x GLOSSARY OF NOTATION

Ha(M)

i

i-th cohomology with support in V (a) 8

Hm(M)

i

i-th local cohomology at a maximal ideal m 8

JY ideal sheaf of Y 9

OX

r

direct sum of r copies of OX 9

k(x) residue field of OX,x 9

PA

d

d-dimensional projective space over a ring A 11

C•(U, F ) alternating

ˇ

Cech complex with respect to a covering U 11

ˇ

H

•(U,

F )

ˇ

Cech cohomology 11

C•(U,

F ) normalized

ˇ

Cech complex 11

C •(U, F ) sheafified

ˇ

Cech complex 12

dim (Krull) dimension of a ring or module or scheme 16

C•[1] complex with degree shift by 1 16

Rd exterior algebra of Rd 22

K•(t, M) homological Koszul complex 22

K•(t,

M) cohomological Koszul complex 23

H•(t, M) homology of K•(t, M) 23

H•(t,

M) homology of

K•(t,

M) 23

MS localization with respect to S 23

∆ transition determinant 26

µx multiplication by x 27

S(R) socle of a zero-dimensional local

or graded ring R 29

lim

−→

direct (injective) limit 29

Φt canonical map M/(t)M → Hm(M)

d

32

m

t

generalized fraction of m over denominator set t 32

ΩR/k

•

algebra of differential forms of R/k 35

ResR residue for a complete reduced local ring R 39, 93

ρt canonical map induced by the residue 40, 89

f(0) constant term of a power series f 41

J, Jf Jacobian determinant (of a set f of polynomials) 43

lim

←−

inverse (projective) limit 45

S+ homogeneous maximal ideal of a graded ring S 47

Proj S projective scheme of a graded ring S 47

D+(f) non-vanishing set of a homogeneous element

f ∈ S on Proj S 47

M(f

)

homogeneous localization of a graded module 47

M(n) graded module shifted by n 47

M∗

sheaf associated to a module 47