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Knus M.-A. — Quadratic and hermitian forms over rings
Knus M.-A. — Quadratic and hermitian forms over rings

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Название: Quadratic and hermitian forms over rings

Автор: Knus M.-A.

Аннотация:

This book presents the theory of quadratic and hermitian forms over rings in a very general setting. It avoids, as far as possible, any restriction on the characteristic and takes full advantage of the functorial properties of the theory. It is not an encyclopedic survey. It stresses the algebraic aspects of the theory and avoids - within reason overlapping with other books on quadratic forms (like those of Lam, Milnor-Husemöller and Scharlau). One important tool is descent theory with the corresponding cohomological machinery. It is used to define the classical invariants of quadratic forms, but also for the study of Azmaya algebras, which are fundamental in the theory of Clifford algebras. Clifford algebras are applied, in particular, to treat in detail quadratic forms of low rank and their spinor groups. Another important tool is algebraic K-theory, which plays the role that linear algebra plays in the case of forms over fields. The book contains complete proofs of the stability, cancellation and splitting theorems in the linear and in the unitary case. These results are applied to polynomial rings to give quadratic analogues of the theorem of Quillen and Suslin on projective modules. Another, more geometric, application is to Witt groups of regular rings and Witt groups of real curves and surfaces.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1991

Количество страниц: 524

Добавлена в каталог: 09.02.2006

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$(\varepsilon,\Lambda)$-unitary module      24
$(\varepsilon,\Lambda)$-unitary space      24 78
$\alpha$-derivation      392
$\varepsilon$-Hermitian form      8 76
$\varepsilon$-hermitian matrix      10
$\varepsilon$-hermitian module      11 76
$\varepsilon$-hermitian space      11 76
$\varepsilon$-hyperbolic space      14
$\varepsilon$-involution      170
2-torsion datum      180
A-filtration      456
A-lattice      456
Additive category      72
Additive functor      74
Adjacent hyperbolic planes      32
Adjoint Hermitian form      56
Adjoint involution      54
Adjoint of a form      7
Admits      44 47
Affine k-algebra      481
Affine line      421
Affine space      455
Albert form      192
Alternating bilinear form      21 398
Alternating element      170 180 186
Alternating matrix      21
Amitsur 1-cocycle      111
Amitsur 1-cohomology set      111
Analytic isomorphism      380
Anisotropic hermitian matrix      401
Anisotropic hermitian module      17
Anisotropic unitary space      27 104
Arf invariant      211 213
Artin — Schreier sequence      132
Axiom Q      416
Azumaya algebra      42 107 134 471
Bimodule      47
Binary form      311
Brauer equivalent algebras      146 156
Brauer group      146 471
Brauer — Wall group      156
Cancellation theorem      333 376
Canonical involution      195
Cartan formula      463
Cartan — Dieudonne — Kneser theorem      378
Cartesian diagram of categories      65
Cartesian diagram of rings      64
Category modulo an ideal      88
Category with duality      76
Cayley algebra      307
Cayley — Hamilton theorem      108
Cech cohomology groups      116 117 124
Centre      1
Checker-board gradation      151
Chern class      463
Classical invariants      490
Clifford algebra      193
Clifford group      228
Clifford invariant      242
Codimension      328
Coherent sheaf      454
Cohomologous cocycles      111 112
Combinatorial dimension      325
Commutative separable algebra      42
Complete ring      89
Completion      89
Composable quadratic module      309
composition      308
Composition algebra      305
Congruent hermitian matrices      401
Congruent matrices      10
Corestriction      265
Cup-product      148
Degree      392
Descended module      110 115
Descent-datum      110
Diagonalizable hermitian module      30
Dickson map      224
Direct product      73
Direct sum      72
Discriminant      12
Discriminant algebra      211 221
Discriminant bundle      470
Discriminant ideal      12
Discriminant module      124
Dominant diagonal      400
Double division ring      36
Dual lattice      456
Duality functor      75
Duality preserving functor      80
Elementary automorphism      333
Elementary isometry      345
Elementary matrix      315
Equivalent algebras      278
Essentially of finite type      481
Etale R-algebra      117
Etale splitting      180
Etale topology      117
Euclidian ring      393
Even $\varepsilon$-hermitian form      8 77
Even $\varepsilon$-hermitian module      11 77
Even $\varepsilon$-hermitian space      11 77
Even involution      173
Exact diagram      109
Extended hermitian module      403
Extended hermitian space      403
Extended unitary space      409
F-rank      322
Faithfully projective module      52
Fibre product      75
Fibre product category      65
Finer topology      116
Finite etale algebra      42
Finite R-algebra      137 326
Form parameter      23 77
fppf-topology      117
Frobenius algebra      435
Galois 1-cochain      112
Galois 1-cocycle      112
Galois 1-cohomology set      113
Galois algebra      107
Galois cohomologous      112
Galois descent-datum      115
Generic alternating matrix      177
Generic quadratic form      207
Global dimension      459
Graded algebra      149
Graded Arf invariant      156 213
Graded Azumaya algebra      152
Graded central algebra      152
Graded centre      152
Graded commutant      152
Graded discriminant module      151
Graded inner automorphism      158
Graded invertible module      150
Graded module      149
Graded separable algebra      149
Graded signed discriminant      213
Graded simple algebra      152
Graded tensor product      149
Group of elementary isometries      345
Group of quadratic algebras      130
Half-discriminant      208
Half-ideal      208
Hermitian bundle      455
Hermitian coherent sheaf      455
Hermitian discriminant      302
Hermitian discriminant module      301
Hermitian form      8
Hermitian matrix      401
Highest common left factor      391
Hilbert syzygy theorem      459
Homogeneous element      149
Homological dimension      431 459
Hopf bundle      498
Horrocks theorem      422 424
Hyperbolic pair      19 27
Hyperbolic ring      3
Hyperbolic space      14 78 469
Hyperbolic unitary space      26 78
I-adic topology      89
Ideal      85
Idempotent      83
Indecomposable object      93
Inner automorphism      144
Invariant ideal      90
Involution      1
Involution of orthogonal type      170
Involution of symplectic type      170
Involution of the first kind      35 169
Involution of the second kind      35 169
Involution trace      40
Irreducible component      325
Irreducible subset      325
Isometric sesquilinear forms      7
Isometry      7 77 468
Isometry of unitary spaces      24
Isometry, up to discriminant      267 268
Isotropic element      17
Isotropic hermitian module      17
Isotropic unitary space      27
Krull dimension      325
Krull — Schmidt theorem      93 98
Lagrangian      19 468
Least common right multiple      391
Left divisor      391
Left root      393
Length      375 391
Length of a resolution      431
Line bundle      470
Local ring      92
Local Witt index      425
Locally free module      454
Locally homogeneous element      149
Mapping cone      432
Metabolic pair      206
Metabolic space      19 468
Metabolic unitary space      26
Module involution      174
Morita equivalence      53
Morphism of forms      7
Morphism of unitary modules      24
Multiplicative norm      3
Multiplier      176 236
Nakayama's lemma      86
Negative elements      490
Noetherian topological subspace      325
Nondegenerate form      11
Nonsingular form      11
Norm      3 163 228
Norm form      161
Norm functor      162
Normal ring      440
Normalized cocycle      167
Normalized involution      367
Opposite module      6
Opposite ring      2
Ordering      490
Orthogonal      468
Orthogonal basis      30
Orthogonal group      8 223
Orthogonal subsets      16
Orthogonal sum      14 25 468
Orthogonal type      170
Patching diagram      64 66
Permutation matrix      315
Pfaffian      179
Pfaffian, adjoint      186
Pfaffian, characteristic polynomial      191
Pfaffian, discriminant      187
Pfaffian, invariant      183
Pfaffian, trace      191
Picard group      122 470
Pioncare — Birkhoff — Witt Theorem      196
Polar of a form      25
Positive definite hermitian space      101
Positive elements      490
Positively graded module      457
Primitive quadratic module      164
Principal ideal domain      390
Product of forms      48
Projection      73
Projective line      75
Projective linear group      145
Projective orthogonal group      177
Projective space      454
Projective symplectic group      176
Proper composition      308
Q-functor      416
Quadratic algebra      4 42
Quadratic form      25
Quadratic module      25
Quadratic space      25
Quaternion algebra      4
Radical      86
Radical of a category      88
Rationally hyperbolic space      481
Real topology      491
Reduced characteristic polynomial      107
Reduced norm      2 42 107
Reduced trace      2 42
Reduction functor      88
Reduction Theorem      91
Reflection      228 364
Reflexive module      11 459
Regular form      11
Regular ring      431
Related hyperbolic planes      32
Residue class form homomorphism      473
Right associate      391
Right divisor      391
Right regular ring      431
Right root      393
Rigidity theorem      420
Same length      348
Scalar extension      37
Scaling      27
Selfdual lattice      456
Semilocal ring      100 312
Semiregular quadratic module      208
Semiregular split module      209
Separability idempotent      136
Separable algebra      41 42 135
Separable field extension      41
Sesquilinear form      7 76
Sesquilinear module      76
Sesquilinear space      76
Sheaf      118
Short exact sequence of sheaves      118
Signature      490
Signed discriminant      213 471
Signed discriminant module      213
Similar elements      391
Similitude      176 236
Simple ring-with-involution      3
Singular maximal ideal      476
Skew polynomial ring      392
Skew-Hermitian form      8
Skew-symmetric form      8
Skew-symmetric form module      11
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