Algebraic K-theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
2008
|
| Edition: | 2. ed., reprint of the 1995 2. ed. |
| Series: | Modern Birkhäuser classics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Originally publ. as volume 90 in the series 'Progress in mathematics' |
| Physical Description: | XVI, 341 S. |
| ISBN: | 9780817647360 |
Staff View
MARC
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| 245 | 1 | 0 | |a Algebraic K-theory |c V. Srinivas |
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| 490 | 0 | |a Modern Birkhäuser classics | |
| 500 | |a Originally publ. as volume 90 in the series 'Progress in mathematics' | ||
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Record in the Search Index
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| adam_text | Contents
Preface
to the First Edition
............................... xi
Preface to the Second Edition
............................. xvii
1.
Classical
К-ТЪеоту
..................................... 1
Review of parts of Milnor s book: definitions of
Ко,
Κι,
Кг
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
Ki of a commutative local ring.
2.
The Plus Construction
.................................... 18
The plus construction; computation that m(BGL(R)+)
S K%(R);
Я
-space structure of BGL(R)+ and products in
Я
-theory {following
Loday); statement of QuiHen s theorem on Ki of a finite field.
3.
The Classifying Space of a Small Gategory
................... 31
Simplicia!
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
Quilîen s Q-Construcfcion
............... 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
К, {Ко
agrees with that
defined classically , theorem on exact sequences of functors, resolu¬
tion theorem,
dévissage
theorem, localization theorem); bare hands
construction of a homomorphism Ko{C)
—*
viii
Algebraic
/f-Theory
5.
The ií-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 K, of finitely gen¬
erated ^-modules, for Noetherian rings A; computations of Gi{A[t]),
Gi(A[i,i ł])
for Noetherian A, and hence
КААЩ),
Кѓ{АЏ,Г1])
for
Noetherian regular A; definition of Kt(X),
G
AX) for schemes, using
vector bundles and coherent sheaves, respectively; construction of di¬
rect image and inverse image maps for K, and G, of Noetherian schemes
for morphisms satisfying appropriate conditions; action of
Ko on
Ki,
Gì
and projection formulas; Ki, Gi commute with filtered direct lim¬
its; localization for
G
і
of a closed subscheme and the open complement;
Mayer-Vietoris for G%
G¡
of
affine
and
projective
space bundles; fil¬
tration by codimension of support and the 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 P1 over a noncommutative ring,
and of Severi-Brauer schemes.
6.
Proofs of the Theorems of Chapter
4 ........................ 89
Proofe of the following theorems: ttiBQC)
й Ко (С);
Theorems A and
В
of Quillen; the theorem on exact sequences of functors; the resolution
theorem; the
dévissage
theorem; the localization theorem.
7.
Comparison of the Plus and Q-Constructions
................ 126
Monoidal categories; localization of the action of a monoidal category
on a small category; computation of the homology of the classifying
space of a localized category; the S~1S construction, viewed as a func-
torial version of the plus construction; construction of the homotopy
equivalence S~1S
—>
QBQC for any exact category
С
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
Aľ-groups
for
projective
modules over a ring.
8.
The Merkurjev-Suslin Theorem
............................ 145
The Galois symbol; statement of the Merkurjev—Suslin theorem;
Hubert s Theorem
90
for
Кг;
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
Fundamentai
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
Ях(/Сг)
of the resolution; com¬
putation of this
Ко
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;
(Α.?)
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.29)-( A.35) Quasi
fibrations (some results of
Dold
and Thom); (A.36)-(A.42) NDR-pairs
and cofibrations (following Steenrod); (A.43)-(A.47) if-spaces; (A.48)-
(A.50) Covering spaces of simplicial sets; (A.51)-(A.54) Hurewicz and
Whitehead theorems for non-simply connected
Jï-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.IHD.M) Sheaves; (D.15)-(D.2O) 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
|
| adam_txt |
Contents
Preface
to the First Edition
. xi
Preface to the Second Edition
. xvii
1.
"Classical"
К-ТЪеоту
. 1
Review of parts of Milnor's book: definitions of
Ко,
Κι,
Кг
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
Ki of a commutative local ring.
2.
The Plus Construction
. 18
The plus construction; computation that m(BGL(R)+)
S K%(R);
Я
-space structure of BGL(R)+ and products in
Я
-theory {following
Loday); statement of QuiHen's theorem on Ki of a finite field.
3.
The Classifying Space of a Small Gategory
. 31
Simplicia!
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
Quilîen's Q-Construcfcion
. 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
К, {Ко
agrees with that
defined "classically", theorem on exact sequences of functors, resolu¬
tion theorem,
dévissage
theorem, localization theorem); "bare hands"
construction of a homomorphism Ko{C)
—*
viii
Algebraic
/f-Theory
5.
The ií-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 K, of finitely gen¬
erated ^-modules, for Noetherian rings A; computations of Gi{A[t]),
Gi(A[i,i"ł])
for Noetherian A, and hence
КААЩ),
Кѓ{АЏ,Г1])
for
Noetherian regular A; definition of Kt(X),
G
AX) for schemes, using
vector bundles and coherent sheaves, respectively; construction of di¬
rect image and inverse image maps for K, and G, of Noetherian schemes
for morphisms satisfying appropriate conditions; action of
Ko on
Ki,
Gì
and projection formulas; Ki, Gi commute with filtered direct lim¬
its; localization for
G
і
of a closed subscheme and the open complement;
Mayer-Vietoris for G%\
G¡
of
affine
and
projective
space bundles; fil¬
tration by codimension of support and the 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 P1 over a noncommutative ring,
and of Severi-Brauer schemes.
6.
Proofs of the Theorems of Chapter
4 . 89
Proofe of the following theorems: ttiBQC)
й Ко (С);
Theorems A and
В
of Quillen; the theorem on exact sequences of functors; the resolution
theorem; the
dévissage
theorem; the localization theorem.
7.
Comparison of the Plus and Q-Constructions
. 126
Monoidal categories; localization of the action of a monoidal category
on a small category; computation of the homology of the classifying
space of a localized category; the S~1S construction, viewed as a "func-
torial" version of the plus construction; construction of the homotopy
equivalence S~1S
—>
QBQC for any exact category
С
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
Aľ-groups
for
projective
modules over a ring.
8.
The Merkurjev-Suslin Theorem
. 145
The Galois symbol; statement of the Merkurjev—Suslin theorem;
Hubert's Theorem
90
for
Кг;
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
"Fundamentai
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
Ях(/Сг)
of the resolution; com¬
putation of this
Ко
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;
(Α.?)
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.29)-( A.35) Quasi
fibrations (some results of
Dold
and Thom); (A.36)-(A.42) NDR-pairs
and cofibrations (following Steenrod); (A.43)-(A.47) if-spaces; (A.48)-
(A.50) Covering spaces of simplicial sets; (A.51)-(A.54) Hurewicz and
Whitehead theorems for non-simply connected
Jï-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.IHD.M) Sheaves; (D.15)-(D.2O) 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, Vasudevan 1958- |
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| callnumber-subject | QA - Mathematics |
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| classification_tum | MAT 552f MAT 189f |
| ctrlnum | (OCoLC)181090572 (DE-599)BVBBV022948632 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 2. ed., reprint of the 1995 2. ed. |
| format | Book |
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| id | DE-604.BV022948632 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T19:01:10Z |
| indexdate | 2024-07-09T21:08:22Z |
| institution | BVB |
| isbn | 9780817647360 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016153147 |
| oclc_num | 181090572 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-20 DE-355 DE-BY-UBR DE-11 DE-188 |
| owner_facet | DE-19 DE-BY-UBM DE-91G DE-BY-TUM DE-20 DE-355 DE-BY-UBR DE-11 DE-188 |
| physical | XVI, 341 S. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Modern Birkhäuser classics |
| spelling | Srinivas, Vasudevan 1958- Verfasser (DE-588)112913695 aut Algebraic K-theory V. Srinivas 2. ed., reprint of the 1995 2. ed. Boston [u.a.] Birkhäuser 2008 XVI, 341 S. txt rdacontent n rdamedia nc rdacarrier Modern Birkhäuser classics Originally publ. as volume 90 in the series 'Progress in mathematics' Algebraic topology K-theory Rings (Algebra) Algebraische K-Theorie (DE-588)4141839-6 gnd rswk-swf Algebra (DE-588)4001156-2 gnd rswk-swf K-Theorie (DE-588)4033335-8 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 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016153147&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, Vasudevan 1958- Algebraic K-theory Algebraic topology K-theory Rings (Algebra) Algebraische K-Theorie (DE-588)4141839-6 gnd Algebra (DE-588)4001156-2 gnd K-Theorie (DE-588)4033335-8 gnd |
| subject_GND | (DE-588)4141839-6 (DE-588)4001156-2 (DE-588)4033335-8 |
| title | Algebraic K-theory |
| title_auth | Algebraic K-theory |
| title_exact_search | Algebraic K-theory |
| title_exact_search_txtP | 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 | Algebraic topology K-theory Rings (Algebra) Algebraische K-Theorie (DE-588)4141839-6 gnd Algebra (DE-588)4001156-2 gnd K-Theorie (DE-588)4033335-8 gnd |
| topic_facet | Algebraic topology K-theory Rings (Algebra) Algebraische K-Theorie Algebra K-Theorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016153147&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT srinivasvasudevan algebraicktheory |