158 INDEX

residue formula

for the multiplicity, 96

for centroids, 97

residue theorem

aﬃne version, 58

for

Pd

, 57

for projective varieties, 104

for projective curves, 107

residues and duality for proper varieties,

106

restriction property of residues, 40

retraction, 74

Riemann-Hurwitz genus formula, 109

right-derived functor, 1

ringed space, 1

scheme at infinity, 63

section of a sheaf, 1

semigroup algebra, 130

of a lattice polytope, 132

separable Noether normalization, 81

of analytic algebras, 82

in the graded case, 100

separated scheme, 16

separating system of parameters, 82

Serre’s finiteness conditions, 49

sheaf

associated to a module, 1, 47

of differential forms, 52

of rational d-forms, 52

of regular d-forms, 84

of Zariski d-forms, 86

simplicial toric variety, 143

singularity degree, 113

skyscraper sheaf, 9

socle

degree, 60, 77, 117

of a zero-dimensional local or graded

ring, 29

standard

covering of

Pd

k

, 57

d-simplex, 134

scalar product, 63

trace, 65

strict complete intersection, 119

support

of a divisor, 108

of a section, 1

T -acyclic object, 3

tangent, 63

tangent hyperplane, 61

tangential center, 99

Tate trace, 73

theorem of

B´ ezout, 97

Cayley-Bacharach, 62

Chasles, 99

G. Humbert, 64

Pappus, 62

Pascal, 62

Reiss, 64

Riemann-Roch for projective curves, 108

toric

aﬃne variety, 131

Euler-Jacobi vanishing theorem, 145, 147

homogeneous coordinates, 133

projective variety, 132

quotient property, 148

residue, 133, 138, 140

residues and resultants, 149

transformation law, 140

torsion of differential forms, 86

torus, 129

torus-invariant Weil divisor, 133

total

complex of a double complex, 14

coordinate ring, 133

ramification number, 109

sum of residues, 124

trace

map of an algebra, 65

of an endomorphism, 65

trace formula

for residues, 93, 95, 104

of Hasse, 96

transformation law for residues, 40

transition determinant, 26

transitive law

for the complementary module and the

different, 67

for dualizing sheaves, 88

for the trace, 65

transitive properties of regular differential

forms, 87, 106

transversal intersection, 61

universal

derivation, 35, 52

module of differentials, 35

universally finite

derivation, 35

module of differentials, 35

vertex of the aﬃne cone, 47, 101

very ample sheaf, 49

volume polynomial, 119

Waring’s problem, 127

Weierstraß preparation theorem, 91

weighted case, 61, 121

Zariski differentials, 86

zero-dimensional complete intersection, 58