COMBINATORIAL COMMUTATIVE ALGEBRA
CONTENTS I MONOMIAL IDEALS SQUAREFREE MONOMIAL IDEALS EQUIVALENT DESCRIPTIONS 1 2 HILBERT SERIES 1 3 SIMPLICIAL COMPLEXES AND HOMOLOGY 1 4 MONOMIAL MATRICES 1 5 BETTI NUMBERS EXERCISES NOTES BOREL-?XED MONOMIAL IDEALS GROUP ACTIONS GENERIC INITIAL IDEALS THE ELIAHOU–KERVAIRE RESOLUTION LEX-SEGMENT IDEALS THREE-DIMENSIONAL STAIRCASES MONOMIAL IDEALS IN TWO VARIABLES AN EXAMPLE WITH SIX MONOMIALS THE BUCHBERGER GRAPH GENERICITY AND DEFORMATIONS THE PLANAR RESOLUTION ALGORITHM CELLULAR RESOLUTIONS CONSTRUCTION AND EXACTNESS 4 2 BETTI NUMBERS AND K -POLYNOMIALS EXAMPLES OF CELLULAR RESOLUTIONS THE HULL RESOLUTION SUBDIVIDING THE SIMPLEX 2 BOREL-XED MONOMIALIDEALS XII CONTENTS EXERCISES ALEXANDER DUALITY SIMPLICIAL ALEXANDER DUALITY GENERATORS VERSUS IRREDUCIBLE COMPONENTS DUALITY FOR RESOLUTIONS COHULL RESOLUTIONS AND OTHER APPLICATIONS PROJECTIVE DIMENSION AND REGULARITY GENERIC MONOMIAL IDEALS TAYLOR COMPLEXES AND GENERICITY THE SCARF COMPLEX GENERICITY BY DEFORMATION BOUNDS ON BETTI NUMBERS COGENERIC MONOMIAL IDEALS II TORIC ALGEBRA SEMIGROUP RINGS SEMIGROUPS AND LATTICE IDEALS A?NE SEMIGROUPS AND POLYHEDRAL CONES HILBERT BASES INITIAL IDEALS OF LATTICE IDEALS MULTIGRADED POLYNOMIAL RINGS MULTIGRADINGS HILBERT SERIES AND K-POLYNOMIALS MULTIGRADED BETTI NUMBERS K-POLYNOMIALS IN NONPOSITIVE GRADINGS MULTIDEGREES SYZYGIES OF LATTICE IDEALS BETTI NUMBERS LAURENT MONOMIAL MODULES FREE RESOLUTIONS OF LATTICE IDEALS GENERICITY AND THE SCARF COMPLEX CONTENTS XIII 10 TORIC VARIETIES 10 1 ABELIAN GROUP ACTIONS 10 2 PROJECTIVE QUOTIENTS 10 3 CONSTRUCTING TORIC VARIETIES 10 4 TORIC VARIETIES AS QUOTIENTS 11 IRREDUCIBLE AND INJECTIVE RESOLUTIONS 11 1 IRREDUCIBLE RESOLUTIONS 11 2 INJECTIVE MODULES 11 3 MONOMIAL MATRICES REVISITED 11 4 ESSENTIAL PROPERTIES OF INJECTIVES 11 5 INJECTIVE HULLS AND RESOLUTIONS 12 EHRHART POLYNOMIALS 12 1 EHRHART FROM HILBERT 12 2 DUALIZING COMPLEXES 12 3 BRION’S FORMULA 12 4 SHORT RATIONAL GENERATING FUNCTIONS 13 LOCAL COHOMOLOGY 13 1 EQUIVALENT DE?NITIONS 13 2 HILBERT SERIES CALCULATIONS 13 3 TORIC LOCAL COHOMOLOGY 13 4 COHEN–MACAULAY CONDITIONS 13 5 EXAMPLES OF COHEN–MACAULAY RINGS III DETERMINANTS 14 PL¨UCKER COORDINATES 14 1 THE COMPLETE ?AG VARIETY 14 2 QUADRATIC PL¨UCKER RELATIONS 14 3 MINORS FORM SAGBI BASES 14 4 GELFAND–TSETLIN SEMIGROUPS XIV CONTENTS 15 MATRIX SCHUBERT VARIETIES 15 1 SCHUBERT DETERMINANTAL IDEALS 15 2 ESSENTIAL SETS 15 3 BRUHAT AND WEAK ORDERS 15 4 BOREL GROUP ORBITS 15 5 SCHUBERT POLYNOMIALS 16 ANTIDIAGONAL INITIAL IDEALS 16 1 PIPE DREAMS 16 2 A COMBINATORIAL FORMULA 16 3 ANTIDIAGONAL SIMPLICIAL COMPLEXES 16 4 MINORS FORM GR¨OBNER BASES 16 5 SUBWORD COMPLEXES 17 MINORS IN MATRIX PRODUCTS 17 1 QUIVER IDEALS AND QUIVER LOCI 17 2 ZELEVINSKY MAP 17 3 PRIMALITY AND COHEN–MACAULAYNESS 17 4 QUIVER POLYNOMIALS 17 5 PIPES TO LACES 18 HILBERT SCHEMES OF POINTS 18 1 IDEALS OF POINTS IN THE PLANE 18 2 CONNECTEDNESS AND SMOOTHNESS 18 3 HAIMAN’S THEORY 18 4 IDEALS OF POINTS IN D-SPACE 18 5 MULTIGRADED HILBERT SCHEMES REFERENCES