Vor Ort
Merkzettel Warenkorb
  • Produktbild: Cohomology Theories for Compact Abelian Groups
  • Produktbild: Cohomology Theories for Compact Abelian Groups

Cohomology Theories for Compact Abelian Groups

51,99 €

inkl. gesetzl. MwSt., Versandkostenfrei

Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

14.12.2011

Verlag

Springer Berlin

Seitenzahl

236

Maße (L/B/H)

24,4/17/1,4 cm

Gewicht

422 g

Auflage

1973

Sprache

Englisch

ISBN

978-3-642-80672-8

Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

14.12.2011

Verlag

Springer Berlin

Seitenzahl

236

Maße (L/B/H)

24,4/17/1,4 cm

Gewicht

422 g

Auflage

1973

Sprache

Englisch

ISBN

978-3-642-80672-8

Herstelleradresse

Springer-Verlag KG
Sachsenplatz 4-6
1201 Wien
AT

Email: GPSR Kontakt

Noch keine Bewertungen vorhanden

Verfassen Sie die erste Bewertung zu diesem Artikel

Helfen Sie anderen Kundinnen und Kunden durch Ihre Meinung.

Kundinnen und Kunden meinen

Bewertungen (0)
  • Produktbild: Cohomology Theories for Compact Abelian Groups
  • Produktbild: Cohomology Theories for Compact Abelian Groups
  • I. Algebraic background.- Section 1. On exponential functors.- Definition 1.1. Multiplicative category, exponential functor and polynomial algebra, Hopf algebra — Definition 1.2. Subadditive and sub-multiplicative functors, compatible natural transformations — Lemma 1.3. E2SA and TE1A are algebras when Ei is exponential, S subadditive, T submultiplicative — Lemma 1.4. ? is a morphism of algebras — Lemma 1.5. About coalgebras — Proposition 1.6. E HomR (—, M) ? HomZ (E —, M) is a morphism of graded algebras for E = ?, P — Lemmas 1.10, 1.11. The structure of Hom(PZn, Z) — Lemmas 1.12, 1.13. More about Hom (E—, M) — Proposition 1.14. The structure of Hom (EA, M), E = ?, P — Definition 1.15. Polynomial algebras with divided powers — Proposition 1.16. E Hom (—, M) ? Hom (E —, M) for E = P ? ? — Lemma 1.17. About the natural map ?A Proposition 1.18. The coalgebra Hom (G, Z) — Corollary 1.19. The duality of polynomial algebras and algebras with divided powers — Theorem 1.22. The map PRA ? R ? RB ? Hom ($$\hat PA \otimes {\text{ }} \wedge {\text{ }}\hat B,R$$) — Proposition 1.23. about HomS (K ? L, A) ? HomS (L, Hom (K, A)) for complexes.- Section 2. The arithmetic of certain spectral algebras.- Definition 2.1. Spectral algebra, edge algebra — Lemmas 2.2, 2.3. The derivations d, d?, — Definition 2.4. The functors E2, E3 - Lemma 2.5. The cohomology map preserves multiplication — Lemma 2.6. Definition of the cohomology map ? — Definition 2.7. The first edge algebra and B2P (?) — Definition 2.8. Integral elements in rings, weakly principal ideal rings — Definition 2.10. The formalism of the derivation d? on E2(?) — Definition 2.11. The elementary morphisms — Proposition 2.12. The structure of the edge terms in E3(?) — Lemma 2.13. The elements of ker d?, — Lemma 2.14. The elements of im d? — Proposition 2.16. u ? a8’ ? ua8’: E3II(?) ? a8’ ? E3(?) is injective — Proposition 2.17. The terms next to the edge terms — An explicit example — Corollary 2.18. The terms next to the edge terms for a principal ideal domain as coefficient ring — Lemma 2.20. Passage to the ring of quotients in the coefficient ring — Proposition 2.21. E2(? ? ?) ? E2(?) ? E2(?) — Proposition 2.24, 2.25. Conditions under which d? is exact — Proposition 2.26. The exactness of d? within the ground ring extension — Lemma 2.31. Elementary morphisms yielding the same E3 - Proposition 2.32. The case that ? is a homothety — Proposition 2.33. Elementary morphisms which differ by a scalar — Proposition 2.34. E3(?1 ? ?2) ? E3(?2) if im d(?1) is flat — Proposition 2.35. An inductive process to compute E3(?) if the ground ring is a principal ideal domain — Theorem I. E3(?) is generated as a (P coker ?)-module by M — Definition 2.39. Definition of ? and E2*(?) — Lemma 2.40. The differential modules (E2(?), d’) — Proposition 2.42. About the structure of E3(?) — Propositions 2.43, 2.44. About the PA-module structure of Er(?) — Propositions 2.47, 2.48. Non-injective elementary morphisms.- Section 3. Some analogues of the results about spectral algebras with dual derivations.- Lemma 3.1. The differential and derivative ?? — Definition 3.2, 3.3. The spectral algebras Er[?], Er{?} — Lemma 3.4. E2 [—] is an exponential functor — Lemma 3.5. About ?d + d? — Proposition 3.6. The edge algebra E3II [?] — Definition 3.6a. R-coalgebras, differential graded co-algebras, differential graded Hopf algebras — Proposition 3.7. E2{?} is a differential bi-graded Hopf algebra relative to d?, and ?? — Lemma 3.8. The cofunctor f ? E2 {Hom8 (f, R)} — Lemma 3.9. About the structure of finite abelian groups — Definition 3.10. Standard resolution of a finite abelian group — Lemma 3.11. The uniqueness of standard resolutions —Lemma 3.12. The four term exact sequence derived from an injection — Lemma 3.13. Isomorphic version of ker ?pHom (f, A) — Proposition 3.14. The edge terms in E3 (Hom (f, R)) — Corollary 3.15. The morphism PR Ext (G, R) ?R Hom (? G, R) ? E3 (Hom (f, R)) — Corollary 3.16. The functoriality of this morphism — Propositions 3.17, 3.18. The isomorphisms H (R/Z ? E2(f)) ? E3(f) ? H (E2(f)?)?..- Section 4. The Bockstein formalism.- Lemmas 4.1, 4.2, 4.3, 4.4. Some diagram chasing — Definition 4.5. The definition of pre-Bockstein diagrams and standard Bockstein diagrams — Lemmas 4.6, 4.7, 4.8. About the Bockstein formalism — Proposition 4.9. An isomorphism of exact sequences — Lemma 4.10. More diagram chasing — Proposition 4.11. Sufficient conditions for the Bockstein formalism for complexes — Proposition 4.12. When is the Bockstein differential a derivation? — Corollaries 4.13, 4.14. The standard situation — Proposition 4.15. The Bockstein formalism for the cohomology of groups and complexes — Proposition 4.16. The Bockstein formalism for the spectral algebras E2(?) of Section 2 — Corollary 4.17. A particular case of 4.16..- II. The cohomology of finite abelian groups.- Section 1. Products.- Definition 1.1. The construction of ? — Definition 1.2. The construction of ? — Lemma 1.3. Tensoring resolutions — Corollary 1.4 — Lemma 1.5. The Künneth theorem — Theorem 1.6. The resolution of augmented Hopf algebras — Theorem II. Cohomology and the tensor product of Hopf algebras — Corollary 1.7. About H(G1 × G2, R) — Corollary 1.8. A Künneth theorem for H(G1 × G2, R) — Corollary 1.9. A special case of 1.8 — Corollary 1.10. H(G1 × G2, R) for cyclic G1 — Corollary 1.11. H(G1, R) ? ? ? H(Gn, R) ? H(G1 × ? × Gn, R) — Corollary 1.12. About the annihilator of H+(G1 × G2, R) — Corollary 1.13. About the exponent of H+(G1 × G2, R) — Corollary 1.14. The exponent of H+(G, M) for a finite abelian group G and arbitrary M — Corollary 1.15. H(G, M) ? N ? H(G, M ? N).- Section 2. Special free resolutions for finite abelian groups.- Definition 2.1. Special elements in the group ring of a finite abelian group — Lemma 2.2. About ?: ? ? ? A+ — Lemma 2.3. d? ?d = 0 — Lemma 2.4. The coderivation D = d + ? — Definition 2.5. E(f) and Ê (f) — Lemma 2.6. Ê is exponential — Lemma 2.7. Ê(f) exact — special case — Lemma 2.8. 0 ? Z ? Ê (f) is a resolution — Lemma 2.9. R ?SA Horns (HomS (A, S), R) — Theorem III. Fundamental theorem about the cohomology of finite abelian groups — Lemma 2.10. Hi (G, R/Z) ? Hi+1(G, Z) — Proposition 2.11. Various isomorphisms involving H(G, R/Z) — Lemma 2.12. A categorical lemma — Theorem 2.13. The morphism ?: PR Ext(G, R) ?R Hom(? G, R) ? H(G, R) — Lemma 2.14. A lemma involving the bar resolution — Proposition and Corollaries 2.15-2.18. A relation between the bar resolution and the bi-resolution.- Section 3. About the cohomology of finite abelian groups in the case of trivial action.- Definition 3.1. Recapitulation of the standard resolution — Lemma 3.2. A group theoretical lemma — Proposition 3.3. A special case of Theorem 2.13 — Definition 3.4. The z-constituent of a group — Proposition 3.5. Splitting the z-constituent in H(G, R) — Theorem 3.6. The cohomology of Z (z)n — Corollary 3.7. The Poincaré series for Theorem 3.6 — Corollary 3.8. The additive structure of H (G, Z (z)) for an arbitrary G whose exponent divides z — Proposition 3.9. Decomposing H (G, Z) — Theorem 3.11. A structure theorem for H(G, Z) — Theorem IV. About the structure of the ring H(G, Z) — Corollaries 3.13–3.15. A minimal generating module for H (G, Z) — Theorem V. The complete structure of H(G, R) if R is a field — Example — Proposition 3.16. H(G1 × G2, R) for groups G1, G2 with relatively prime order — Propositions 3.17, 3.18. About the Bocksteins in low dimension — Proposition 3.19. A global version of the previous results.- Section 4. Appendix to Section 3: The low dimensions.- Proposition 4.1. A list for Hi (G, R) for i < 6 — Proposition 4.2 (? G)? ? ?? — Proposition 4.3. A list for Hi (G, R/Z) for i < 4 — A remark about Schur’s multiplicator — Two dimensional cohomology and central extensions.- III. The cohomology of classifying spaces of compact groups.- Section 1. The functor h.- Definition 1.1. The join of two spaces, the iterated join — Proposition 1.2. A Künneth theorem for the join and the relation with the standard Künneth theorem — Corollaries 1.3, 1.4. The acyclicity of iterated joins — Definition 1.5. The spectrum of universal spaces for G and the spectrum of classifying spaces for G. Classifying spaces up to n — Lemma 1.6. The existence of spectra of universal spaces — Definition 1.7. The Milnor spectrum of universal spaces (resp. classifying spaces) for G — Proposition 1.8. Properties of the Milnor spectrum — Definition 1.9. The definition of the functor h — Proposition 1.10. The independence of h from the choice of universal spaces — Proposition 1.11, 1.12. h transforms projective limits into direct limits — Proposition 1.13, 1.14. The Künneth thoerem for h — Corollaries 1.15, 1.16. Comments on h (G’ × G, R) — Propositions 1.17-1.20. The Bockstein formalism for h.- Section 2. The functor h for finite groups.- Definition 2.1. Simplicial objects — Proposition 2.2. Products, equalizers, etc. for simplicial sets — Definition 2.3. Group actions on simplicial sets — Lemma 2.4, 2.5. Free simplicial modules — Proposition 2.6. The equivalence of h(G, R) with H(G, R) for finite G — Corollary 2.7. The computation of h(G, R) for totally disconnected compact G.- IV. Kan extensions of functors on dense categories.- Section 1. Dense categories and continuous functors.- Lemma 1.1. The functor LIM — Lemma 1.2. The functor SD — Definition 1.3. D -continuous functors — Definition 1.4. The comma category — Example 1.5. The category of Lie groups is dense in the category of compact groups — Lemma 1.6. A uniqueness statement for natural transformations — Definition 1.7. Dense subcategories — Example 1.8. Continuation of Example 1.5 — Definition 1.9. Extendable functors — Definition 1.10. Compatible functors — Definition 1.11. Strictly dense subcategories — Proposition 1.12. Extending extendable functors — Definition 1.13. Kan extensions — Theorem 1.14. The Kan extension existence theorem — Theorem 1.15. Density theorem for the category of compact groups.- Section 2. Multiplicative Hopf extensions.- Definition 2.1. Freely generated categories — Theorem 2.2. The existence and uniqueness of Hopf extensions — Corollary 2.3. Hopf extensions of functors on compact abelian Lie groups — Corollary 2.4. Hopf extensions of functors on compact connected Lie groups — Corollary 2.5. A uniqueness theorem for exponential functors on compact abelian groups — Proposition 2.6. The exterior algebra functor for compact abelian groups — Lemma 2.7. The properties of the exterior algebra functor — Lemma 2.8. About the functor Hom (? —, R) — Lemma 2.9. The dual of the exterior algebra of a compact abelian group — Lemma 2.10. R ? Hom (G, K) ? Hom (G, R).- V. The cohomological structure of compact abelian groups.- Section 1. The cohomologies of connected compact abelian groups.- Lemmas 1.1.-1.6. Continuous exponential functors on compact connected abelian groups — Lemmas 1.7, 1.8. Change of coefficients — Theorem 1.9. The structure theorem for cohomology theories on compact connected abelian groups — Theorem 1.10. The singular cohomology on compact connected abelian groups — Corollary 1.11. The algebraic cohomology of a finitely generated abelian group.- Section 2. The space cohomology of arbitrary compact abelian groups.- Theorem VI. The structure theorem for topological cohomology.- Section 3. The canonical embedding of ? in hG.- Theorem 3.1. ? = h2(G, Z)..- Section 4. Cohomology theories for compact groups over fields as coefficient domains.- Lemmas 4.1, 4.2. Exponential functors on compact abelian groups — Theorem VII. The algebraic cohomology of a compact abelian group over a field — Corollary 4.3. The algebraic cohomology of a compact abelian group with real coefficients — Theorem 4.4. The algebraic cohomology over a finite prime field and the Bockstein differential.- Section 5. The structure of h for arbitrary compact abelian groups and integral coefficients.- Proposition 5.1. Splitting a connected group — Proposition 5.2. The cohomology of compact abelian Lie groups — Propositions 5.3, 5.4. The maps induced in cohomology by the inclusion of the connected identity component and its cokernel — Theorem VIII. The principal theorem for integral cohomology — Lemma 5.5. Reducing an abelian group — Lemma 5.6. The cohomology of a p-adic group — Proposition 5.7. Classification of compact abelian groups with compact classifying space.- VI. Appendix. Another construction of the functor h.- Proposition 1. About the graph of < for a topological monoid acting on a space — Proposition 2. Properties of the Dold-Lashof spectrum.- List of notations.