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Marshall M. — Spaces of Orderings and Abstract Real Spectra, Vol. 163 |
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Предметный указатель |
61
, 33
residue space 64
, 93 94
, 144 145
, 83
, , objects associated to an ordering P 12
, objects associated to a real place 50
87
, , Zariski dimension, spectral dimension, mixed dimension 138
idempotents of G 92
-module 106
19 20
92
5
168
, , , , orderings on 86
positive cone of x 101
, , 94
118
89
101
localization at S 85
stability index at 136
relative stability index at 136
, , 93
, , 90
23
111
165
54
, , objects associated to a real place 13
, 57
partial ordering on a spectral space 113
minimal elements of X 115
set of orderings of A containing T 91
set of orderings of k containing T 19
74
126
, 93
92
20
character group of a group of exponent two 8
50
15 16 50
support of x 101
boundary of Y 139
89
isometry of forms 25 26 105
Witt equivalence 29
derived form of 47 134
space of connected components 115
91
19
5
patch topology on a spectral space 112
, 176 178
, 12 128
Anisotropic form 29
Archimedian ordering 11
Artinean abstract real spectrum 121
Axioms for abstract real spectra 99
Axioms for abstract real spectra, alternate axioms 128
Axioms for abstract real spectra, Broecker's axioms 155
Axioms for spaces of orderings 21
Axioms for spaces of orderings, alternate axioms 26
Basic sets in a space of orderings 23 32—34 47—50 69
Basic sets in an abstract real spectrum 102 110—112 133—144
Basic sets in real algebraic geometry 89—91
Boolean space 10
Boolean space, connection to SAP 45
C(x) connected component of x 78
Chain length of a space of orderings 65
Chain length of an abstract real spectrum 167
Character group of a group of exponent two 8—10
Character group of a group of exponent two, closed subgroups of 9
cl(X, G) chain length 65
Compatibility of orderings and valuations 54
Connected components of a space of orderings 66 72 76 78 164 166
Connected components of a topological space 10 115
Connectivity relation 66 76
Constructible sets 89—91 110—113 134—150
Cont(X, ) 31
Cont(X, {-1, 1}) 44
D<a, b>, binary value sets 99
dim(D) the Krull dimension of a ring 139
Dimension of a form 20 24 95 105
Dimension of an abstract real spectrum, mixed 138
Dimension of an abstract real spectrum, spectral 138
Dimension of an abstract real spectrum, Zariski 138
Dimension, -dimension of a group of exponent two 8
Dimension, Krull dimension of a ring 138
Direct sum of spaces of orderings 61 63 179—181
Direct sum of spaces of orderings, direct summand 179—181
Discriminant of a form 20 24 95 105
E(X, x) all non-fans containing x 78
Fan 37 39
Fan in an abstract real spectrum 162
Fan, trivial fan 39
Finite spaces of orderings 39 46 65—66
Finiteness theorem 91
Form notations: , , , , , 105
Form notations: , , , , 24
Form operations: , , , 24 105
Formally real field 5
G* unit group of G 92
gr(X, G) translation group 64
Group extension 62
H real holomorphy ring of k 50
| Harrison (or spectral) topology 110
Hilbert's 17th problem 7
Idempotents in an abstract real spectrum 92 101
Isometry of forms 25—26 95 105
Isotropic form 29
k( ) residue field at 83
k((t)), power series fields 16
k* = k\{0} 5
Localization in an abstract real spectrum 119—121
Localization in an abstract real spectrum, at a prime or a finite set of primes 120 152—155
M* maximal elements in M 170
Many units property for a ring 153
Many units property for an abstract real spectrum 152
Minimal generating set 39
Morphism of abstract real spectra 103
Morphism of spaces of orderings 23 165 173
Morphism, preserving P-structure 172
Noetherian abstract real spectrum 119
Operations on forms 24 105
Ordered field 5
Ordering in a space of orderings 22
Ordering in an abstract real spectrum 101
Ordering on a field 5
Ordering on a ring 83
Ordering, maximal, minimal 115
Ordering, positive cone of 22 101
Ordering, support of 83 101
p(s), t(s) 148 149
P-structure 52—53 168—172
P-structure, existence of 53 79
P-structure, Hausdorff 52 168
P-structure, relationship to chain length 65 67 80 170
P-structure, relationship to connected components 76 79
P-structure, relationship to stability index 51—53
Pfister form 24 105
Pfister form, derived form of 47 134
Pfister form, relationship to subspaces and preorderings 34—35 121—124
Place 11
Power series field 16
Preordering in a field 5
Preordering in a ring 84
Preordering in a space of orderings 33
Preordering in an abstract real spectrum 121
Prime ideal 83
Prime ideal in an abstract real spectrum 117
Prime ideal, real 84
Prime ideal, T-compatible 93
Pullback of an ordering 54
Pushdown of an ordering 54
Radical of an ideal, classical 94
Radical of an ideal, Q-radical 94
Radical of an ideal, T-radical 94 118
Real closure of a field 6
Real closure of a ring 178
Real closure of an abstract real spectrum 177
Real holomorphy ring 50 87—88
Represented elements of a form 20 22 24 95 105
Residue field at a prime ideal of a ring 83
Residue field of a valuation ring 11
Residue forms 62
Residue space 64
Residue space at a prime ideal 102 126
Residue space of a space of orderings 64
Ring of continuous functions into 50—51 87 144 178
Ring of continuous functions into 31 40—41 44 47 161
Sat(Y) saturation of Y 130
Saturated set 126
Semi-algebraic sets 89—91 136—137 141
Separating ideal 163
Separation of disjoint closed sets 44 142
Signature of a form 20 24 95 105
Singleton space 61
Spec(A) prime spectrum 83
Specialization 113—116 163—164
Specialization and real places 116 169
Spectral space 112—113
Spectral space, completely normal 114
Spectrum of a ring, prime (or Zariski) 83
Spectrum of a ring, real 84
Sper(A) real spectrum 84
stab(X, G) stability index 47
Stability index 47 136
Stability index for a finitely generated extension of an ordered field 57
Stability index, local 49 137
Stability index, relative 49 136
Strong approximation property 44—46
Support of an abstract real spectrum 101
Support of an ordering 83 101
T*=T\{0} 19
Tarski's transfer principle 7
Translation group 64
trdeg(F : k) 8
Tychonoff (or patch) topology 111
U(a), 23
U(a), Z(a), W(a) 102 103
U(f;e) 146
Units in an abstract real spectrum 92 101
Universal form 29
Valuation 13
Valuation ring 11
Value set of a form 20 22 24 95 99 105
Value set of a form, transversal value set 96 99 105
W=W(X, G) the Witt ring 29
Witt, cancellation 27
Witt, equivalence 29
Witt, ring 29
X* maximal elements of X 115
z-cl(Y) Zariski closure 118
~ connectivity relation 66
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