COHOMOLOGY OF VECTOR BUNDLES AND SYZYGIES
CONTENTS INTRODUCTORY MATERIAL MULTILINEAR ALGEBRA AND COMBINATORICS HOMOLOGICAL AND COMMUTATIVE ALGEBRA DETERMINANTS OF COMPLEXES SCHUR FUNCTORS AND SCHUR COMPLEXES SCHUR FUNCTORS AND WEYL FUNCTORS SCHUR FUNCTORS AND HIGHEST WEIGHT THEORY PROPERTIESOFSCHURFUNCTORS CAUCHYFORMULAS,LITTLEWOOD– RICHARDSON RULE, AND PLETHYSM THE SCHUR COMPLEXES EXERCISES FOR CHAPTER GRASSMANNIANS AND FLAG VARIETIES THE PL¨UCKER EMBEDDINGS THE STANDARD OPEN COVERINGS OF FLAG MANIFOLDS AND THE STRAIGHTENING LAW THE HOMOGENEOUS VECTOR BUNDLES ON FLAG MANIFOLDS BOTT’S THEOREM THE FORMULATION OF BOTT’S THEOREM FOR THE GENERAL LINEAR GROUP THE PROOF OF BOTT’S THEOREM FOR THE GENERAL LINEAR GROUP BOTT’S THEOREM FOR GENERAL REDUCTIVE GROUPS THE GEOMETRIC TECHNIQUE THE FORMULATION OF THE BASIC THEOREM THE PROOF OF THE BASIC THEOREM THE PROOF OF PROPERTIES OF COMPLEXES F(V) THE G-EQUIVARIANT SETUP THE DIFFERENTIALS IN COMPLEXES F(V) DEGENERATION SEQUENCES THE DETERMINANTAL VARIETIES THE LASCOUX RESOLUTION THE RESOLUTIONS OF DETERMINANTAL IDEALS IN POSITIVE CHARACTERISTIC THE DETERMINANTAL IDEALS FOR SYMMETRIC MATRICES THE DETERMINANTAL IDEALS FOR SKEW SYMMETRIC MATRICES MODULES SUPPORTED IN DETERMINANTAL VARIETIES MODULES SUPPORTED IN SYMMETRIC DETERMINANTAL VARIETIES MODULES SUPPORTED IN SKEW SYMMETRIC DETERMINANTAL VARIETIES HIGHER RANK VARIETIES BASIC PROPERTIES RANK VARIETIES FOR SYMMETRIC TENSORS RANK VARIETIES FOR SKEW SYMMETRIC TENSORS THE NILPOTENT ORBIT CLOSURES THE CLOSURES OF CONJUGACY CLASSES OF NILPOTENT MATRICES THE EQUATIONS OF THE CONJUGACY CLASSES OF NILPOTENT MATRICES THE NILPOTENT ORBITS FOR OTHER SIMPLE GROUPS CONJUGACY CLASSES FOR THE ORTHOGONAL GROUP CONJUGACY CLASSES FOR THE SYMPLECTIC GROUP RESULTANTS AND DISCRIMINANTS THE GENERALIZED RESULTANTS THE RESULTANTS OF MULTIHOMOGENEOUS POLYNOMIALS THE GENERALIZED DISCRIMINANTS THE HYPERDETERMINANTS REFERENCES INTRODUCTORY MATERIAL 1 1 MULTILINEAR ALGEBRA AND COMBINATORICS 1 1 1 EXTERIOR, DIVIDED, AND SYMMETRIC POWERS; MULTIPLICATION AND DIAGONAL MAPS LET K BE A COMMUTATIVE RING, AND LET E BE A FREE K-MODULE WITH A BASIS {E1, , EN} WE DE?NE THE R-TH EXTERIOR POWER R E OF E TO BE THE R-TH TENSOR POWER E?R OF E DIVIDED BY THE SUBMODULE GENERATED BY THE ELEMENTS: U1 ? ?UR ?(?1)SGN ?U?(1) ? ?U?(R) FOR ALL ? ?R, U1, , UR ?E WE DENOTE THE COSET OF U1 ? ?UR BY U1 ? ?UR THE FOLLOWING BASIC PROPERTIES OF EXTERIOR POWERS ARE PROVED IN [L, CHAP- TER XIX, SECTION 1] (1 1 1) PROPOSITION (A) LET {E1, , EN} BE AN ORDERED BASIS OF E THEN THE ELEMENTS EI1 ? ? EIR FOR 1 ?I1 < < IR ?N FORM A BASIS OF R E IN PARTICULAR, R E IS A FREE K-MODULE OF DIMENSION N R (B) (UNIVERSALITY PROPERTY OF EXTERIOR POWERS) WE HAVE A FUNCTORIAL ISO- MORPHISM ?M : ALTR(ER, M) ?HOMK R E, M WHERE ALTR(ER, M) DENOTES THE SET OF MULTILINEAR ALTERNATING MAPS FROM E×R TO M,GIVENBYTHEFORMULA?R M( F )(U1 ? ?UR) = F (U1, , UR)