156 INDEX

degree

of a generalized fraction, 42

of a zero-dimensional scheme, 96

of the socle, 60, 77

degree form, 58

δ-functor, 3, 11

denominator of a generalized fraction, 32

description of cohomology by generalized

fractions, 48

diameter, 99

of complete intersection curves, 99

of subschemes (cycles) in aﬃne space, 99

differential

forms, 35, 52

operator, 115

differentiation

in the

ˇ

Cech complex, 11

in the Koszul complex, 22, 23

of differential forms, 35

differents and Jacobians, 72

direct image of a sheaf, 5

double complex, 14

double dual, 85

dual

of a lattice, 130

of a sheaf, 85

of a polyhedral cone, 130

duality theorem

for

Pd

, 55

for projective varieties, 102

dualizing

complex, 114

form, 119

module, 40, 65, 84

sheaf, 85

effective computation, 125

embedding the semigroup algebra into the

total coordinate ring, 134

endomorpphism ring of the canonical

module, 112

equality of generalized fractions, 32, 34

´ etale, 66

Euler

characteristic, 50

derivation, 100

sequence, 53

Euler-Cramer paradox, 62

Euler-Jacobi

formula, 62

vanishing condition, 122

toric generalization, 145, 147

Euler’s formula, 54

evaluation at the origin, 116

exact differentials, 37

exactness of the Koszul complex, 26

exchange lemma

for separating systems of parameters, 82

extension of a sheaf by zero, 2

extension rule for generalized fractions, 33

exterior

algebra, 22

differentiation, 52

Ext groups of sheaves, 110

facet normal, 134

field

of Laurent series, 36

of rational functions, 52

finiteness theorem for cohomology, 49

finite derivation, 35

Fitting ideal, 70

flasque

module, 1

resolution, 2

sheaf, 1

flasque sheaf associated to a sheaf, 2

form of maximal degree (degree form), 58

Frobenius

algebra, 77

map, 38

fundamental class of a variety, 86

generalized

B´ ezoutian, 70

fraction, 32

Jacobian, 74

generator of the socle, 29

germ of a section, 1

geometric genus, 103

global

membership, 123

residue, 143

Gorenstein

algebra, 69

duality, 77, 78

ideal, 77, 116, 117

point, variety, 86

gradient, 63

Grothendieck residue symbol, 39

group of sections with support in Y , 1

Hilbert polynomial, 51

homogeneous cohomology class

generalized fraction, 41, 48

homogenization, 54

in a toric geometry, 135

H¨ ubl, R., 63

hyperplane at infinity, 63

incremental quotient, 119

injective

object, 1

OX -module, 2

R-module, 6

sheaf, 3