Progress in Commutative Algebra 1

Christopher Francisco, Oklahoma State University, Stillwater, Oklahoma, USA; Lee C. Klingler, Florida Atlantic University, Boca Raton, Florida, USA; Sean M. Sather-Wagstaff, North Dakota State University, Fargo, North Dakota, USA; Janet Vassilev, University of New Mexico, Albuquerque, New Mexico, USA.

• Preface1 Boij-Soederberg Theory: Introduction and SurveyGunnar Fløystad1.1 Introduction1.2 The Boij-Söderberg Conjectures1.2.1 Resolutions and Betti Diagrams1.2.2 The Positive Cone of Betti Diagrams1.2.3 Herzog-Kühl Equations1.2.4 Pure Resolutions1.2.5 Linear Combinations of Pure Diagrams1.2.6 The Boij-Söderberg Conjectures1.2.7 Algorithmic Interpretation1.2.8 Geometric Interpretation1.3 The Exterior Facets of the Boij-Söderberg Fan and their Supporting Hyperplanes1.3.1 The Exterior Facets1.3.2 The Supporting Hyperplanes1.3.3 Pairings of vector Bundles and Resolutions1.4 The Existence of Pure Free Resolutions and of Vector Bundles with Supernatural Cohomology1.4.1 The Equivariant Pure Free Resolution1.4.2 Equivariant Supernatural Bundles1.4.3 Characteristic Free Supernatural Bundles1.4.4 The Characteristic Free Pure Resolution1.4.5 Pure Resolutions Constructed from Generic Matrices1.5 Cohomology of Vector Bundles on Projective Spaces1.5.1 Cohomology Tables1.5.2 The Fan of Cohomology Tables of Vector Bundles1.5.3 Facet Equations1.6 Extensions to Non-Cohen-Macaulay Modules and to Coherent Sheaves1.6.1 Betti Diagrams of Graded Modules in General1.6.2 Cohomology of Coherent Sheaves1.7 Further Topics1.7.1 The Semigroup of Betti Diagrams of Modules1.7.2 Variants on the Grading1.7.3 Poset Structures1.7.4 Computer Packages1.7.5 Three Basic Problems2 Hilbert Functions of Fat Point Subschemes of the Plane: the Two-fold WayAnthony V. Geramita, Brian Harbourne, and Juan Migliore2.1 Introduction2.2 Approach I: Nine Double Points2.3 Approach I: Points on Cubics2.4 Approach II: Points on Cubics3 Edge Ideals: Algebraic and Combinatorial PropertiesSusan Morey and Rafael H. Villarreal3.1 Introduction3.2 Algebraic and Combinatorial Properties of Edge Ideals3.3 Invariants of Edge Ideals: Regularity, Projective Dimension, Depth3.4 Stability of Associated Primes4 Three Simplicial ResolutionsJeff Mermin4.1 Introduction4.2 Background and Notation4.2.1 Algebra4.2.2 Combinatorics4.3 The Taylor Resolution4.4 Simplicial Resolutions4.5 The Scarf Complex4.6 The Lyubeznik Resolutions4.7 Intersections4.8 Questions5 A Minimal Poset Resolution of Stable IdealsTimothy B. P. Clark5.1 Introduction5.2 Poset Resolutions and Stable Ideals5.3 The Shallability of PN5.4 The Topology of PN and Properties of D(PN)5.5 Proof of Theorem 2.45.6 A Minimal Cellular Resolution of R/N6 Subsets of Complete Intersections and the EGH ConjectureSusan M. Cooper6.1 Introduction6.2 Preliminary Definitions and Results6.2.1 The Eisenbud-Green-Harris Conjecture and Complete Intersections6.3 Rectangular Complete Intersections6.4 Some Key Tools6.4.1 Pairs of Hilbert Functions and Maximal Growth6.4.2 Ideals Containing Regular Sequences6.5 Subsets of Complete Intersections in P26.6 Subsets of C.I.(2, d2, d3) with d2 = d36.7 Subsets of C.I.(3, d2, d3) with d3 = d26.8 An Application: The Cayley-Bacharach Property7 The Homological ConjecturesPaul C. Roberts7.1 Introduction7.2 The Serre Multiplicity Conjectures7.2.1 The Vanishing Conjecture7.2.2 Gabber's Proof of the Nonnegativity Conjecture7.2.3 The Positivity Conjecture7.3 The Peskine-Szpiro Intersection Conjecture7.3.1 Hochester's Metatheorem7.4 Generalizations of the Multiplicity Conjectures7.4.1 The Graded Case7.4.2 The Generalized Rigidity Conjecture7.5 The Monomial, Direct Summand, and Canonical Element Conjectures7.6 Cohen-Macaulay Modules and Algebras7.6.1 Weakly Functorial Big Cohen-Macaulay Algebras7.7 The Syzygy Conjecture and the Improved New Intersection Conjecture7.8 Tight Closure Theory7.9 The Strong Direct Summand Conjecture7.10 Almost Cohen-Macaulay Algebras7.11 A Summary of Open Questions7.11.1 The Serre Positivity Conjecture7.11.2 Partial Euler Characteristics7.11.3 Strong Multiplicity Conjectures7.11.4 Cohen-Macaulay Modules and Related Conjectures7.11.5 Almost Cohen-Macaulay Algebras8 The Compatibility, Independence, and Linear Growth PropertiesYongwei Yao8.1 Introduction8.2 Primary Decomposition8.3 Compatibility of Primary Components8.4 Maximal Primary Components, Independence8.5 Linear Growth of Primary Components8.6 Linear growth of {?Tor?_c^R (?(M/(I^m M )),?(N/(J^n N)))}8.7 Secondary Representation8.8 Compatibility of Secondary Components8.9 Applying a Result of Sharp on Artinian Modules8.10 Independence8.11 Minimal Secondary Components8.12 Linear Growth of Secondary Components9 Recent Progress in Coherent Rings: A Homological PerspectiveLivia Hummel9.1 Introduction9.2 Coherent Rings and Grade9.2.1 Coherent Rings and ?(FP)?_?^R Modules9.2.2 Non-Noetherian Grade9.3 Cohen-Macaulay Rings9.4 Gorenstein Dimensions and the Auslander-Bridger Property9.4.1 Gorenstein Dimenstions9.4.2 The Auslander-Bridger Formula9.5 Gorenstein Rings and Injective Dimensions9.6 Foundations for Coherent Complete Intersections10 Non-commutative Crepant Resolutions: Scenes from Categorical GeometryGraham J. Leuschke10.1 Introduction10.2 Morita Equivalence10.3 (Quasi)coherent Sheaves10.4 Derived Categories of Modules10.5 Derived Categories of Sheaves10.6 Example: Tilting on Projective Space10.7 The Non-existence of Non-commutative Spaces10.8 Resolutions of Singularities10.9 The Minimal Model Program10.10. Categorical Desingularizations10.11 Example: the McKay Correspondence10.12 Non-commutative Crepant Resolutions10.13 Example: Normalization10.14 MCM Endomorphism Rings10.15 Global Dimension of Endomorphism Rings10.16 Rational Singularities10.17 Examples: Finite Representation Type10.18 Example: the Generic Determinant10.19 Non-commutative Blowups10.20 Omissions and Open Questions