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Rosenberg J. — Algebraic K-Theory and Its Applications
Rosenberg J. — Algebraic K-Theory and Its Applications



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Название: Algebraic K-Theory and Its Applications

Автор: Rosenberg J.

Аннотация:

Algebraic K-theory plays an important role in many areas of modern mathematics: most notably algebraic topology, number theory, and algebraic geometry, but even including operator theory. The broad range of these topics has tended to give the subject an aura of inapproachability. This book, based on a course at the University of Maryland in the fall of 1990, is intended to enable graduate students or mathematicians working in other areas not only to learn the basics of algebraic K-theory, but also to get a feel for its many applications. The required prerequisites are only the standard one-year graduate algebra course and the standard introductory graduate course on algebraic and geometric topology. Many topics from algebraic topology, homological algebra, and algebraic number theory are developed as needed. The final chapter gives a concise introduction to cyclic homology and its interrelationship with K-theory.


Язык: en

Рубрика: Математика/Геометрия и топология/Алгебраическая и дифференциальная топология/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1994

Количество страниц: 392

Добавлена в каталог: 16.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$K_0(R)$      1.1.5
$K_0(R)$, behavior under Cartesian product      1.2.8
$K_0(R)$, continuity of      1.2.5
$K_0(R)$, Morita invariance of      1.2.4
$K_0(R)$, of division ring      1—1.6
$K_0(R)$, of local ring      1.3.11
$K_0(R)$, of PID      13.1
$K_0(R)$, ring structure on      1.1—9
$K_1(R)$      2.1.5
$K_1(R)$, behavior under Cartesian product      2.1.6
$K_1(R)$, continuity of      2.1.9
$K_1(R)$, Morita invariance of      2.1.8
$K_1(R)$, of a field      2.2.1
$K_1(R)$, of Dedekind domain      2.3.5
$K_1(R)$, of division ring      2.2.6
$K_1(R)$, of Euclidean domain      2.3.2
$K_1(R)$, of local ring      2.2.6
$K_2(R)$      4.2.2 4.2.10
$K_2(R)$, behavior under Cartesian product      4.3.18
$K_2(R)$, Morita invariance of      4.2.23
$K_2(R)$, of $\mathbb Q$      4.4.9
$K_2(R)$, of $\mathbb Z$      4.3.20
$K_2(R)$, of $\mathbb Z/(m)$      4.3.19
$K_2(R)$, of (commutative) field      4.3.15
$K_2(R)$, of finite field      4.3.13 4.3.14
$K_2(R)$, of rings of functions      4.4.19
$K_i(R)$      5.2.6
$K_i(R)$, of $\mathbb R$ and $\mathbb C$      5.3.9
$K_i(R)$, of $\mathbb Z$      5.3.10—14
$K_i(R)$, of algebraically closed field      5.3.8
$K_i(R)$, of division algebra      5.3.40
$K_i(R)$, of finite field      5.3.2
$K_i(R)$, of purely transcendental extension field      5.3.36 5.3.37
$NK_0(R)$      3.2.14
$NK_0(R)$ and $K_0$ of Laurent polynomial rings      3.3.2
$NK_0(R)$, non-triviality of      3.2.24
$NK_1(R)$      3.2.14
$NK_1(R)$ and $K_1$ of Laurent polynomial rings      3.2.22
$NK_1(R)$ and Nil R      3.2.22
$NK_1(R)$, non-finite generation of      3.2.29
$NK_1(R)$, non-triviality of      3.2.23
Abelian category      remarks preceding 3.1.1
Acyclic complex      1.7.1
Acyclic space      5.3.31
Adams operation      5.3.2
Additive category      remarks preceding 3.1.1
Alexander trick      remarks following 2.4.5
Almost-commuting algebra      4.4.20
Alternating group      4.1.27 5.2.19
Assembly map in cyclic homology      6.2.11
Assembly map in Hochschild homology      6.2.11
Assembly map in K-theory      6.3.18 6.3.21
Augmented algebra      6.1.27(3) 6.1.40
Baer sum      4.1.16
Banach algebra      remarks following 1.6.5 3.3.8
Bass — Heller — Swan Theorem      3.2.20 3.2.22
Bass — Heller — Swan Theorem, analogue in cyclic homology      6.1.49
Bass — Heller — Swan Theorem, analogue in Hochschild homology      6.1.48
Beilinson conjectures      remarks following 5.3.14
Bernoulli numbers      remarks following 5.3.14
Bloch's formula      remarks following 5.3.30
Block sum      1.2.3 2.1.5 5.2.12
Borel's Theorem on K-groups of rings of integers      5.3.13
Bott Periodicity Theorem      1.6.13 3.3.8(2)
Boundaries      1.7.1
Bounded h-cobordism      remarks following 3.3.5
Brauer group      4.4.13
Brauer group and Galois cohomology      4.4.14
Brauer lift      5.3.4
Brauer's Theorem on Characters      5.3.4
Brown — Douglas — Fillmore invariant      4.4.24
C*-algebra      3.3.8(4) 6.3.8
C*-algebra of a group      6.3.9
Campbell — Baker — Hausdorff formula      2.2.10 4.4.30(4)
Category with exact sequences      3.1.1
Cayley graph      6.3.15
Cellular approximation theorem      remarks following 2.4.9 5.1.12
Cellular homology      remarks following 5.1.13
Central extension      4.1.1
Central extension of Lie groups      4.1.5
Central simple algebra      4.4.13
Chain complex      1.7.1
Chain homotopy      1.7.2
Chapman's Theorem      remarks following 2.4.9
Chern character for $K_0$      6.2.7
Chern character for Chow ring      remarks following 5.3.30
Chern character for higher K-groups      6.2.14
Chern character for lower K-groups      6.2.24
Chern character relative      6.2.22
Chern character with coefficients      6.3.26
Chern character, classical      beginning of 6.2
Chinese remainder theorem      1.3.14 1.4.22 2.3.5 4.3.19
Chow ring      remarks following 5.3.30
Class group      1.4.3
Class group of cubic number field      6.3.7
Class group of cyclotomic number field      3.3.5(b)
Class group of quadratic number field      1.4.24 6.3.6
Class group, finiteness theorem for      1.4.19
Classifying space, for topological K-theory      5.3.2 5.3.9
Classifying space, of a group      5.1.16
Classifying space, of a small category      5.3.15
Classifying space, universal property of      5.1.32
Clutching      2.5.4
Coherent sheaf      3.1.2(7) remarks
Cohomology, cyclic      6.1.34
Cohomology, de Rham      beginning of 6.2
Cohomology, exceptional (or generalized)      beginning of 1.6 6.3.19
Cohomology, group      4.1.7
Cohomology, Hochschild      6.1.33
Cohomology, sheaf      remarks following 3.1.16
Compact operators      2.2.10 6.3.4
Compact supports, Jf-theory with      1.6.14
Congruence Subgroup Problem      2.5.21
Connes periodicity operator      6.1.18 6.1.34
Contractible      1.7.2
Corestriction      4.1.21
Covering map      5.1.5 5.1.25
Crossed product      4.4.14
CW-complex      remarks preceding 2.4.6 5.1.7
Cycles      1.7.1
Cyclic homology      6.1.12
Cyclic homology, behavior under Cartesian product      6.1.44
Cyclic homology, calculation for $k[v, v^{-1}]$      6.1.43 6.1.49
Cyclic homology, calculation for $\mathbb Z[\sqrt{D}]$      6.2.10(b)
Cyclic homology, calculation for dual numbers      6.1.27(b)
Cyclic homology, calculation for ground ring      6.1.16
Cyclic homology, calculation for group ring      6.2.11
Cyclic homology, calculation for k[t]      6.1.20(b) 6.1.42
Cyclic homology, Morita invariance of      6.1.37
Dedekind domain      1.4.2
Dedekind domain, $K_0$ of      1.4.12
Dedekind domain, $K_1$ of      2.3.5
Dedekind domain, characterization of      1.4.17
Dedekind domain, class group of      1.4.3
Dedekind domain, examples of      1.4.18 1.4.20 1.4.23 1.4.24
Dennis trace map      6.2.14
Determinant, class (of operators)      2.2.10(4)
Determinant, Dieudonne      2.2.5
Determinant, of an operator      2.2.10
Devissage Theorem      3.1.8 5.3.24
Dirichlet Theorem on primes in arithmetic progressions      2.5.17 4.4.1
Dirichlet unit theorem      2.3.8
Divisor (in algebraic geometry)      remarks following 3.1.16 3.1.25
Dolbeault's theorem      remarks following 3.1.18
Double complex      1.7.17 6.1.10 6.1.31
Dual numbers      6.1.7(b)
Elementary collapse      2.4.7 2.4.8
Elementary matrix      2.1.1
Elementary row and column operations      2.1.5 2.2.5
End (of a space)      1.7.14
Euclidean Domain      2.3.1
Euler characteristic      1.7.9 remarks
Euler-Poincare Principle      1.7.10 3.1.10
Exact sequence of Connes      6.1.19 6.1.34
Exact sequence of K-groups      1.5.5 2.5.4 3.3.4 4.3.1
Exact sequence, localization      5.3.27 5.3.28
Excision, failure for $K_1$      2.5.20
Excision, Theorem for $K_0$      1.5.9
Factorization into prime ideals      1.4.7
Fibration      5.1.19
Fibration, fiber of      5.1.23
Fibration, Hopf      5.1.26
Fibration, local nature of      5.1.22
Fibration, long exact homotopy sequence of      5.1.24
Finitely dominated space      beginning of 1.7 remarks
Flat ring extension      remarks following 3.2.6 5.3.23
Fractional ideal      1.4.1
Free group      5.1.17 remarks
Fundamental Theorem of Arithmetic      remarks preceding 1.4.7 4.4.9
Fundamental Theorem of homological algebra      1.7.6
Fundamental Theorem of K-theory (see also Bass — Heller — Swan Theorem)      3.3.3 5.3.30
G-groups      3.1.6 3.1.16 5.3.23 5.3.26
G-module      4.1.6
Galois field extension      1.4.18 2.3.8 4.4.6 4.4.14—4.4.18
Genus (of a curve)      remarks following 3.1.16
Grothendieck group      1.1.3
Grothendieck's Theorem on $K_0$ of polynomial extensions      3.2.12 3.2.13
Grothendieck's Theorem on $K_1$ of polynomial extensions      3.2.16 3.2.17
Grothendieck's Theorem on the map from $G_i$ to $R_i$      3.1.16
Group completion      1.1.3
h-cobordism      2.4.4
H-group      5.1.11
H-space      5.1.11
Halo      5.1.1
Helton — Howe determinant invariant      4.4.23 4.4.24
Helton — Howe trace invariant      6.3.29
Hilbert basis theorem      3.2.1
Hilbert symbol      4.3.3
Hilbert syzygy theorem      3.2.3
Hilbert's Nullstellensatz      5.3.34(1)
Hilbert's Theorem 90      4.4.16
Hochschild complex      6.1.1
Hochschild complex, reduced      6.1.38
Hochschild homology      6.1.1
Hochschild homology, behavior under Cartesian product      6.1.44
Hochschild homology, calculation for $k[v, v^{-1}]$      6.1.43 6.1.48
Hochschild homology, calculation for dual numbers      6.1.7(a)
Hochschild homology, calculation for group ring      6.1.45
Hochschild homology, calculation for k[t]      6.1.7(a)
Hochschild homology, Morita invariance of      6.1.36
Homology, and direct and inverse limits      4.1.29
Homology, calculation for a finite cyclic group      4.1.25
Homology, calculation for a free abelian group      4.1.31
Homology, cyclic      6.1.12
Homology, generalized      remarks following 6.3.19
Homology, Hochschild      6.1.1
Homology, non-commutative de Rham      6.1.39
Homology, of a chain complex      1.7.1
Homology, of a group      4.1.7
Homotopy Extension Theorem      5.1.8
Homotopy group      5.1.6
Homotopy invariance      1.6.11 remarks
Hurewicz Theorem      5.1.14
Hurewicz Theorem with coefficients      5.3.7
Icosahedral group      4.1.27 5.2.16
Idempotent      beginning of 1.2
Inflation-restriction sequence      4.1.20
Jacobson radical      1.3.6
Join of spaces      5.1.15
Jordan-Holder Theorem      3.1.2(5)
Kadison Conjecture      6.3.17
Karoubi Density Theorem      1.6.16
Laurent polynomial ring      beginning of 3.2
Laurent power series ring      3.3.8(3)
Laurent power series ring in commutative algebra      3.2.2 3.2.4
Laurent power series ring, cyclic homology of      6.1.43 6.1.49
Laurent power series ring, Hochschild homology of      6.1.43 6.1.48
Laurent power series ring, K-theory of      3.2.12 3.3.3 5.3.30
Left regular ring      3.1.2(4)
Left regular ring, Grothendieck's Theorem on K-theory of      3.1.16
Lichtenbaum conjecture      remarks following 5.3.14
Local field      4.4.6
Local ring      1.3.3
Local ring, $K_0$ of      1.3.11
Local ring, $K_1$ of      2.2.6
Local ring, characterization of      1.3.4
Localization theorem      5.3.27 5.3.30
Loday product      5.3.1 6.3.22
Logarithmic derivative      6.1.43 6.2.16 6.2.18 6.3.4
Loop space      5.1.14
Mapping cone      1.7.5
Mapping cylinder      remarks following 2.4.9 5.1.13
Matsumoto's Theorem      4.3.15
Mennicke symbol      2.3.6
Mennicke's Theorem on relative $SK_1$      2.5.17 4.4.1
Mercurjev — Suslin Theorem      remarks following 4.4.18
Mittag — Leffler condition      1.7.14 6.1.22
Mixed complex      6.1.28
Morita invariance      1.2.4 2.1.8 4.2.23 6.1.36 6.1.37
Multitrace      6.1.34
n-connected      5.1.6
n-regular ring      3.2.26
Nakayama's lemma      1.3.9 1.3.13
Negative K-groups      3.3.1
Negative K-groups of $(\mathbb Z, (m))$      3.3.10
Negative K-groups of a Banach algebra      3.3.8
Negative K-groups of group rings      3.3.10
Non-commutative de Rham homology      6.1.39
Non-commutative de Rham homology and cyclic homology      6.1.40
Non-commutative differential forms      6.1.38
Norm for a field extension      1.4.19 1.4.24(3) 2.3.8 4.4.4—4.4.9 4.4.16—4.4.18 6.3.6—6.3.7
Norm in a Euclidean domain      2.3.1
Norm of an ideal      1.4.19
Norm of an operator      2.2.10
Norm residue symbol      4.4.18 4.4.28 4.4.29
Norm, Schatten      2.2.10
Orthogonal group      1.6.13 5.1.17
Orthogonal projection      6.3.8
Paracompact      5.1.4 5.1.22
Partition of unity      1.6.3 5.1.4
Path space      5.1.20
Perfect group      2.3.9(2) 4.1.3 5.2.2
Periodic cyclic homology      6.1.12
Periodicity operator      6.1.18 6.1.34
PID (principal ideal domain)      1.3.1 remarks 3 3.1.20
Plus-construction (+-construction)      5.2.2
Plus-construction (+-construction), functoriality of      5.2.4
Poincare conjecture      remarks following 2.4.5 5.2.5
Poincare homology 3-sphere      4.1.27 5.2.5 5.2.16
Projection (self-adjoint)      6.3.8
Projective module      1.1.1
Projective module, characterization of      1.1.2
Projective variety      3.1.2(7) remarks remarks
Property T      remarks following 2.3.8
Pseudo-isotopy      remarks following 4.4.25
Pull-back (of vector bundle)      1.6.2
Q-construction      5.3.19
Quadratic reciprocity law      4.4.10
Quaternion algebra symbol      4.4.28
Quaternions      2.2.9 4.4.11
Quillen's Theorem, comparing the +-construction and Q-construction      5.3.20
Quillen's Theorem, on finite generation of K-groups      5.3.12
Quillen's Theorem, on if-theory of finite fields      5.3.2
Quillen's Theorem, on localization      5.3.27
Radical (of a ring)      1.3.6
Ramified field extension      4.4.6 4.4.9
Ramified prime ideal      1.4.24
Rank (of a module)      1.3.1 1.3.11 1.3.12
Reduced C*-algebra      6.3.9
Reduced cyclic complex      6.1.40
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