Marshall M. — Spaces of Orderings and Abstract Real Spectra, Vol. 163 :: Электронная библиотека попечительского совета мехмата МГУ

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Название: Spaces of Orderings and Abstract Real Spectra, Vol. 163

Автор: Marshall M.

Аннотация:

This book is of interest to students as well as experts in the area of real algebraic geometry, quadratic forms, orderings, valuations, lattice ordered groups and rings, and in model theory. The original motivation comes from orderings on fields and commutative rings. This is explained as is the important application to minimal generation of semi-algebraic sets. Many results in the new theory of abstract real spectra (also called spaces of signs) appear here for the first time. The reader needs elementary knowledge of commutative rings, ordered fields and real closed fields and valuations.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1996

Количество страниц: 204

Добавлена в каталог: 04.07.2008

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Предметный указатель

$(X_{1}, G_{1})\oplus...\oplus(X_{n}, G_{n})$       61
$(Y, G|_{Y})$ , $\phi|_{Y}$       33
$(\tilde{X}, \tilde{G})$ residue space      64
$A[\frac{1}{a}]$ , $T[\frac{1}{a^{2}}]$       93 94
$a^{+}$ , $a^{-}$       144 145
$A^{2}$ , $\sum A^{2}$       83
$B_{P}$ , $m_{P}$ , $\lambda(P)$ objects associated to an ordering P      12
$B_{\alpha}=\alpha^{-1}(\mathbb{R})$ , $m_{\alpha}=\alpha^{-1}(0)$ objects associated to a real place $\alpha$       50
$Cont(Y, \mathbb{R})$       87
$d_{1}(Y)$ , $d_{0}(Y)$ , Zariski dimension, spectral dimension, mixed dimension      138
$G^{2}$ idempotents of G      92
$G^{2}$ -module      106
$G_{T}=k^{*}/T^{*}\cong\{\bar{a}|a\in k^{*}\}$       19 20
$G_{T}={\bar{a}|a\in A}$       92
$k^{2}=\{a^{2}|a\in k \}$       5
$M_{\mathfrak{p}}=\lambda(X_{\mathfrak{p}})$       168
$P_{a}$ , $P^{+}_{a}$ , $P^{-}_{a}$ , $P_{\infty +}$ , $P_{\infty -}$ orderings on $\mathbb{R}[t]$       86
$P_{x}$ positive cone of x      101
$rad(\mathfrak{a})$ , $rad_{Q}(\mathfrak{a})$ , $rad_{T}(\mathfrak{a})$       94
$rad_{T}(\mathfrak{a})$       118
$Sper_{Q}(A)$       89
$Supp(X)=\{\mathfrak{p}_{x}|x\in X\}$       101
$S^{-1} A$ localization at S      85
$s_{\mathfrak{p}}$ stability index at $\mathfrak{p}$       136
$s_{\mathfrak{p}}(Y_{\mathfrak{p}})$ relative stability index at $\mathfrak{p}$       136
$T/\mathfrak{a}$ , $S^{-2}T$ , $T(\mathfrak{p})$       93
$U(\bar{f})$ , $Z(\bar{f})$ , $W(\bar{f})$       90
$u:X\hookrightarrow\chi(G)$       23
$u:X\hookrightarrow\{-1, 0, 1\}^{G}$       111
$u:X_{\mathfrak{pi}}\rightarrow X_{\mathfrak{q}}, \mathfrak{q}=\mathfrak{q}(\mathfrak{p}, \mathfrak{i})$       165
$U^{+}=U^{+}(Q)$       54
$U_{\alpha}$ , $U_{\alpha}^{+}$ , $v_{\alpha}$ objects associated to a real place $\alpha$       13
$v_{Q}$ , $\delta_{Q}$       57
$x\leq y$ partial ordering on a spectral space      113
$X_{*}$ minimal elements of X      115
$X_{T}$ set of orderings of A containing T      91
$X_{T}$ set of orderings of k containing T      19
$X_{t}, t\in\chi(G)$       74
$Y_{\mathfrak{p}}=Y\cap X_{\mathfrak{p}}$       126
$\alpha^{*}$ , $\alpha_{*}$       93
$\bar{a}=\bar{a}_{T}:X_{T}\rightarrow\{-1, 0, 1\}$       92
$\bar{a}=\bar{a}_{T}:X_{T}\rightarrow\{-1, 1\}$       20
$\chi(G)=Hom(G, \{-1, 1\})$ character group of a group of exponent two      8
$\hat{}:H\rightarrow Cont(M, \mathbb{R})$       50
$\lambda: X\rightarrow M$       15 16 50
$\mathfrak{p}_{x}$ support of x      101
$\partial Y$ boundary of Y      139
$\Phi:R^{n}\rightarrow Sper_{Q}(k[t_{1}, ... , t_{n}])$       89
$\phi\cong\psi$ isometry of forms      25 26 105
$\phi\thicksim\psi$ Witt equivalence      29
$\phi^{'}$ derived form of $\phi$       47 134
$\pi_{0}(X)$ space of connected components      115
$\sum A^{2}Q$       91
$\sum F^{2}Q$       19
$\sum k^{2}$       5
$\tau$ patch topology on a spectral space      112
$\tilde{G}$ , $\tilde{A}$       176 178
$\tilde{\phi}$ , $\tilde{\phi}_{\mathfrak{p}}$       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, $\mathbb{Z}$ )      31
Cont(X, {-1, 1})      44
D<a, b>, $D^{t}<a, b>$ 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, $\mathbb{Z}_2$ -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 $X_{t}, t\neq1$ 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: $dim(\phi)$ , $disc(\phi)$ , $\phi(x)$ , $D(\phi)$ , $D^{t}(\phi)$ , $\phi=<a_{1}, ... , a_{n}>$       105
Form notations: $dim(\phi)$ , $disc(\phi)$ , $\phi(x)$ , $D(\phi)$ , $\phi=<a_{1}, ... , a_{n}>$       24
Form operations: $\phi\oplus\psi$ , $c\phi$ , $\phi\otimes\psi$ , $k\times\phi$       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( $\mathfrak{p}$ ) residue field at $\mathfrak{p}$       83
k((t)), $k((\Gamma))$ 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 $\mathbb{R}$       50—51 87 144 178
Ring of continuous functions into $\mathbb{Z}$       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), $U(a_{1}, ..., a_{n})$       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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