Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Chicago u.a.
Univ. of Chicago Press
1994
|
| Series: | Chicago lectures in mathematics science
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 229 S. graph. Darst. |
| ISBN: | 0226742024 0226742032 |
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MARC
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| 100 | 1 | |a Schwartz, Lionel |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture |c Lionel Schwartz |
| 264 | 1 | |a Chicago u.a. |b Univ. of Chicago Press |c 1994 | |
| 300 | |a X, 229 S. |b graph. Darst. | ||
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| 490 | 0 | |a Chicago lectures in mathematics science | |
| 650 | 7 | |a Modules (Algèbre) |2 ram | |
| 650 | 7 | |a Point fixe, théorème du |2 ram | |
| 650 | 7 | |a Steenrod, algèbre de |2 ram | |
| 650 | 4 | |a Fixed point theory | |
| 650 | 4 | |a Modules (Algebra) | |
| 650 | 4 | |a Steenrod algebra | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Introduction 1
Parti
The algebraic structure of the category 14 and the functor Ty
Chapter 1. Recollections concerning the Steenrod algebra
and unstable A modules . 17
1.1. The Steenrod algebra 17
1.2. Generators for the Steenrod algebra 18
1.3. The instability condition 20
1.4. Unstable ^. algebras 21
1.5. Notation and examples 22
1.6. Free objects in the category U 23
1.7. Instability and the Adem relations 26
1.8. The category U is locally noetherian 29
1.9. Proof of Proposition 1.6.3 and of Corollary 1.8.5 31
1.10. Appendix on Milnor s dual A* of the Steenrod algebra
and on Milnor s derivations . 32
Chapter 2. Algebraic Brown Gitler technology 35
2.1. Generalities 35
2.2. A representability statement 37
2.3. Brown Gitler modules 39
2.4. The bigraded module J* 41
2.5. The relation between J* and Milnor s algebra A* 45
2.6. Carlsson s modules K(i) and reduced unstable A modules .. 46
2.7. Carlsson s bigraded algebra K* 50
2.8. Tensor products of injective unstable A modules 52
2.9. The unstable modules K(i) and binary trees 55
Chapter 3. U Injectivity of the mod p cohomology of elementary
abelian p groups and Lannes functor Ty . 61
3.1. U lnjectivity of H*V®J(n) 61
3.2. Lannes functor Ty 63
vii
viii Contents
3.3. First examples of Ty computations . 67 3.4. Commutation of Ty
and I 70
3.5. Commutation of Ty with tensor products 73
3.6. The case of an odd prime 76
3.7. Comments and exercise 78
3.8. The functor Tv and unstable algebras 80
3.9. Further examples of Tv computations 86
3.10. The formula for TVH*BG 89
3.11. The classification theorem for injective unstable A modules . . 90
3.12. Reduced indecomposable U injectives 93
3.13. The general case 96
3.14. Applications 98
Part 2
Deeper algebraic structure
Chapter 4. The structure of indecomposable reduced U injectives . . 105
4.1. The structure of the set L 105
4.2. Results on the Poincare series
of indecomposable reduced U injectives . . 107
4.3. The decomposition of the Carlsson modules K(i) 112
4.4. Information about the A module structure
of the reduced indecomposable IX injectives . . 113
Chapter 5. The category 1X/N/7, analytic functors,
and representations of the symmetric groups ..115
5.1. The category UfNil and the functor / 115
5.2. Analytic functors 118
5.3. Injective objects in the category 3 ^ 120
5.4. Proof of theorem 5.2.6 125
5.5. The functor pn : U/Wl Mod^Q j and the filtration
on IX/N/7 . . 126
5.6. Simple objects of IX/X/7 131
5.7. Proof of formula 4.3.1 134
5.8. Comments on the Weyl correspondence 136
5.9. The case of an odd prime in the preceding sections 137
5.10. The Grothendieck ring of IX 138
Contents ix
Chapter 6. Subcategories of U 139
6.1. The categories WIe and the quotient categories Nilt/Nil(+ . 139
6.2. The category B of locally finite unstable A modules
and the categories Nil^ . . 144
6.3. Localization away from N/7^ 147
6.4. The categories N/7^ and the functors Tor 149
Part 3
The Sullivan conjecture and the cohomology of mapping spaces
Chapter 7. Non Abelian homological algebra
and Andre Quillen cohomology . . 155
7.1. Simplicial resolutions in the categories % and Alg 155
7.2. Andre Quillen cohomology of unstable A algebras 160
7.3. A connectivity result for Andre Quillen cohomology 163
7.4. Proof of Proposition 7.3.3 165
7.5. Miller s spectral sequence 170
7.6. A change of rings theorem 171
7.7. Derivations 173
7.8. Derived functors of derivations and Lannes theorem 175
Chapter 8. On homotopy classes of maps from B V 179
8.1. Miller s conjecture 179
8.2. Bousfield Kan s and Bousfield s theorems 179
8.3. Resolutions of spaces and mapping spaces 182
8.4. Proof of Theorem 8.1.1 184
8.5. Comments on the connected components of the total space . .. 188
8.6. H. Miller s theorem and a converse 189
8.7. Applications of the Eilenberg Moore spectral sequence 192
8.8. A topological characterization of spaces
such that H*X e Wlk .. 198
Chapter 9. The generalized Sullivan conjecture
and the cohomology of mapping spaces . . 199
9.1. The Sullivan conjecture 199
9.2. A comparison theorem 201
9.3. The case of the Borel construction 202
9.4. Proof of the Sullivan conjecture 204
x Contents
9.5. The case of the action of a finite p group it 206
9.6. The space map(£V, BG) 207
9.7. The cohomology of mapping spaces with source B V 209
9.8. A special case 212
9.9. The Eilenberg Moore spectral sequence 214
9.10. A lemma of Bousfield 217
References 219
Index of notation 228
Index 229
Frequently used notation
A — is the mod p Steenrod algebra
A* — is the dual of the mod p Steenrod algebra
Alg — is the category of unital, commutative, N graded ¥p algebras
Alga — is the category of augmented objects in Alg
£ — is the category of ¥p vector spaces
£gr — is the category of graded Fp vector spaces
H*X — is the mod p cohomology of the space X
H*X — is the reduced mod p cohomology of the space X
H*V = H*BV — is the mod p cohomology of V
H V — is the reduced mod p cohomology of V
X — is the category of unstable .A algebras
%a — is the category of augmented objects in DC
A/® , a e N (or more generally a cardinal) — denotes the direct sum
of a copies of the object M of £, or of £gr, or of U etc.
0 — denotes always a forgetful functor
K — is the category of unstable A modules
W — is the category of evenly graded unstable A modules
V, W, ... — denote finite dimensional Fp vector spaces
a(n), n e N — is the sum of the coefficients in the p adic expansion
of the integer n
» — always denotes an epimorphism
—*¦ — always denotes a monomorphism
|
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| language | English |
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| publishDate | 1994 |
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| publisher | Univ. of Chicago Press |
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| series2 | Chicago lectures in mathematics science |
| spelling | Schwartz, Lionel Verfasser aut Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture Lionel Schwartz Chicago u.a. Univ. of Chicago Press 1994 X, 229 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Chicago lectures in mathematics science Modules (Algèbre) ram Point fixe, théorème du ram Steenrod, algèbre de ram Fixed point theory Modules (Algebra) Steenrod algebra Modul (DE-588)4129770-2 gnd rswk-swf Fixpunkttheorie (DE-588)4293945-8 gnd rswk-swf Steenrod-Algebra (DE-588)4182999-2 gnd rswk-swf Steenrod-Algebra (DE-588)4182999-2 s Fixpunkttheorie (DE-588)4293945-8 s Modul (DE-588)4129770-2 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006531836&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Schwartz, Lionel Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture Modules (Algèbre) ram Point fixe, théorème du ram Steenrod, algèbre de ram Fixed point theory Modules (Algebra) Steenrod algebra Modul (DE-588)4129770-2 gnd Fixpunkttheorie (DE-588)4293945-8 gnd Steenrod-Algebra (DE-588)4182999-2 gnd |
| subject_GND | (DE-588)4129770-2 (DE-588)4293945-8 (DE-588)4182999-2 |
| title | Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture |
| title_auth | Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture |
| title_exact_search | Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture |
| title_full | Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture Lionel Schwartz |
| title_fullStr | Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture Lionel Schwartz |
| title_full_unstemmed | Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture Lionel Schwartz |
| title_short | Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture |
| title_sort | unstable modules over the steenrod algebra and sullivan s fixed point set conjecture |
| topic | Modules (Algèbre) ram Point fixe, théorème du ram Steenrod, algèbre de ram Fixed point theory Modules (Algebra) Steenrod algebra Modul (DE-588)4129770-2 gnd Fixpunkttheorie (DE-588)4293945-8 gnd Steenrod-Algebra (DE-588)4182999-2 gnd |
| topic_facet | Modules (Algèbre) Point fixe, théorème du Steenrod, algèbre de Fixed point theory Modules (Algebra) Steenrod algebra Modul Fixpunkttheorie Steenrod-Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006531836&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT schwartzlionel unstablemodulesoverthesteenrodalgebraandsullivansfixedpointsetconjecture |