COMPUTATIONAL ALGEBRAIC GEOMETRY
CONTENTS 1 BASICS OF COMMUTATIVE ALGEBRA 1 1 IDEALS AND VARIETIES 1 2 NOETHERIAN RINGS AND THE HILBERT BASIS THEOREM 1 3 ASSOCIATED PRIMES AND PRIMARY DECOMPOSITION 1 4 THE NULLSTELLENSATZ AND ZARISKI TOPOLOGY 2 PROJECTIVE SPACE AND GRADED OBJECTS 2 1 PROJECTIVE SPACE AND PROJECTIVE VARIETIES 2 2 GRADED RINGS AND MODULES, HILBERT FUNCTION AND SERIES 2 3 LINEAR ALGEBRA FLASHBACK, HILBERT POLYNOMIAL 3 FREE RESOLUTIONS AND REGULAR SEQUENCES 3 1 FREE MODULES AND PROJECTIVE MODULES 3 2 FREE RESOLUTIONS 3 3 REGULAR SEQUENCES, MAPPING CONE 4 GR¨OBNER BASES AND THE BUCHBERGER ALGORITHM 4 1 GR¨OBNER BASES 4 2 MONOMIAL IDEALS AND APPLICATIONS 4 3 SYZYGIES AND GR¨OBNER BASES FOR MODULES 4 4 PROJECTION AND ELIMINATION 5 COMBINATORICS, TOPOLOGY AND THE STANLEY–REISNER RING 5 1 SIMPLICIAL COMPLEXES AND SIMPLICIAL HOMOLOGY 5 2 THE STANLEY–REISNER RING 5 3 ASSOCIATED PRIMES AND PRIMARY DECOMPOSITION 6 FUNCTORS: LOCALIZATION, HOM, AND TENSOR 6 1 LOCALIZATION 6 2 THE HOM FUNCTOR 6 3 TENSOR PRODUCT 7 GEOMETRY OF POINTS AND THE HILBERT FUNCTION 7 1 HILBERT FUNCTIONS OF POINTS, REGULARITY CONTENTS 7 2 THE THEOREMS OF MACAULAY AND GOTZMANN 7 3 ARTINIAN REDUCTION AND HYPERSURFACES 8 SNAKE LEMMA, DERIVED FUNCTORS, TOR AND EXT 8 1 SNAKE LEMMA, LONG EXACT SEQUENCE IN HOMOLOGY 8 2 DERIVED FUNCTORS, TOR 8 3 EXT 8 4 DOUBLE COMPLEXES 9 CURVES, SHEAVES, AND COHOMOLOGY 9 1 SHEAVES 9 2 COHOMOLOGY AND GLOBAL SECTIONS 9 3 DIVISORS AND MAPS TO PN 9 4 RIEMANN–ROCH AND HILBERT POLYNOMIAL REDUX 10 PROJECTIVE DIMENSION, COHEN–MACAULAY MODULES, UPPER BOUND THEOREM 10 1 CODIMENSION, DEPTH, AUSLANDER–BUCHSBAUM THEOREM 10 2 COHEN–MACAULAY MODULES AND GEOMETRY 10 3 THE UPPER BOUND CONJECTURE FOR SPHERES