{"product_id":"progress-in-commutative-algebra-1-combinatorics-and-homology-9783110250343","title":"Progress in Commutative Algebra 1: Combinatorics and Homology","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003eThis is the first of two volumes of a state-of-the-art survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: non-noetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry).\u003c\/p\u003e \u003cp\u003eThis volume contains combinatorial and homological surveys. The combinatorial papers document some of the increasing focus in commutative algebra recently on the interaction between algebra and combinatorics. Specifically, one can use combinatorial techniques to investigate resolutions and other algebraic structures as with the papers of Fløystad on Boij-Söderburg theory, of Geramita, Harbourne and Migliore, and of Cooper on Hilbert functions, of Clark on minimal poset resolutions and of Mermin on simplicial resolutions. One can also utilize algebraic invariants to understand combinatorial structures like graphs, hypergraphs, and simplicial complexes such as in the paper of Morey and Villarreal on edge ideals.\u003c\/p\u003e \u003cp\u003eHomological techniques have become indispensable tools for the study of noetherian rings. These ideas have yielded amazing levels of interaction with other fields like algebraic topology (via differential graded techniques as well as the foundations of homological algebra), analysis (via the study of D-modules), and combinatorics (as described in the previous paragraph). The homological articles the editors have included in this volume relate mostly to how homological techniques help us better understand rings and singularities both noetherian and non-noetherian such as in the papers by Roberts, Yao, Hummel and Leuschke.\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003cp\u003ePreface\u003c\/p\u003e \u003cp\u003e\u003cstrong\u003e1 Boij-Soederberg Theory: Introduction and Survey\u003cbr\u003e\u003c\/strong\u003e\u003cem\u003eGunnar Fløystad\u003cbr\u003e\u003c\/em\u003e1.1 Introduction\u003cbr\u003e1.2 The Boij-Söderberg Conjectures\u003cbr\u003e1.2.1 Resolutions and Betti Diagrams\u003cbr\u003e1.2.2 The Positive Cone of Betti Diagrams\u003cbr\u003e1.2.3 Herzog-Kühl Equations\u003cbr\u003e1.2.4 Pure Resolutions\u003cbr\u003e1.2.5 Linear Combinations of Pure Diagrams\u003cbr\u003e1.2.6 The Boij-Söderberg Conjectures\u003cbr\u003e1.2.7 Algorithmic Interpretation\u003cbr\u003e1.2.8 Geometric Interpretation\u003cbr\u003e1.3 The Exterior Facets of the Boij-Söderberg Fan and their Supporting Hyperplanes\u003cbr\u003e1.3.1 The Exterior Facets\u003cbr\u003e1.3.2 The Supporting Hyperplanes\u003cbr\u003e1.3.3 Pairings of vector Bundles and Resolutions\u003cbr\u003e1.4 The Existence of Pure Free Resolutions and of Vector Bundles with Supernatural Cohomology\u003cbr\u003e1.4.1 The Equivariant Pure Free Resolution\u003cbr\u003e1.4.2 Equivariant Supernatural Bundles\u003cbr\u003e1.4.3 Characteristic Free Supernatural Bundles\u003cbr\u003e1.4.4 The Characteristic Free Pure Resolution\u003cbr\u003e1.4.5 Pure Resolutions Constructed from Generic Matrices\u003cbr\u003e1.5 Cohomology of Vector Bundles on Projective Spaces\u003cbr\u003e1.5.1 Cohomology Tables\u003cbr\u003e1.5.2 The Fan of Cohomology Tables of Vector Bundles\u003cbr\u003e1.5.3 Facet Equations\u003cbr\u003e1.6 Extensions to Non-Cohen-Macaulay Modules and to Coherent Sheaves\u003cbr\u003e1.6.1 Betti Diagrams of Graded Modules in General\u003cbr\u003e1.6.2 Cohomology of Coherent Sheaves\u003cbr\u003e1.7 Further Topics\u003cbr\u003e1.7.1 The Semigroup of Betti Diagrams of Modules\u003cbr\u003e1.7.2 Variants on the Grading\u003cbr\u003e1.7.3 Poset Structures\u003cbr\u003e1.7.4 Computer Packages\u003cbr\u003e1.7.5 Three Basic Problems\u003c\/p\u003e \u003cp\u003e\u003cstrong\u003e2 Hilbert Functions of Fat Point Subschemes of the Plane: the Two-fold Way\u003cbr\u003e\u003c\/strong\u003e\u003cem\u003eAnthony V. Geramita, Brian Harbourne, and Juan Migliore\u003cbr\u003e\u003c\/em\u003e2.1 Introduction\u003cbr\u003e2.2 Approach I: Nine Double Points\u003cbr\u003e2.3 Approach I: Points on Cubics\u003cbr\u003e2.4 Approach II: Points on Cubics\u003c\/p\u003e \u003cp\u003e\u003cstrong\u003e3 Edge Ideals: Algebraic and Combinatorial Properties\u003cbr\u003e\u003c\/strong\u003e\u003cem\u003eSusan Morey and Rafael H. Villarreal\u003cbr\u003e\u003c\/em\u003e3.1 Introduction\u003cbr\u003e3.2 Algebraic and Combinatorial Properties of Edge Ideals\u003cbr\u003e3.3 Invariants of Edge Ideals: Regularity, Projective Dimension, Depth\u003cbr\u003e3.4 Stability of Associated Primes\u003c\/p\u003e \u003cp\u003e\u003cstrong\u003e4 Three Simplicial Resolutions\u003cbr\u003e\u003c\/strong\u003e\u003cem\u003eJeff Mermin\u003cbr\u003e\u003c\/em\u003e4.1 Introduction\u003cbr\u003e4.2 Background and Notation\u003cbr\u003e4.2.1 Algebra\u003cbr\u003e4.2.2 Combinatorics\u003cbr\u003e4.3 The Taylor Resolution\u003cbr\u003e4.4 Simplicial Resolutions\u003cbr\u003e4.5 The Scarf Complex\u003cbr\u003e4.6 The Lyubeznik Resolutions\u003cbr\u003e4.7 Intersections\u003cbr\u003e4.8 Questions\u003c\/p\u003e \u003cp\u003e\u003cstrong\u003e5 A Minimal Poset Resolution of Stable Ideals\u003cbr\u003e\u003c\/strong\u003e\u003cem\u003eTimothy B. P. Clark\u003cbr\u003e\u003c\/em\u003e5.1 Introduction\u003cbr\u003e5.2 Poset Resolutions and Stable Ideals\u003cbr\u003e5.3 The Shallability of PN\u003cbr\u003e5.4 The Topology of PN and Properties of D(PN)\u003cbr\u003e5.5 Proof of Theorem 2.4\u003cbr\u003e5.6 A Minimal Cellular Resolution of R\/N\u003c\/p\u003e \u003cp\u003e\u003cstrong\u003e6 Subsets of Complete Intersections and the EGH Conjecture\u003cbr\u003e\u003c\/strong\u003e\u003cem\u003eSusan M. Cooper\u003cbr\u003e\u003c\/em\u003e6.1 Introduction\u003cbr\u003e6.2 Preliminary Definitions and Results\u003cbr\u003e6.2.1 The Eisenbud-Green-Harris Conjecture and Complete Intersections\u003cbr\u003e6.3 Rectangular Complete Intersections\u003cbr\u003e6.4 Some Key Tools\u003cbr\u003e6.4.1 Pairs of Hilbert Functions and Maximal Growth\u003cbr\u003e6.4.2 Ideals Containing Regular Sequences\u003cbr\u003e6.5 Subsets of Complete Intersections in P2\u003cbr\u003e6.6 Subsets of C.I.(2, d2, d3) with d2 = d3\u003cbr\u003e6.7 Subsets of C.I.(3, d2, d3) with d3 = d2\u003cbr\u003e6.8 An Application: The Cayley-Bacharach Property\u003c\/p\u003e \u003cp\u003e\u003cstrong\u003e7 The Homological Conjectures\u003cbr\u003e\u003c\/strong\u003e\u003cem\u003ePaul C. Roberts\u003cbr\u003e\u003c\/em\u003e7.1 Introduction\u003cbr\u003e7.2 The Serre Multiplicity Conjectures\u003cbr\u003e7.2.1 The Vanishing Conjecture\u003cbr\u003e7.2.2 Gabber's Proof of the Nonnegativity Conjecture\u003cbr\u003e7.2.3 The Positivity Conjecture\u003cbr\u003e7.3 The Peskine-Szpiro Intersection Conjecture\u003cbr\u003e7.3.1 Hochester's Metatheorem\u003cbr\u003e7.4 Generalizations of the Multiplicity Conjectures\u003cbr\u003e7.4.1 The Graded Case\u003cbr\u003e7.4.2 The Generalized Rigidity Conjecture\u003cbr\u003e7.5 The Monomial, Direct Summand, and Canonical Element Conjectures\u003cbr\u003e7.6 Cohen-Macaulay Modules and Algebras\u003cbr\u003e7.6.1 Weakly Functorial Big Cohen-Macaulay Algebras\u003cbr\u003e7.7 The Syzygy Conjecture and the Improved New Intersection Conjecture\u003cbr\u003e7.8 Tight Closure Theory\u003cbr\u003e7.9 The Strong Direct Summand Conjecture\u003cbr\u003e7.10 Almost Cohen-Macaulay Algebras\u003cbr\u003e7.11 A Summary of Open Questions\u003cbr\u003e7.11.1 The Serre Positivity Conjecture\u003cbr\u003e7.11.2 Partial Euler Characteristics\u003cbr\u003e7.11.3 Strong Multiplicity Conjectures\u003cbr\u003e7.11.4 Cohen-Macaulay Modules and Related Conjectures\u003cbr\u003e7.11.5 Almost Cohen-Macaulay Algebras\u003c\/p\u003e \u003cp\u003e\u003cstrong\u003e8 The Compatibility, Independence, and Linear Growth Properties\u003cbr\u003e\u003c\/strong\u003e\u003cem\u003eYongwei Yao\u003cbr\u003e\u003c\/em\u003e8.1 Introduction\u003cbr\u003e8.2 Primary Decomposition\u003cbr\u003e8.3 Compatibility of Primary Components\u003cbr\u003e8.4 Maximal Primary Components, Independence\u003cbr\u003e8.5 Linear Growth of Primary Components\u003cbr\u003e8.6 Linear growth of {?Tor?_c^R (?(M\/(I^m M )),?(N\/(J^n N)))}\u003cbr\u003e8.7 Secondary Representation\u003cbr\u003e8.8 Compatibility of Secondary Components\u003cbr\u003e8.9 Applying a Result of Sharp on Artinian Modules\u003cbr\u003e8.10 Independence\u003cbr\u003e8.11 Minimal Secondary Components\u003cbr\u003e8.12 Linear Growth of Secondary Components\u003c\/p\u003e \u003cp\u003e\u003cstrong\u003e9 Recent Progress in Coherent Rings: A Homological Perspective\u003cbr\u003e\u003c\/strong\u003e\u003cem\u003eLivia Hummel\u003cbr\u003e\u003c\/em\u003e9.1 Introduction\u003cbr\u003e9.2 Coherent Rings and Grade\u003cbr\u003e9.2.1 Coherent Rings and ?(FP)?_?^R Modules\u003cbr\u003e9.2.2 Non-Noetherian Grade\u003cbr\u003e9.3 Cohen-Macaulay Rings\u003cbr\u003e9.4 Gorenstein Dimensions and the Auslander-Bridger Property\u003cbr\u003e9.4.1 Gorenstein Dimenstions\u003cbr\u003e9.4.2 The Auslander-Bridger Formula\u003cbr\u003e9.5 Gorenstein Rings and Injective Dimensions\u003cbr\u003e9.6 Foundations for Coherent Complete Intersections\u003c\/p\u003e \u003cp\u003e\u003cstrong\u003e10 Non-commutative Crepant Resolutions: Scenes from Categorical Geometry\u003cbr\u003e\u003c\/strong\u003e\u003cem\u003eGraham J. Leuschke\u003cbr\u003e\u003c\/em\u003e10.1 Introduction\u003cbr\u003e10.2 Morita Equivalence\u003cbr\u003e10.3 (Quasi)coherent Sheaves\u003cbr\u003e10.4 Derived Categories of Modules\u003cbr\u003e10.5 Derived Categories of Sheaves\u003cbr\u003e10.6 Example: Tilting on Projective Space\u003cbr\u003e10.7 The Non-existence of Non-commutative Spaces\u003cbr\u003e10.8 Resolutions of Singularities\u003cbr\u003e10.9 The Minimal Model Program\u003cbr\u003e10.10. Categorical Desingularizations\u003cbr\u003e10.11 Example: the McKay Correspondence\u003cbr\u003e10.12 Non-commutative Crepant Resolutions\u003cbr\u003e10.13 Example: Normalization\u003cbr\u003e10.14 MCM Endomorphism Rings\u003cbr\u003e10.15 Global Dimension of Endomorphism Rings\u003cbr\u003e10.16 Rational Singularities\u003cbr\u003e10.17 Examples: Finite Representation Type\u003cbr\u003e10.18 Example: the Generic Determinant\u003cbr\u003e10.19 Non-commutative Blowups\u003cbr\u003e10.20 Omissions and Open Questions\u003c\/p\u003e","brand":"De Gruyter","offers":[{"title":"Default Title","offer_id":53516433457495,"sku":"9783110250343","price":164.82,"currency_code":"GBP","in_stock":true}],"url":"https:\/\/bookcurl.com\/products\/progress-in-commutative-algebra-1-combinatorics-and-homology-9783110250343","provider":"Book Curl","version":"1.0","type":"link"}