Algebraic K-theory:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
Boston ; Basel ; Berlin
Birkhäuser
1996
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Progress in mathematics
90 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. 339 - 431 |
| Beschreibung: | XVI, 341 S. |
| ISBN: | 3764337028 0817637028 |
Internformat
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| 100 | 1 | |a Srinivas, V. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Algebraic K-theory |c V. Srinivas |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Boston ; Basel ; Berlin |b Birkhäuser |c 1996 | |
| 300 | |a XVI, 341 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Progress in mathematics |v 90 | |
| 500 | |a Literaturverz. S. 339 - 431 | ||
| 650 | 7 | |a Algebra homologica |2 larpcal | |
| 650 | 4 | |a K-theory | |
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Datensatz im Suchindex
| _version_ | 1804125088546553857 |
|---|---|
| adam_text | Contents
Preface to the First Edition xi
Preface to the Second Edition xvii
1. Classical /( Theory 1
Review of parts of Milnor s book: definitions of Kg, K , K2 of rings;
computation of K of a noncommutative local ring; definition of sym¬
bols; statement of Matsumoto s theorem; examples of symbols (norm
residue symbol, Galois symbol, differential symbol); presentation for
A 2 of a commutative local ring.
2. The Plus Construction 18
The plus construction; computation that n2{BGL(R)^) S K2(fl);
// space structure of BGL(R)* and products in A theory (following
Loday); statement of Quillen s theorem on K of a finite field.
3. The Classifying Space of a Small Category 31
Simplicial sets; geometric realization; classifying space of a small cate¬
gory; elementary theorems about classifying spaces (compatibility with
products, natural transformations give homotopies, adjoint functors
give homotopy inverses, filtering categories are contractible); example
of the classifying space of a discrete group as the classifying space of
the category with one object, whose endomorphisms equal the group.
4. Exact Categories and Quillen s Q Construction 38
Exact categories; admissible mono and epimorphisms; definition of
QC for a small exact category C; definition of Ki(C) for a small ex¬
act category C; statements of theorems about K, (Ko agrees with that
defined classically , theorem on exact sequences of functors, resolu¬
tion theorem, devissage theorem, localization theorem); bare hands
construction of a homomorphism Ka{C) —» Tt BQC).
viii Algebraic ff Theory
5. The /( Theory of Rings and Schemes 46
Statement of the theorem comparing the definitions of Ki of a ring using
the plus and Q constructions; definition of Gi(A) as Ki of finitely gen¬
erated /1 modules, for Noetherian rings A; computations of Gj( 4[ ]),
GiiAlt.t 1]) for Noetherian A, and hence Ki(A t]), Ki(A t,t~ ]) for
Noetherian regular A; definition of Ki(X), Gi(X) for schemes, using
vector bundles and coherent sheaves, respectively; construction of di¬
rect image and inverse image maps for Ki and Gi of Noetherian schemes
for morphisms satisfying appropriate conditions; action of Ko on Ki,
G, and projection formulas; Ki, Gi commute with filtered direct lim¬
its; localization for G, of a closed subscheme and the open complement;
Mayer Vietoris for Gc, Gi of affine and projective space bundles; fil¬
tration by codimension of support and (he BGQ spectral sequence;
Gersten s conjecture for power series rings, and semilocal rings of fi¬
nite sets of points on a smooth variety over an infinite field; Bloch s
formula; Ki of projective bundles, of P over a noncommutative ring,
and of Severi Brauer schemes.
6. Proofs of the Theorems of Chapter 4 89
Proofs of the following theorems: mBQC) S K0(C); Theorems A and B
of Quillen; the theorem on exact sequences of functors; the resolution
theorem; the devissage theorem; the localization theorem.
7. Comparison of the Plus and Q Constructions 126
Monoidal categories; localization of the action of a monoida) category
on a small category; computation of the homology of the classifying
space of a localized category; the S~ S construction, viewed as a func
torial version of the plus construction; construction of the homotopy
equivalence S~ S — QBQC for any exact category C in which all ex¬
act sequences are split, where S is the category of isomorphisms in C;
corollary that the plus and Q constructions yield the same A groups
for projective modules over a ring.
8. The Merkurjev Suslin Theorem 145
The Galois symbol; statement of the Mcrkurjev Suslin theorem;
Hilbert s Theorem 90 for Ki; proof of the Merkurjev Suslin theorem;
torsion in Ki torsion in CH2.
9. Localization for Singular Varieties 194
Quillen s localization theorem for the complement of an effective Cartier
divisor in a quasi projective scheme with affine complement; discussion
of naturality of this sequence (after Swan); proof of the Fundamental
Theorem on Ki of polynomial and Laurent polynomial rings; Levine s
localization theorem; computation of Ko of the category of modules
of finite length and finite projective dimension over the local ring of a
normal surface singularity, in terms of H (K2) of the resolution; com¬
putation of this Ko for quotient singularities; Chow groups of surfaces
with quotient singularities.
Contents ix
Appendix A. Results from Topology 230
(A.I) Compactly generated spaces; (A.2) (A.6) Homotopy groups,
Hurewicz theorems; (A.7) Products; (A.8) (A.12) CW complexes,
Whitehead theorem, Milnor s theorem on the homotopy type of map¬
ping spaces, comparison of singular and cellular homology and cohomol
ogy; (A.13) (A.15) Local coefficients, homology and cohomology with
local coefficients for CW complexes via cellular chains; (A. 16) Obstruc¬
tion theory for maps and homotopies between CW complexes (which
may not be simply connected); (A. 17) (A.22) Fibrations, the homotopy
lifting property, long exact homotopy sequence, fiber homotopy equiv¬
alence, fibrations over a contractible base are fiber homotopy equiva¬
lent to a product, local coefficient systems of the homology and coho¬
mology groups of the fibers of a fibration; (A.23) (A.26) Leray Serre
spectral sequence for homology and cohomology of a fibration over a
CW complex; (A.27) Homotopy fibers; (A.28) Spectral sequences for
the homology and cohomology of a covering space; (A.2!)) (A35) Quasi
fibrations (some results of Dold and Thoni); (A.36) (A.42) NDR pairs
and cofibrations (following Steenrod); (A.43) (A.47) // spaces; (A.48)
(A.50) Covering spaces of simplicial sets; (A.51) (A.54) Hurewicz and
Whitehead theorems for non simply connected // spaces; (A.55) Mil¬
nor s theorem on the geometric realization of a product of simplicial
sets.
Appendix B. Results from Category Theory 276
Small categories; equivalences; Abelian categories; construction of the
quotient of a small Abelian category by a Serre subcategory; examples
of quotients; adjoint functors; filtering categories and direct limits.
Appendix C. Exact Couples 287
The spectral sequence of an exact couple; bigraded couples; elementary
discussion of convergence; the BGQ spectral sequence; the spectral
sequence of a filtered complex.
Appendix D. Results from Algebraic Geometry 295
(D.1) (D.14) Sheaves; (D.15) (D.20) Schemes; (D.21) (D.41) Some
properties of schemes; (D.42) (D.59) Coherent and quasi coherent
sheaves; (D.60) (D.66) Cohomology and direct images of quasi coherent
and coherent sheaves; (D.67) (D.7O) Some miscellaneous topics.
Bibliography 339
|
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| author | Srinivas, V. |
| author_facet | Srinivas, V. |
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| callnumber-subject | QA - Mathematics |
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| dewey-full | 512/.55 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.55 |
| dewey-search | 512/.55 |
| dewey-sort | 3512 255 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV010613250 |
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| indexdate | 2024-07-09T17:56:00Z |
| institution | BVB |
| isbn | 3764337028 0817637028 |
| language | German |
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| physical | XVI, 341 S. |
| publishDate | 1996 |
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| publisher | Birkhäuser |
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| series | Progress in mathematics |
| series2 | Progress in mathematics |
| spelling | Srinivas, V. Verfasser aut Algebraic K-theory V. Srinivas 2. ed. Boston ; Basel ; Berlin Birkhäuser 1996 XVI, 341 S. txt rdacontent n rdamedia nc rdacarrier Progress in mathematics 90 Literaturverz. S. 339 - 431 Algebra homologica larpcal K-theory K-Theorie (DE-588)4033335-8 gnd rswk-swf Algebra (DE-588)4001156-2 gnd rswk-swf Algebraische K-Theorie (DE-588)4141839-6 gnd rswk-swf Algebraische K-Theorie (DE-588)4141839-6 s DE-604 K-Theorie (DE-588)4033335-8 s 1\p DE-604 Algebra (DE-588)4001156-2 s 2\p DE-604 Progress in mathematics 90 (DE-604)BV000004120 90 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007080863&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Srinivas, V. Algebraic K-theory Progress in mathematics Algebra homologica larpcal K-theory K-Theorie (DE-588)4033335-8 gnd Algebra (DE-588)4001156-2 gnd Algebraische K-Theorie (DE-588)4141839-6 gnd |
| subject_GND | (DE-588)4033335-8 (DE-588)4001156-2 (DE-588)4141839-6 |
| title | Algebraic K-theory |
| title_auth | Algebraic K-theory |
| title_exact_search | Algebraic K-theory |
| title_full | Algebraic K-theory V. Srinivas |
| title_fullStr | Algebraic K-theory V. Srinivas |
| title_full_unstemmed | Algebraic K-theory V. Srinivas |
| title_short | Algebraic K-theory |
| title_sort | algebraic k theory |
| topic | Algebra homologica larpcal K-theory K-Theorie (DE-588)4033335-8 gnd Algebra (DE-588)4001156-2 gnd Algebraische K-Theorie (DE-588)4141839-6 gnd |
| topic_facet | Algebra homologica K-theory K-Theorie Algebra Algebraische K-Theorie |
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| volume_link | (DE-604)BV000004120 |
| work_keys_str_mv | AT srinivasv algebraicktheory |