Cohomology of Vector Bundles and Syzygies (Cambridge Tracts in Mathematics)

The central theme of this book is a detailed exposition of the geometric technique of calculating syzygies. While this is an important tool in algebraic geometry, Jerzy Weyman has elected to write from the point of view of commutative algebra in order to avoid being tied to special cases from geometry. No prior knowledge of representation theory is assumed. Chapters on several applications are included, and numerous exercises will give the reader insight into how to apply this important method. Cover......Page 1 Half-title......Page 3 Title......Page 7 Copyright......Page 8 Dedication......Page 9 Contents......Page 11 Preface......Page 13 1.1.1. Exterior, Divided, and Symmetric Powers; Multiplication and Diagonal Maps......Page 17 1.1.2. Partitions, Skew Partitions. Combinatorics of Z-Graded Tableaux.......Page 24 1.2.1. Regular Sequences, Koszul Complexes, Depth......Page 28 1.2.2. Cohen–Macaulay Rings and Modules, Gorenstein Rings......Page 30 1.2.3. Minimal Resolutions......Page 33 1.2.4. Effective Calculation of Normalization......Page 35 1.2.5. Duality for Proper Morphisms and Rational Singularities......Page 36 1.3. Determinants of Complexes......Page 43 2.1. Schur Functors and Weyl Functors......Page 48 2.2. Schur Functors and Highest Weight Theory......Page 65 2.3. Properties of Schur Functors. Cauchy Formulas, Littlewood–Richardson Rule, and Plethysm......Page 73 2.4. The Schur Complexes......Page 82 Definition of Schur Modules......Page 94 Schur and Weyl Modules in Positive Characteristic......Page 95 Littlewood–Richardson Rule......Page 97 Acyclicity Properties of Schur Complexes......Page 99 3.1. The Plücker Embeddings......Page 101 3.2. The Standard Open Coverings of Flag Manifolds and the Straightening Law......Page 107 3.3. The Homogeneous Vector Bundles on Flag Manifolds......Page 114 Isotropic Grassmannians......Page 120 Combinatorial Proof of Littlewood–Richardson Rule......Page 122 Canonical Bundles on Flag Varieties......Page 125 4.1. The Formulation of Bott’s Theorem for the General Linear Group......Page 126 4.2. The Proof of Bott’s Theorem for the General Linear Group......Page 133 4.3. Bott’s Theorem for General Reductive Groups......Page 139 The General Linear Group......Page 148 Other Classical Groups......Page 149 Tensor Product Multiplicities......Page 150 5 The Geometric Technique......Page 152 5.1. The Formulation of the Basic Theorem......Page 153 5.2. The Proof of the Basic Theorem......Page 157 5.3. The Proof of Properties of Complexes F(Nu)......Page 162 5.4. The G-Equivariant Setup......Page 165 5.5. The Differentials in Complexes F(Nu)......Page 168 5.6. Degeneration Sequences......Page 170 Cones over Nonsingular Curves......Page 172 The Representations of SL(2). Binary Forms......Page 173 Highest Weight Vector Orbit Varieties......Page 174 6 The Determinantal Varieties......Page 175 6.1. The Lascoux Resolution......Page 176 6.2. The Resolutions of Determinantal Ideals in Positive Characteristic......Page 184 6.3. The Determinantal Ideals for Symmetric Matrices......Page 191 6.4. The Determinantal Ideals for Skew Symmetric Matrices......Page 203 6.5. Modules Supported in Determinantal Varieties......Page 211 6.6. Modules Supported in Symmetric Determinantal Varieties......Page 225 6.7. Modules Supported in Skew Symmetric Determinantal Varieties......Page 229 The Symplectic Group......Page 234 The Orthogonal Group......Page 235 The First Fundamental Theorem for the General Linear Group......Page 236 Differentials in the Resolutions of Ideals of Minors of a Generic Matrix......Page 237 Differentials in the Resolutions of Ideals of Pfaffians of a Generic Skew Symmetric Matrix......Page 238 Maximal Cohen–Macaulay Modules with Linear Resolutions......Page 239 Resolutions of K(Phi)......Page 240 Resolutions of Powers of the Ideal of 2t × 2t Pfaffians of a (2t + 1) × (2t + 1) Skew Symmetric Matrix......Page 242 7.1. Basic Properties......Page 244 7.2. Rank Varieties for Symmetric Tensors......Page 250 7.3. Rank Varieties for Skew Symmetric Tensors......Page 255 Minimal Resolutions of the Ideal.........Page 261 The Isotropic Grassmannian IGrass(3, 6)......Page 265 8 The Nilpotent Orbit Closures......Page 267 8.1. The Closures of Conjugacy Classes of Nilpotent Matrices......Page 268 8.2. The Equations of the Conjugacy Classes of Nilpotent Matrices......Page 279 8.3. The Nilpotent Orbits for Other Simple Groups......Page 294 8.4. Conjugacy Classes for the Orthogonal Group......Page 299 8.5. Conjugacy Classes for the Symplectic Group......Page 312 Type B.......Page 325 Type C.......Page 326 Type D.......Page 327 9 Resultants and Discriminants......Page 329 9.1. The Generalized Resultants......Page 330 9.2. The Resultants of Multihomogeneous Polynomials......Page 334 9.3. The Generalized Discriminants......Page 344 9.4. The Hyperdeterminants......Page 348 Exercises for Chapter 9......Page 371 Discriminants of Adjoint Representations......Page 372 Determinantal Expressions for Powers of the Resultant......Page 374 References......Page 375 Notation Index......Page 383 Subject Index......Page 385 The central theme of this book is an exposition of the geometric technique of calculating syzygies. It is written from a point of view of commutative algebra, and without assuming any knowledge of representation theory the calculation of syzygies of determinantal varieties is explained. The starting point is a definition of Schur functors, and these are discussed from both an algebraic and geometric point of view. Then a chapter on various versions of Bott's Theorem leads on to a careful explanation of the technique itself, based on a description of the direct image of a Koszul complex. Applications to determinantal varieties follow, plus there are also chapters on applications of the technique to rank varieties for symmetric and skew symmetric tensors of arbitrary degree, closures of conjugacy classes of nilpotent matrices, discriminants and resultants. Numerous exercises are included to give the reader insight into how to apply this important method. Jerzy Weyman. Includes Bibliographical References (p. 359-366) And Indexes.