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Rb.2 Roberts, P., On the construction of generic resolutions of determinantal ideals,
Ast´risque 87/88 (1981), 353“378.
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Index of Notations


Notations which seem to be completely standard or have been used in accordance
with [Mt], have not been listed.

N set of non-negative integers
N+ set of positive integers
Z ring of integers
Q ¬eld of rational numbers
C ¬eld of complex numbers
rk M rank of a module, 1, 204
rk f rank of (the image of) a homomorphism, 2
»(M ) length of a module, 2
M— dual of a module, 2
f— dual of a homomorphism, 2
i
M i-th exterior power of a module, 2
Sj (M ) j-th symmetric power of a module, 2

eI , eI 2
σ(I1 , . . . , In ), σ(i1 , . . . , in ) signum of a permutation, 2
|I| cardinality of a set, 2
S(m, I) 2
1, . . . , i, . . . , n i is to be omitted from 1, . . . , n
[a1 , . . . , at |b1 , . . . , bt ] t-minor of a matrix, 3
[a1 , . . . , am ] maximal minor of a matrix, 3
It (U ) ideal generated by the t-minors of a matrix, 3
Cof U matrix of cofactors of a matrix, 3
Rt (X) determinantal ring, 4
Lt (V, W ), L(V, W ) determinantal variety, 4
PN (K) projective N -space over the ¬eld K
i
f i-th exterior power of the homomorphism f
P(V ) projective space of the vector space V
AN (K) a¬ne N -space over the ¬eld K
G(X) B-subalgebra of B[X] generated by the maximal minors of X, 7
Gm (V ) Grassmann variety, 8
GL(V ) group of automorphisms of a vector space, 8
„¦(a1 , . . . , am ) Schubert variety, 8
S(M ) symmetric algebra of the module M
230 Index of Notations

C(g) 17

Ci (g), Ci (g) 17
Di (g) 17
C(X) 21
Mn (A) set of n — n matrices with entries in a ring, 22
G(U ) Gulliksen-Neg˚ complex, 22
ard
grade M grade of a module, 206
ωA canonical module of a Cohen-Macaulay ring, 210
GrI A, GrI M associated graded ring, module, 30
x— leading form, 30
U— form module, 30
“(X) set of maximal minors of a matrix, 46
∆(X) set of all minors of a matrix, 46
I(X; δ), I(x; δ) determinantal ideal, 51
R(X; δ) determinantal ring, 51
∆(X; δ) 51
I(X; γ) ideal de¬ning a Schubert cycle, 52
G(X; γ) Schubert cycle, 52
“(X; γ) 52
rk ξ rank of an element in a poset, 55
rk „¦ rank of a subset of a poset, 55
ara I arithmetical rank of an ideal, 61
Σ(X; γ) 67
Ξ(X; δ) 69
group of invertible r — r matrices over a ring, 74
GL(r, B)
AG subring of invariants, 74
group of r — r matrices with determinant 1, 74
SL(r, B)
VG subspace of invariants, 81
Hi (A) cohomology with support in an ideal, 81
I
MG module of invariants, 88
grade(I, M ) grade of an ideal with respect to a module, 206
Cl(S) divisor class group of a normal domain, 93
cl(I) divisor class of a fractionary ideal, 94
div(I) divisor of a fractionary ideal, 94
RI (A) Rees algebra, 108
RI (A) extended Rees algebra, 108
grad x 108
Π— 108
Π„¦ 109
vP valuation associated with a divisorial prime ideal, 116
l(I) analytic spread of an ideal in a local ring, 117
231
Index of Notations

GrF A associated graded ring with respect to a ¬ltration, 118
γt (δ) 123
()
GrP A symbolic graded ring, 124
()
RP A extended symbolic Rees ring, 124
e(j, t), ej 126
F(i, j) 126
|µ| shape of a monomial, 136
(σ)
I 136
σ
I 137
Σ (Young) tableau, 137
(Σ, T) bitableau, 138
c(Σ), c(Σ, T) content of a tableau, bitableau, 138
Kσ , Kσ 139
Λσ , Λσ 139
(σ)
I> 139
σ L, Lσ 141
+
group of upper (lower) triangular n — n matrices with entry 1
U (n, K), (U (n, K)) ’


on all diagonal positions, 141
group of invertible diagonal n — n matrices, 143
D(n, K)
Vω isotypic component, 143
Mσ 143
Iσ 146
I(S) 147
Rad S radical of a D-ideal, 147
I(s0 , . . . , sr ) 155
I(s0 , . . . , sr ; v) 156
•f,r 165
D(M ) Auslander-Bridger dual of a module, 214
„¦1 , „¦ module of K¨hler di¬erentials, 174
a
R/B
δst 175
I(X; δs+1,t , δs,t+1 ), I(x; δs+1,t , δs,t+1 ) 175
∆(X; δs+1,t , δs,t+1 ) 175
d±, d± di¬erential of ±, 177
M(s, t), M(s, t) 178
•, χ, ψ 184
Nkl 186
DS Auslander-module, 198
µ(M ) minimal number of generators of a module, 203
( Sn ) Serre type condition for modules, 213
Subject Index



absolute invariant, 74, 86 bitableau, 138
left (right) ¬nal, 139
absolute semi-invariant, 76
left (right) initial, 139
acyclicity criterion, 207, 208
standard, 138
algebra generated by the t-minors, 132,
133, 151
canonical (divisor) class, 93
algebra with straightening law (ASL), 38
of a determinantal ring, 97, 98, 103
algebraic quotient, 80
of a Schubert cycle, 97, 102
analytic spread, 117
canonical module, 210
arithmetical rank, 61
of a determinantal ring, 22, 97, 98,
of a determinantal ideal, 62, 81
103, 115
of an ideal de¬ning a Schubert cycle,
of a Schubert cycle, 97, 102, 115
62
class group, s. divisor class group
of an ideal generated by a poset ideal,
cofactors, matrix of, 3
62
Cohen-Macaulay
in a symmetric ASL, 62
module, 209
ASL, 38
ring, 209
discrete, 62
type of a determinantal ring, 115
symmetric, 43
Cohen-Macaulay property
ASL-property of
and perfection, 210
a determinantal ring, 52
of a determinantal ring, 13, 25, 60
a graded ring derived from a straight-
of a graded ring derived from an ideal
ening closed ideal, 108, 110
of maximal minors, 112
a Schubert cycle, 53

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