GLOSSARY OF NOTATION xi

dimk dimension of a k-vector space 49

Γ∗F graded module associated to a sheaf 49

Γ(X, F ) module of global sections 50

χ(F ) Euler characteristic of F 50

Supp(F ) support of a sheaf F 50

PF Hilbert polynomial of F 50

pa(X) arithmetic genus of X 51

ΩX/A

p

sheaf of p-forms on a scheme X/A 52

R(X) field of rational functions on

an integral scheme X/A 52

X

integral for a projective variety X 55, 101

δF duality isomorphism 55, 94, 102

R, M completion of a local ring R or an R-module M 56

Resx residue at a closed point x 56, 95

X(k) set of k-rational points of X 57

V+(I) zero-set of a homogeneous ideal I in Proj S 58

Ak

d

aﬃne space over k 58

G(f) = Gf degree form of a polynomial f 58

TP (V ) tangent space of a variety V at P 61

grad f gradient of a polynomial f 63

, standard scalar product 63

H∞ hyperplane at infinity 63

fXi =

∂f

∂Xi

partial derivative of f with respect to Xi 63

tr(ϕ) trace of an endomorphism 65

trS/R trace map of an algebra 65

ωS/R canonical (dualizing) module 65, 84

Q(R) full ring of quotients of a ring R 66

CS/R Dedekind complementary module 66

dD(S/R) Dedekind different 66

fT/S conductor of an algebra T/S 67

k(SP) residue field of SP 68

µ(M) minimal number of generators of M 68

µP(M) minimal number of generators of MP 68

rP Cohen-Macaulay type at P 68

Max S set of maximal ideals of S 69

δ canonical multiplication map S ⊗R S → S 70

dN (S/R) Noether different 70

dK (S/R) K¨ ahler different 70

∆x

t

B´ ezoutian 72