Preface -- Boij-Söderberg Theory: Introduction and Survey -- 1 The Boij-Söderberg Conjectures -- 1.1 Resolutions and Betti Diagrams -- 1.2 The Positive Cone of Betti Diagrams -- 1.3 Herzog-Kühl Equations -- 1.4 Pure resolutions -- 1.5 Linear Combinations of Pure Diagrams -- 1.6 The Boij-Söderberg Conjectures -- 1.7 Algorithmic Interpretation -- 1.8 Geometric Interpretation -- 2 The Exterior Facets of the Boij-Söderberg Fan and Their Supporting Hyperplanes -- 2.1 The Exterior Facets -- 2.2 The Supporting Hyperplanes -- 2.3 Pairings of Vector Bundles and Resolutions -- 3 The Existence of Pure Free Resolutions and of Vector Bundles with Supernatural Cohomology -- 3.1 The Equivariant Pure Free Resolution -- 3.2 Equivariant Supernatural Bundles -- 3.3 Characteristic Free Supernatural Bundles -- 3.4 The Characteristic Free Pure Resolutions -- 3.5 Pure Resolutions Constructed from Generic Matrices -- 4 Cohomology of Vector Bundles on Projective Spaces -- 4.1 Cohomology Tables -- 4.2 The Fan of Cohomology Tables of Vector Bundles -- 4.3 Facet Equations -- 5 Extensions to Non-Cohen-Macaulay Modules and to Coherent Sheaves -- 5.1 Betti Diagrams of Graded Modules in General -- 5.2 Cohomology of Coherent Sheaves -- 6 Further Topics -- 6.1 The Semigroup of Betti Diagrams of Modules -- 6.2 Variations on the Grading -- 6.3 Poset Structures -- 6.4 Computer Packages -- 6.5 Three Basic Problems -- Hilbert Functions of Fat Point Subschemes of the Plane: the Two-fold Way -- 1 Introduction -- 2 Approach I: Nine Double Points -- 3 Approach I: Points on Cubics -- 4 Approach II: Points on Cubics -- Edge Ideals: Algebraic and Combinatorial Properties -- 1 Introduction -- 2 Algebraic and Combinatorial Properties of Edge Ideals -- 3 Invariants of Edge Ideals: Regularity, Projective Dimension, Depth -- 4 Stability of Associated Primes.

Three Simplicial Resolutions -- 1 Introduction -- 2 Background and Notation -- 2.1 Algebra -- 2.2 Combinatorics -- 3 The Taylor Resolution -- 4 Simplicial Resolutions -- 5 The Scarf Complex -- 6 The Lyubeznik Resolutions -- 7 Intersections -- 8 Questions -- A Minimal Poset Resolution of Stable Ideals -- 1 Introduction -- 2 Poset Resolutions and Stable Ideals -- 3 The Shellability of PN -- 4 The topology of PN and properties of D (PN) -- 5 Proof of Theorem 2.4 -- 6 A Minimal Cellular Resolution of R/N -- Subsets of Complete Intersections and the EGH Conjecture -- 1 Introduction -- 2 Preliminary Definitions and Results -- 2.1 The Eisenbud-Green-Harris Conjecture and Complete Intersections -- 2.2 Some Enumeration -- 3 Rectangular Complete Intersections -- 4 Some Key Tools -- 4.1 Pairs of Hilbert Functions and Maximal Growth -- 4.2 Ideals Containing Regular Sequences -- 5 Subsets of Complete Intersections in ℙ2 -- 6 Subsets of C.I. (2, d2,d3) with d2 = d3 -- 7 Subsets of C.I. (3, d2, d3) with d3 = d2 -- 8 An Application: The Cayley-Bacharach Property -- The Homological Conjectures -- 1 Introduction -- 2 The Serre Multiplicity Conjectures -- 2.1 The Vanishing Conjecture -- 2.2 Gabber's Proof of the Nonnegativity Conjecture -- 2.3 The Positivity Conjecture -- 3 The Peskine-Szpiro Intersection Conjecture -- 3.1 Hochsterzs Metatheorem -- 4 Generalizations of the Multiplicity Conjectures -- 4.1 The Graded Case -- 4.2 The Generalized Rigidity Conjecture -- 5 The Monomial, Direct Summand, and Canonical Element Conjectures -- 6 Cohen-Macaulay Modules and Algebras -- 6.1 Weakly Functorial Big Cohen-Macaulay Algebras -- 7 The Syzygy Conjecture and the Improved New Intersection Conjecture -- 8 Tight Closure Theory -- 9 The Strong Direct Summand Conjecture -- 10 Almost Cohen-Macaulay Algebras -- 11 A Summary of Open Questions.

11.1 The Serre Positivity Conjecture -- 11.2 Partial Euler Characteristics -- 11.3 Strong Multiplicity Conjectures -- 11.4 Cohen-Macaulay Modules and Related Conjectures -- 11.5 Almost Cohen-Macaulay Algebras -- The Compatibility, Independence, and Linear Growth Properties -- 1 Introduction -- 2 Primary decomposition -- 3 Compatibility of Primary Components -- 4 Maximal Primary Components, Independence -- 5 Linear growth of primary components -- 6 Linear Growth of {TorRc (M/ImM, N/JnN)} -- 7 Secondary Representation -- 8 Compatibility of Secondary Components -- 9 Applying a Result of Sharp on Artinian Modules -- 10 Independence -- 11 Minimal Secondary Components -- 12 Linear Growth of Secondary Components -- Recent Progress in Coherent Rings: a Homological Perspective -- 1 Introduction -- 2 Coherent Rings and Grade -- 2.1 Coherent Rings and (FP)R∞ Modules -- 2.2 Non-Noetherian Grade -- 3 Cohen-Macaulay Rings -- 4 Gorenstein Dimensions and the Auslander-Bridger Property -- 4.1 Gorenstein Dimensions -- 4.2 The Auslander-Bridger Formula -- 5 Gorenstein Rings and Injective Dimensions -- 6 Foundations for Coherent Complete Intersections -- Non-commutative Crepant Resolutions: Scenes from Categorical Geometry -- 1 Introduction -- 2 Morita Equivalence -- 3 (Quasi)coherent Sheaves -- 4 Derived Categories of Modules -- 5 Derived Categories of Sheaves -- 6 Example: Tilting on Projective Space -- 7 The Non-existence of Non-commutative Spaces -- 8 Resolutions of Singularities -- 9 The Minimal Model Program -- 10 Categorical Desingularizations -- 11 Example: the McKay Correspondence -- 12 Non-commutative Crepant Resolutions -- 13 Example: Normalization -- 14 MCM Endomorphism Rings -- 15 Global Dimension of Endomorphism Rings -- 16 Rational Singularities -- 17 Examples: Finite Representation Type.

18 Example: the Generic Determinant -- 19 Non-commutative Blowups -- 20 Omissions and Open Questions.