Topic
Spectral sequence
About: Spectral sequence is a research topic. Over the lifetime, 2471 publications have been published within this topic receiving 50292 citations.
Papers published on a yearly basis
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Papers
Book•
27 Oct 1995TL;DR: The landscape of homological algebra has evolved over the last half-century into a fundamental tool for the working mathematician as discussed by the authors, which is suitable for second or third year graduate students.
Abstract: The landscape of homological algebra has evolved over the last half-century into a fundamental tool for the working mathematician This book provides a unified account of homological algebra as it exists today The historical connection with topology, regular local rings, and semi-simple Lie algebras are also described This book is suitable for second or third year graduate students The first half of the book takes as its subject the canonical topics in homological algebra: derived functors, Tor and Ext, projective dimensions and spectral sequences Homology of group and Lie algebras illustrate these topics Intermingled are less canonical topics, such as the derived inverse limit functor lim1, local cohomology, Galois cohomology, and affine Lie algebras The last part of the book covers less traditional topics that are a vital part of the modern homological toolkit: simplicial methods, Hochschild and cyclic homology, derived categories and total derived functors By making these tools more accessible, the book helps to break down the technological barrier between experts and casual users of homological algebra
4,088 citations
Book•
31 Jan 2013
TL;DR: This paper presents a meta-thesis on the basis of a model derived from the model developed in [Bouchut-Boyaval, M3AS (23) 2013] that states that the mode of action of the Higgs boson is determined by the modulus of the E-modulus.
Abstract: I De Rham Theory.- II The ?ech-de Rham Complex.- III Spectral Sequences and Applications.- IV Characteristic Classes.- References.- List of Notations.
3,733 citations
Book•
31 Dec 1971
TL;DR: In this paper, the authors propose an extension of the Kunneth Theorem for Abelian groups, which is based on the notion of double complexes, and they use it to define the (co-)homology of groups.
Abstract: I. Modules.- 1. Modules.- 2. The Group of Homomorphisms.- 3. Sums and Products.- 4. Free and Projective Modules.- 5. Projective Modules over a Principal Ideal Domain.- 6. Dualization, Injective Modules.- 7 Injective Modules over a Principal Ideal Domain.- 8. Cofree Modules.- 9. Essential Extensions.- II. Categories and Functors.- 1. Categories.- 2. Functors.- 3. Duality.- 4. Natural Transformations.- 5. Products and Coproducts Universal Constructions.- 6. Universal Constructions (Continued) Pull-backs and Push-outs.- 7. Adjoint Functors.- 8. Adjoint Functors and Universal Constructions.- 9. Abelian Categories.- 10. Projective, Injective, and Free Objects.- III. Extensions of Modules.- 1. Extensions.- 2. The Functor Ext.- 3. Ext Using Injectives.- 4. Computation of some Ext-Groups.- 5. Two Exact Sequences.- 6. A Theorem of Stein-Serre for Abelian Groups.- 7. The Tensor Product.- 8. The Functor Tor.- IV. Derived Functors.- 1. Complexes.- 2. The Long Exact (Co) Homology Sequence.- 3. Homotopy.- 4. Resolutions.- 5. Derived Functors.- 6. The Two Long Exact Sequences of Derived Functors.- 7. The Functors Extn? Using Projectives.- 8. The Functors % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbWexLMBb50ujbqegm0B % 1jxALjharqqr1ngBPrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY- % Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq % 0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaae % aaeaaakeaadaqdaaqaaGqaaiaa-veacaWF4bGaa8hDaaaadaqhaaWc % baacciGae43MdWeabaGaamOBaaaaaaa!40A3! $$ \overline {Ext} _\Lambda ^n $$ Using Injectives.- 9. Extn and n-Extensions.- 10. Another Characterization of Derived Functors.- 11. The Functor Torn?.- 12. Change of Rings.- V. The Kiinneth Formula.- 1. Double Complexes.- 2. The Kunneth Theorem.- 3. The Dual Kunneth Theorem.- 4. Applications of the Kunneth Formulas.- VI. Cohomology of Groups.- 1. The Group Ring.- 2. Definition of (Co) Homology.- 3. H0, H0.- 4. H1, H1 with Trivial Coefficient Modules.- 5. The Augmentation Ideal, Derivations, and the Semi-Direct Product.- 6. A Short Exact Sequence.- 7. The (Co) Homology of Finite Cyclic Groups.- 8. The 5-Term Exact Sequences.- 9. H2, Hopf's Formula, and the Lower Central Series.- 10. H2 and Extensions.- 11. Relative Projectives and Relative Injectives.- 12. Reduction Theorems.- 13. Resolutions.- 14. The (Co) Homology of a Coproduct.- 15. The Universal Coefficient Theorem and the (Co)Homology of a Product.- 16. Groups and Subgroups.- VII. Cohomology of Lie Algebras.- 1. Lie Algebras and their Universal Enveloping Algebra.- 2. Definition of Cohomology H0, H1.- 3. H2 and Extensions.- 4. A Resolution of the Ground Field K.- 5. Semi-simple Lie Algebras.- 6. The two Whitehead Lemmas.- 7. Appendix : Hubert's Chain-of-Syzygies Theorem.- VIII. Exact Couples and Spectral Sequences.- 1. Exact Couples and Spectral Sequences.- 2. Filtered Differential Objects.- 3. Finite Convergence Conditions for Filtered Chain Complexes.- 4. The Ladder of an Exact Couple.- 5. Limits.- 6. Rees Systems and Filtered Complexes.- 7. The Limit of a Rees System.- 8. Completions of Filtrations.- 9. The Grothendieck Spectral Sequence.- IX. Satellites and Homology.- 1. Projective Classes of Epimorphisms.- 2. ?-Derived Functors.- 3. ?-Satellites.- 4. The Adjoint Theorem and Examples.- 5. Kan Extensions and Homology.- 6. Applications: Homology of Small Categories, Spectral Sequences.- X. Some Applications and Recent Developments.- 1. Homological Algebra and Algebraic Topology.- 2. Nilpotent Groups.- 3. Finiteness Conditions on Groups.- 4. Modular Representation Theory.- 5. Stable and Derived Categories.
1,110 citations
Book•
10 Sep 2002
TL;DR: In this article, Kac-Moody Lie Algebra Homology and Cohomology has been studied in the context of representation theory of kac-moody groups.
Abstract: Introduction * Kac--Moody Algebras -- Basic Theory * Representation Theory of Kac--Moody Algebras * Lie Algebra Homology and Cohomology * An Introduction to ind-Varieties and pro-Groups * Tits Systems -- Basic Theory * Kac--Moody Groups -- Basic Theory * Generalized Flag Varieties of Kac--Moody Groups * Demazure and Weyl--Kac Character Formulas * BGG and Kempf Resolutions * Defining Equations of G/P and Conjugacy Theorems * Topology of Kac-Moody Groups and Their Flag Varieties * Smoothness and Rational Smoothness of Schubert Varieties * An Introduction to Affine Kac-Moody Lie Algebras and Groups * Appendix A. Results from Algebraic Geometry * Appendix B. Local Cohomology * Appendix C. Results from Topology * Appendix D. Relative Homological Algebra * Appendix E. An Introduction to Spectral Sequences * Bibliography * Index of Notation * Index
1,003 citations
Book•
1 Jan 1974
TL;DR: In this article, the authors present an algebraic topology for complex manifolds with the purpose of studying the properties of stable homotopy and generalized cohomology operations in the context of K-theory.
Abstract: Preface Pt. I: S.P. Novikov's Work on Operations on Complex Cobordism 2: Cobordism groups 3: Homology 4: The Conner-Floyd Chern classes 5: The Novikov operations 6: The algebra of all operations 7: Scholium on Novikov's exposition 8: Complex manifolds Pt. II: Quillen's Work on Formal Groups and Complex Cobordism 1: Formal groups 2: Examples from algebraic topology 3: Reformulation 4: Calculations in E-homology and cohomology 5: Lazard's universal ring 6: More calculations in E-homology 7: The structure of Lazard's universal ring L 8: Quillen's theorem 9: Corollaries 10: Various formulae in [pi][subscript *](MU) 11: MU[subscript *](MU) 12: Behaviour of the Bott map 13: K[subscript *](K) 14: The Hattori-Stong theorem 15: Quillen's idempotent cohomology operations 16: The Brown-Peterson spectrum 17: KO[subscript *](KO) Pt. III: Stable Homotopy and Generalised Homology 2: Spectra 3: Elementary properties of the category of CW-spectra 4: Smash products 5: Spanier-Whitehead duality 6: Homology and cohomology 7: The Atiyah-Hirzebruch spectral sequence 8: The inverse limit and its derived functors 9: Products 10: Duality in manifolds 11: Applications in K-theory 12: The Steenrod algebra and its dual 13: A universal coefficient theorem 14: A category of fractions 15: The Adams spectral sequence 16: Applications to [pi][subscript *](bu[actual symbol not reproducible]X): modules over K[x, y] 17: Structure of [pi][subscript *](bu[actual symbol not reproducible]bu)~
941 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 21 |
| 2024 | 34 |
| 2023 | 97 |
| 2022 | 181 |
| 2021 | 102 |
| 2020 | 117 |