Women in Commutative Algebra: Proceedings of the 2019 WICA Workshop (Association for Women in Mathematics Series, 29)

This volume features contributions from the Women in Commutative Algebra (WICA) workshop held at the Banff International Research Station (BIRS) from October 20-25, 2019, run by the Pacific Institute of Mathematical Sciences (PIMS). The purpose of this meeting was for groups of mathematicians to work on joint research projects in the mathematical field of Commutative Algebra and continue these projects together long-distance after its close. The chapters include both direct results and surveys, with contributions from research groups and individual authors. The WICA conference was the first of its kind in the large and vibrant area of Commutative Algebra, and this volume is intended to showcase its important results and to encourage further collaboration among marginalized practitioners in the field. It will be of interest to a wide range of researchers, from PhD students to senior experts. Preface Acknowledgments Contents On Gerko's Strongly Tor-independent Modules 1 Introduction 2 DG Homological Algebra 3 Perfect DG Modules and Tensor Products 4 Syzygies and Strongly Tor-independent DG Modules 5 Proof of Theorem 1.1 References Properties of the Toric Rings of a Chordal Bipartite Family of Graphs 1 Introduction 2 The Family of Toric Rings 2.1 The Family F of Graphs 2.2 Toric Rings for F 2.3 Distinction from Join-Meet Ideals of Lattices 3 Properties of the Family of Toric Rings 3.1 Dimension and System of Parameters 3.2 Length, Multiplicity, and Regularity References An Illustrated View of Differential Operators of a Reduced Quotient of an Affine Semigroup Ring 1 Introduction 2 Background and Notation 3 Differential Operators on the Rational Normal Curve of Degree 2 4 Differential Operators on the Rational Normal Curve of Degree 3 5 Differential Operators on a Rational Normal Curve 6 Higher Dimensional Examples 7 Non-normal Examples References A Hypergraph Characterization of Nearly Complete Intersections 1 Introduction 2 Hypergraphs and Squarefree Monomial Ideals 3 NCI-hypergraphs 4 Resolutions of NCIs 5 Appendix: Vertex-Weighted Graphs References The Shape of Hilbert–Kunz Functions 1 Motivation and Outline 2 History in Brief 2.1 Multiplicity 2.2 Functions 3 Techniques in Hilbert–Kunz Functions 3.1 Via Representation Rings and p-Fractals 3.2 Via the Divisor Class Group 3.3 Via Sheaf Theory 3.4 Via Local Chern Characters 3.5 Via Bruns-Gubeladze (BG) Decomposition 3.6 Via Combinatorics 4 Normal Affine Semigroup Rings and Ehrhart Theory 5 Examples in Normal Affine Semigroup Rings References Standard Monomial Theory and Toric Degenerations of Richardson Varieties in Flag Varieties 1 Introduction 1.1 Outline of the Paper 2 Preliminaries 2.1 Flag Varieties 2.2 Richardson Varieties 2.3 Ideals of Flag Varieties and Richardson Varieties 2.4 Gröbner Degenerations of In 2.5 Gröbner Degenerations of I(Xwv) 2.6 Permutations, Tableaux and Their Combinatorial Properties 2.7 Standard Monomials 3 Standard Monomials 3.1 Block Structure on Compatible Permutations 3.2 Proof of Main Result 4 Monomial-Free Ideals 5 Toric Degenerations References Simplicial Resolutions for the Second Power of Square-Free Monomial Ideals 1 Introduction 2 Background 3 The Quasi-Trees Lq2 and L2(I) 4 A Bound on the Betti Numbers of I2 References Cohen–Macaulay Fiber Cones and Defining Ideal of Rees Algebras of Modules 1 Introduction 2 Preliminaries 2.1 Rees Algebras and Fiber Cones of Modules 2.2 Generic Bourbaki Ideals 2.3 Iterated Jacobian Duals 3 A Deformation Condition for the Rees Algebra of a Module 4 Cohen–Macaulay Property of Fiber Cones of Modules 4.1 Modules with Cohen–Macaulay Fiber Cone 5 Defining ideal of Rees Algebras 5.1 Almost Linearly Presented Modules of Projective Dimension One References Principal Matrices of Numerical Semigroups 1 Introduction 2 Principal Matrix and Look Alikes 3 Embedding Dimension 4 and 5 4 Sufficient Condition for Principal Matrix 5 Examples 6 Appendix References A Survey on the Koszul Homology Algebra 1 Introduction 2 Preliminaries 3 Canonical Bases 4 Classical Characterizations 4.1 Complete Intersection and Gorenstein Rings 4.2 Golod Rings 5 Classifications of Koszul Homology Algebras 6 Recent Progess: Koszul Algebras 6.1 Generation by the Linear Strand 6.2 Upper Bounds on Betti Numbers 6.3 Subadditivity of Syzygies 6.4 Towards a Characterization References Canonical Resolutions over Koszul Algebras 1 Introduction 2 Koszul Algebras and the Priddy Resolution 3 Resolutions via the Enveloping Algebra 4 The Case of Koszul Algebras 5 Minimal Resolutions for Powers of the Maximal Ideal 6 Examples References Well Ordered Covers, Simplicial Bouquets, and Subadditivity of Betti Numbers of Square-Free Monomial Ideals 1 Introduction 2 Background 2.1 Simplicial Complexes and Facet Ideals 2.2 Simplicial Resolutions 3 (Well Ordered) Covers 4 The Subadditivity Property 5 Simplicial Bouquets 6 Reordering Well Ordered Covers References A Survey on the Eisenbud-Green-Harris Conjecture 1 Introduction 2 Preliminaries and Macaulay's Theorem on Hilbert Functions 3 The EGH Conjecture 4 Results on the EGH Conjecture 4.1 EGH Depending on the Degrees (a1,..., an) 4.2 EGH via Liaison 4.3 EGH via the Structure of the Regular Sequence 4.4 When a1=...= an=2 5 Open Cases of EGH and More Connections References The Variety Defined by the Matrix of Diagonals is F-Pure 1 Introduction and Preliminaries 2 A System of Parameters 3 The Variety Defined by P(X) Is F-Pure References Classification of Frobenius Forms in Five Variables 1 Introduction 2 Matrix Factorization of Frobenius Forms 3 Quadratic Forms 4 Frobenius Forms in Five Variables: Classification Statements 5 Classification of Frobenius Forms in Five Variables: Proofs References Projective Dimension of Hypergraphs 1 Introduction 2 Preliminaries 2.1 Hypergraph of a Square-Free Monomial Ideal 2.2 Colon Ideals: Key tool 2.3 Splittings: Key Tool 3 Strings with Higher Dimensional Edges 4 Cycles with Higher Dimensional Edges References A Truncated Minimal Free Resolution of the Residue Field 1 Introduction 2 Multiplicative Structure on the Homology of the Koszul Algebra 2.1 Products in Degree Two 2.2 Products in Degree Three 2.3 Massey Products in Degree Four 2.4 Products in Degree Four 2.5 Summary 3 Truncated Minimal Free Resolution of the Residue Field 3.1 Construction 4 Applications of the Construction 5 An Example Illustrating the Construction References On Gerko's Strongly Tor-independent Modules (H. Altmann) -- Properties of the Toric Rings of a Chordal Bipartite Family of Graphs (L. Ballard) -- An illustrated view of differential operators of a reduced quotient of an affine semigroup ring (C. Berkesch) -- A hypergraph characterization of nearly complete intersections (R. Gibbons) -- The Shape Of Hilbert-Kunz Functions (C-Y. Jean Chan) -- Standard monomial theory and toric degenerations of Richardson varieties in flag varieties (F. Mohammadi) -- Simplicial resolutions for the second power of square-free monomial ideals (S. Faridi) -- Cohen-Macaulay fiber cones and defining ideal of Rees algebras of modules (A. Costantini) -- Principal Matrices of Numerical Semigroups (H. Srinivasan) -- A survey on the Koszul homology algebra (N. Diethorn) -- Canonical Resolutions over Koszul Algebras (A. Seceleanu) -- Well Ordered Covers, Simplicial Bouquets, and Subadditivity of Betti Numbers of Square-Free Monomial Ideals (S. Farid) -- A survey on the Eisenbud-Green-Harris Conjecture (S. Güntürkün) -- The variety defined by the matrix of diagonals is f-pure (Z. Kadyrsizova) -- Classification of Frobenius Forms in five variables (E. Witt) -- Projective dimension of hypergraphs (Kuei-Nuan Lin) -- A truncated minimal free resolution of the residue field (O. Veliche).