| Preface | 6 |
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| Boij-Söderberg Theory: Introduction and Survey | 14 |
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| 1 The Boij-Söderberg Conjectures | 18 |
| 1.1 Resolutions and Betti Diagrams | 18 |
| 1.2 The Positive Cone of Betti Diagrams | 19 |
| 1.3 Herzog-Kühl Equations | 20 |
| 1.4 Pure resolutions | 21 |
| 1.5 Linear Combinations of Pure Diagrams | 22 |
| 1.6 The Boij-Söderberg Conjectures | 24 |
| 1.7 Algorithmic Interpretation | 25 |
| 1.8 Geometric Interpretation | 25 |
| 2 The Exterior Facets of the Boij-Söderberg Fan and Their Supporting Hyperplanes | 26 |
| 2.1 The Exterior Facets | 26 |
| 2.2 The Supporting Hyperplanes | 28 |
| 2.3 Pairings of Vector Bundles and Resolutions | 34 |
| 3 The Existence of Pure Free Resolutions and of Vector Bundles with Supernatural Cohomology | 37 |
| 3.1 The Equivariant Pure Free Resolution | 37 |
| 3.2 Equivariant Supernatural Bundles | 42 |
| 3.3 Characteristic Free Supernatural Bundles | 42 |
| 3.4 The Characteristic Free Pure Resolutions | 43 |
| 3.5 Pure Resolutions Constructed from Generic Matrices | 46 |
| 4 Cohomology of Vector Bundles on Projective Spaces | 48 |
| 4.1 Cohomology Tables | 48 |
| 4.2 The Fan of Cohomology Tables of Vector Bundles | 50 |
| 4.3 Facet Equations | 51 |
| 5 Extensions to Non-Cohen-Macaulay Modules and to Coherent Sheaves | 54 |
| 5.1 Betti Diagrams of Graded Modules in General | 55 |
| 5.2 Cohomology of Coherent Sheaves | 56 |
| 6 Further Topics | 58 |
| 6.1 The Semigroup of Betti Diagrams of Modules | 58 |
| 6.2 Variations on the Grading | 62 |
| 6.3 Poset Structures | 63 |
| 6.4 Computer Packages | 63 |
| 6.5 Three Basic Problems | 64 |
| Hilbert Functions of Fat Point Subschemes of the Plane: the Two-fold Way | 68 |
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| 1 Introduction | 68 |
| 2 Approach I: Nine Double Points | 71 |
| 3 Approach I: Points on Cubics | 80 |
| 4 Approach II: Points on Cubics | 87 |
| Edge Ideals: Algebraic and Combinatorial Properties | 98 |
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| 1 Introduction | 98 |
| 2 Algebraic and Combinatorial Properties of Edge Ideals | 99 |
| 3 Invariants of Edge Ideals: Regularity, Projective Dimension, Depth | 103 |
| 4 Stability of Associated Primes | 117 |
| Three Simplicial Resolutions | 140 |
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| 1 Introduction | 140 |
| 2 Background and Notation | 141 |
| 2.1 Algebra | 141 |
| 2.2 Combinatorics | 142 |
| 3 The Taylor Resolution | 143 |
| 4 Simplicial Resolutions | 145 |
| 5 The Scarf Complex | 148 |
| 6 The Lyubeznik Resolutions | 149 |
| 7 Intersections | 151 |
| 8 Questions | 152 |
| A Minimal Poset Resolution of Stable Ideals | 156 |
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| 1 Introduction | 156 |
| 2 Poset Resolutions and Stable Ideals | 160 |
| 3 The Shellability of PN | 163 |
| 4 The topology of PN and properties of D (PN) | 168 |
| 5 Proof of Theorem 2.4 | 173 |
| 6 A Minimal Cellular Resolution of R/N | 176 |
| Subsets of Complete Intersections and the EGH Conjecture | 180 |
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| 1 Introduction | 180 |
| 2 Preliminary Definitions and Results | 181 |
| 2.1 The Eisenbud-Green-Harris Conjecture and Complete Intersections | 181 |
| 2.2 Some Enumeration | 184 |
| 3 Rectangular Complete Intersections | 186 |
| 4 Some Key Tools | 190 |
| 4.1 Pairs of Hilbert Functions and Maximal Growth | 190 |
| 4.2 Ideals Containing Regular Sequences | 191 |
| 5 Subsets of Complete Intersections in P2 | 192 |
| 6 Subsets of C.I. (2, d2,d3) with d2 = d3 | 194 |
| 7 Subsets of C.I. (3, d2, d3) with d3 = d2 | 199 |
| 8 An Application: The Cayley-Bacharach Property | 207 |
| The Homological Conjectures | 212 |
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| 1 Introduction | 212 |
| 2 The Serre Multiplicity Conjectures | 214 |
| 2.1 The Vanishing Conjecture | 217 |
| 2.2 Gabber’s Proof of the Nonnegativity Conjecture | 218 |
| 2.3 The Positivity Conjecture | 219 |
| 3 The Peskine-Szpiro Intersection Conjecture | 220 |
| 3.1 Hochsterzs Metatheorem | 222 |
| 4 Generalizations of the Multiplicity Conjectures | 224 |
| 4.1 The Graded Case | 224 |
| 4.2 The Generalized Rigidity Conjecture | 227 |
| 5 The Monomial, Direct Summand, and Canonical Element Conjectures | 228 |
| 6 Cohen-Macaul
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