -- 9.1.3 Interval Subdivision -- 9.2 Betti Numbers -- 10 Resolutions Supported on Subdivisions -- 10.1 Barycentric Subdivision -- 10.2 Edgewise Subdivision -- 10.3 Interval Subdivision -- References
  • Intro -- Introduction -- Contents -- Koszul Algebras and Computations -- 1 Introduction -- 2 Presentation of Algebras -- 2.1 Some Families of Subalgebras Generated by Monomials -- 3 Quadratic and G-Quadratic Algebras -- 3.1 Generic Initial Ideals -- 4 Resolutions -- 4.1 Ideals with 2-Linear Resolution -- 5 Koszul Algebras and Hilbert Series -- 6 G-Quadratic and LG-Quadratic Algebras -- 7 Koszul Filtrations -- 8 Complete Intersection of Quadrics -- 9 Koszul Algebras in a Nutshell -- Primary Decompositions -- 1 Computation of Primary Decompositions -- 1.1 Introduction to Primary Ideals and Primary Decompositions -- 1.2 Computing Radicals and Primary Decompositions -- 1.3 Computer Experiments: Using Macaulay2 to Obtain Primary Decompositions -- 2 Expanded Lectures on Binomial Ideals -- 2.1 Binomial Ideals in S = k[X1,…, Xn, X1-1,…, Xn-1] =k[X1,…, Xn]X1 . . . Xn -- 2.2 Associated Primes of Binomial Ideals Are Binomial -- 2.3 Primary Decomposition of Binomial Ideals -- 2.4 The Radical of a Binomial Ideal Is Binomial -- 3 Primary Decomposition in Algebraic Statistics -- 3.1 Conditional Independence -- 3.2 Intersection Axiom -- 3.3 A Version of the Hammersley-Clifford Theorem -- 3.4 Summary/Unification of Some Recent Papers -- 3.5 A Related Game -- 3.6 Binomial Edge Ideals with Macaulay2 -- 3.7 A Short Excursion Into Networks Using Monomial Primary Decompositions -- Combinatorics and Algebra of Geometric Subdivision Operations -- 1 Introduction -- 2 Abstract and Geometric Simplicial Complexes -- 3 Subdivisions of Simplicial Complexes -- 3.1 Barycentric Subdivision -- 3.2 Edgewise Subdivision -- 3.3 Interval Subdivision -- 4 The Stanley-Reisner Ring -- 5 The f- and h-Vector Transformations -- 5.1 The f-Vector Transformation -- 5.1.1 Barycentric Subdivision -- 5.1.2 Edgewise Subdivision -- 5.1.3 Interval Subdivision -- 5.2 The h-Vector Transformation