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