<<

. 47
( 47 .)



of an ASL, 59
B[X] on ∆(X), 48
of a Schubert cycle, 60
G(X) on “(X), 41
complete intersection, 198
associated graded module, 30 content of a tableau (bitableau), 138
associated graded ring 30, 108
derived from a straightening-closed decomposition of K[X] into irreducible
ideal, 109 G-modules, 143, 144
derived from an ideal of maximal mi- de¬ning ideal
nors, 112, 114 of an ASL, 40
derived from the ideal of t-minors, 132 of G(X), 43
associated prime ideals, 1 dehomogenization, 211
of a G-stable ideal 148, 149 and being reduced, 212
of powers of a determinantal ideal, 132 determinantal ring as a, 53
Auslander-Bridger dual, 214 integrity, 212
Auslander-Buchsbaum equation, 1 normality, 212
Auslander-module, 198 of a homogeneous ideal, 212, 213
and rigidity, 199, 200 B[X] as a, 48
233
Subject Index

depth, s. also grade extended symbolic Rees ring, 124
derived from a determinantal ideal,
of a module, 1, 206
124
derivations, module of, 184
extraspecial pair, 43
almost perfection, 188
perfection, 192
factoriality of a
system of generators, 188
determinantal ring, 95
syzygetic behaviour, 197
Schubert cycle, 69, 95
syzygies, 189
Fitting invariants of a module, 203
determinantal ideal, 5, 51, 155, 156, 175
form ideal, form module, 30
of a homomorphism, 202
free resolution, 18, 21
determinantal ring, 4, 5
of an ideal
as a ring of absolute invariants, 76, 86,
of maximal minors, 18
87, 160
of submaximal minors in a quadratic
determinantal variety, 4, 6 matrix, 25
diagram, 136 of a generic module, 172
D-ideal, 145 of symmetric and exterior powers, 18,
primary, 147 21
prime, 147 of the image of a generic map, 172
principal, 146 full semigroup, 150
radical, 147
G-decomposition of K[X], 143, 144
di¬erentials, module of, 174
generically perfect
depth, 180, 181, 183
ideal, 27
free locus, 178
module, 27
grade, 180, 183
generic map, 162
presentation as a second syzygy, 186
free resolution of the image, 172
syzygetic behaviour, 182
perfection of the image, 164, 165
syzygies, 181
generic module, 162
dimension
almost perfection, 164
of a determinantal ring, 57
free resolution, 172
of an ASL, 55
perfection, 18, 167, 170
of a Schubert cycle, 57
projective dimension, 171
discrete ASL, 62
re¬‚exivity, 162
divisor class group
syzygetic behaviour, 171, 172
of a determinantal ring, 95, 96, 160
generic perfection, 27
of a Schubert cycle, 95
and substitution of indeterminates, 31,
dual
32
basis, 2 of a determinantal ideal, 61
of a homomorphism, 2 of a Schubert cycle, 61
of a module, 2 of the canonical module, 30
dualizing module, 210 generic point, 73
of a determinantal ring, 74, 85, 87
Eagon-Northcott complex, 16 of a Schubert cycle, 83, 85
extended Rees algebra, 108 G-module, 80
derived from Gorenstein property, 210
an ideal of maximal minors, 112, 114 of a determinantal ring, 22, 25, 98, 104
a straightening-closed ideal, 108, 109 of a Schubert cycle, 102
234 Subject Index

Gorenstein ring, 210 invariant, 80
grade, 206 absolute, 74, 86
of a determinantal ideal, 12, 61, 159 theory, main problems, 80
of a module, 206 versus absolutely invariant, 75
of an ideal, 206 isotopic component, 142
and projective dimension, 207
behaviour along exact sequences, 207 K¨hler di¬erentials, s. di¬erentials
a
of an ideal de¬ning a Schubert cycle, Koszul complex, 17
61
homological description, 206
leading form, 30, 108
local description, 207
linear independence of standard monomi-
grade estimates
als in G(X), 43, 84
for modules of invariants, 89
linearly reductive, 80, 81
for powers of ideals of maximals mi-
localization argument for
nors, 90, 118, 119
determinantal rings, 11, 66, 122
for symbolic powers of a determinantal
Schubert cycles, 64
ideal, 125
locally upper semi-modular poset, 58
Grassmann variety, 6, 8
G-stable ideals of K[X], 146
G-submodule generated by a bitableau, maximal minors, 3, 105
145 ideals of, 10, 105
Gulliksen-Neg˚ complex, 22
ard powers of ideals of, 106, 114
subalgebra of, 106, 107, 112
height of minimal primes of an ideal generated by
a determinantal ideal, 12, 61, 159, 175 a poset ideal
an ideal generated by minors of a ma- in ∆(X), 66
trix, 10, 11 in “(X), 66
an ideal de¬ning a Schubert cycle, 61 minor of a matrix, 3
a specialization of a perfect ideal, 36 module
Hilbert-Burch theorem, 218 torsionfree, 213
homogenization of an ideal, 212, 213 n-torsionfree, 213
n-th syzygy, 214
ideal in a partially ordered set, 50
n-torsionless, 214
cogenerated by a subset, 51
of derivations, s. derivations
generated by a subset, 51
of di¬erentials, s. di¬erentials
ideals of maximal minors, 10, 105
re¬‚exive, 214
free resolution, 18
multiplicity of a (maximal minors) deter-
indecomposability ob the subspace of
minantal ring, 16
(right) initial bitableaux, 142
induction argument for determinantal
normality criterion, 212
rings, s. localization argument
normality of a
integral closure of a G-stable ideal, 149
determinantal ring, 15, 61, 65, 150, 160
integrity of a
graded ring derived from an ideal gen-
determinantal ring, 14, 65, 158
erated by maximal minors, 114
graded ring derived from an ideal of
Schubert cycle, 65
maximal minors, 114
Schubert cycle, 65 normal subsemigroup, 150
235
Subject Index

perfect Rees algebra, 108
derived from
ideal, 209
module, 209 an ideal of maximal minors, 112, 114
a straightening-closed ideal, 110, 111
perfection, 209
extended, 108
and Cohen-Macaulay property, 210
re¬‚exive module, 214
of a determinantal ideal, 13, 18, 25, 61,
159, 175 regular elements in an ASL, 55, 56
of a Schubert cycle, 61 Reynolds operator, 81, 88
of a specialization, 28, 29 rigidity, 198, 200
Pl¨cker
u ring of invariants
coordinates, 6 being noetherian, 81
map, 6 normality, 81, 115, 149
relations, 41 rational singularities, 81, 149
poset, 39 Cohen-Macaulay property, 81
powers of existence of a Reynolds operator, 81
an ideal generated by a regular se- ring of U -invariants, 149
quence, 15, 29 in a determinantal ring, 150
an ideal of maximal minors, 79, 114
determinantal ideals, 79, 114, 130, 149 Schubert cycle, 6, 9
primary decomposition of as a ring of absolute invariants, 76, 86
a G-stable ideal, 148, 149 Schubert variety, 6, 8
products of determinantal ideals, 126, Segre product of ASLs, 107
130, 149 self-covering poset ideal, 110
prime elements in a determinantal ring, Serre™s condition
97 (R2 ) for a determinantal ring, 71
principal radical system, 155 (R2 ) for a Schubert cycle, 71
of determinantal ideals, 158 (Sn ) for an ASL, 60
products of determinantal ideals, 126, Serre type criterion for torsionfree mod-
137 ules, 213
shape of a monomial, 135
radical singular locus of a
of a D-ideal, 147 determinantal ring, 12, 70, 71, 160
subsemigroup, 150 Schubert cycle, 68
rank of a special pair, 43
free module, 1 spectrum of a ring, 1
module, 1, 204 standard
poset element, 55 basis, 38
subset of a poset, 55 bitableau, 138
rational representation, 80 monomial, 38
rational singularity, 81, 150 tableau, 138
reduced standard monomials as generators of an
ASL, 54 ASL, 39
determinantal ring, 15, 55 standard representation, 39
ring, 1 of a bitableau, 138, 139
Schubert cycle, 55 of a monomial, 39
236 Subject Index

straightening symmetric ASL, 43
syzygies of
law, 38
a generic module, 166
relations, 38
an ideal generated by a poset ideal, 53
straightening-closed ideal, 108
determinantal ideals, 164, 175
powers of a, 108
syzygy module, 214
extended Rees-algebra derived from a,
108
tableau, 137
submaximal minor, 3
¬nal, 139
support of a module, 1
initial, 139
support of a monomial, 131
nested, 145
symbolic graded ring, 124
standard, 139
derived from a determinantal ideal,
torsionfree module, 213
124
torsionless module, 214
symbolic powers of determinantal ideals,
122, 123, 124, 126, 129, 130, 149
wonderful poset, 58
intersection of, 136, 137
symmetric algebra of a straightening-
Young diagram, 136
closed ideal, 111

<<

. 47
( 47 .)