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index of algebraic geometry (Topic)

This is a contributed entry in progress

1 Index of Algebraic Geometry

Algebraic Geometry (AG), and non-commutative geometry/. On the other hand, there are also close ties between algebraic geometry and number theory.

1.1 Outline

1.2 Disciplines in algebraic geometry

  1. Birational geometry, Dedekind domains and Riemann-Roch theorem
  2. Homology and cohomology theories
  3. Algebraic groups: Lie groups, matrix group schemes,group machines, linear groups, generalizing Lie groups, representation theory
  4. Abelian varieties
  5. Arithmetic algebraic geometry
  6. duality
  7. category theory applications in algebraic geometry
  8. indexes of category, functors and natural transformations
  9. Grothendieck’s Descent theory
  10. ‘Anabelian geometry’
  11. Categorical Galois theory
  12. higher dimensional algebra (HDA)
  13. Quantum Algebraic Topology (QAT)
  14. Quantum Geometry
  15. computer algebra systems; an example is: explicit projective resolutions for finitely-generated modules over suitable rings

1.3 Cohomology

Cohomology is an essential theory in the study of complex manifolds. computations in cohomology studies of complex manifolds in algebraic geometry utilize similar computations to those of cohomology theory in algebraic topology: spectral sequences, excision, the Mayer-Vietoris sequence, etc.

  1. cohomology groups are defined and then cohomology functors associate Abelian groups to sheaves on a scheme; one may view such Abelian groups them as cohomology with coefficients in a scheme.
  2. Cohomology functors
  3. fundamental cohomology theorems
  4. A basic type of cohomology for schemes is the sheaf cohomology
  5. Whitehead groups, torsion and towers
  6. xyz

1.4 Seminars on Algebraic Geometry and Topos Theory (SGA)

  1. SGA1
  2. SGA2
  3. SGA3
  4. SGA4
  5. SGA5
  6. SGA6
  7. SGA7

1.5 Algebraic varieties and the GAGA principle

  1. new1x
  2. new2y
  3. new3z

1.6 Number theory applications

1.7 Cohomology theory

  1. Cohomology group
  2. Cohomology sequence
  3. DeRham cohomology
  4. new4

1.8 Homology theory

  1. homology group
  2. Homology sequence
  3. Homology complex
  4. new4

1.9 Duality in algebraic topology and category theory

  1. Tanaka-Krein duality
  2. Grothendieck duality
  3. categorical duality
  4. tangled duality
  5. DA5
  6. DA6
  7. DA7

1.10 Category theory applications

  1. Abelian categories
  2. topological category
  3. fundamental groupoid functor
  4. Categorical Galois theory
  5. Non-Abelian algebraic topology
  6. Group category
  7. groupoid category
  8. 𝒯 op category
  9. topos and topoi axioms
  10. generalized toposes
  11. Categorical logic and algebraic topology
  12. meta-theorems
  13. Duality between spaces and algebras

1.11 Examples of Categories

The following is a listing of categories relevant to algebraic topology:

  1. Algebraic categories
  2. Topological category
  3. Category of sets, Set
  4. Category of topological spaces
  5. category of Riemannian manifolds
  6. Category of CW-complexes
  7. Category of Hausdorff spaces
  8. category of Borel spaces
  9. Category of CR-complexes
  10. Category of graphs
  11. Category of spin networks
  12. Category of groups
  13. Galois category
  14. Category of fundamental groups
  15. Category of Polish groups
  16. Groupoid category
  17. category of groupoids (or groupoid category)
  18. category of Borel groupoids
  19. Category of fundamental groupoids
  20. Category of functors (or functor category)
  21. double groupoid category
  22. double category
  23. category of Hilbert spaces
  24. category of quantum automata
  25. R-category
  26. Category of algebroids
  27. Category of double algebroids
  28. Category of dynamical systems

1.12 Index of functors

The following is a contributed listing of functors:

  1. Covariant functors
  2. Contravariant functors
  3. Adjoint functors
  4. preadditive functors
  5. Additive functor
  6. representable functors
  7. Fundamental groupoid functor
  8. Forgetful functors
  9. Grothendieck group functor
  10. Exact functor
  11. Multi-functor
  12. section functors
  13. NT2
  14. NT3

1.13 Index of natural transformations

The following is a contributed listing of natural transformations:

  1. Natural equivalence
  2. Natural transformations in a 2-category
  3. NT3
  4. NT1

1.14 Grothendieck proposals

  1. Esquisse d’un Programme
  2. Pursuing Stacks
  3. S2
  4. S3

1.15 Descent theory

  1. D1
  2. D2
  3. D3

1.16 Higher Dimensional Algebraic Geometry (HDAG)

  1. Categorical groups and supergroup algebras
  2. Double groupoid varieties
  3. Double algebroids
  4. Bi-algebroids
  5. R-algebroid
  6. 2-category
  7. n-category
  8. super-category
  9. weak n-categories of algebraic varieties
  10. Bi-dimensional Algebraic Geometry
  11. Anabelian Geometry
  12. Noncommutative geometry
  13. Higher-homology/cohomology theories
  14. H1
  15. H2
  16. H3
  17. H4

1.16.1 Axioms of cohomology theory

  1. A1
  2. A2
  3. A3

1.16.2 Axioms of homology theory

  1. A1
  2. A2
  3. A3

1.17 Quantum algebraic topology (QAT)

(a). Quantum algebraic topology is described as the mathematical and physical study of general theories of quantum algebraic structures from the standpoint of algebraic topology, category theory and their non-Abelian extensions in higher dimensional algebra and supercategories

  1. quantum operator algebras (such as: involution, *-algebras, or ∗-algebras, von Neumann algebras, , JB- and JL- algebras, C∗ - or C*- algebras,
  2. Quantum von Neumann algebra and subfactors; Jone’s towers and subfactors
  3. Kac-Moody and K-algebras
  4. categorical groups
  5. Hopf algebras, quantum Groups and quantum group algebras
  6. quantum groupoids and weak Hopf C∗-algebras
  7. groupoid C*-convolution algebras and *-convolution algebroids
  8. quantum spacetimes and quantum fundamental groupoids
  9. Quantum double Algebras
  10. quantum gravity, supersymmetries, supergravity, superalgebras and graded ‘Lie’ algebras
  11. Quantum categorical algebra and higher–dimensional, Ł − Mn- Toposes
  12. Quantum R-categories, R-supercategories and spontaneous symmetry breaking
  13. non-Abelian Quantum Algebraic Topology (NA-QAT): closely related to NAAT and HDA.

1.18 Quantum Geometry

  1. Quantum Geometry overview
  2. Quantum non-commutative geometry

1.19 2x

  1. new1x
  2. new2y

1.20 13

  1. new1x
  2. new2y

1.21 14

1.22 Textbooks and bibliograpies

Bibliography on Category theory, AT and QAT

1.22.1 Textbooks and Expositions:

  1. A Textbook1
  2. A Textbook2
  3. A Textbook3
  4. A Textbook4
  5. A Textbook5
  6. A Textbook6
  7. A Textbook7
  8. A Textbook8
  9. A Textbook9
  10. A Textbook10
  11. A Textbook11
  12. A Textbook12
  13. A Textbook13
  14. new1x

References

[1]   Alexander Grothendieck and J. Dieudonné.: 1960, Eléments de geometrie algébrique., Publ. Inst. des Hautes Etudes de Science, 4.

[2]   Alexander Grothendieck. Séminaires en Géometrie Algèbrique- 4, Tome 1, Exposé 1 (or the Appendix to Exposée 1, by ‘N. Bourbaki’ for more detail and a large number of results. AG4 is freely available in French; also available here is an extensive Abstract in English.

[3]   Alexander Grothendieck. 1962. Séminaires en Géométrie Algébrique du Bois-Marie, Vol. 2 - Cohomologie Locale des Faisceaux Cohèrents et Théorèmes de Lefschetz Locaux et Globaux. , pp.287. (with an additional contributed exposé by Mme. Michele Raynaud)., Typewritten manuscript available in French; see also a brief summary in English . Available for free downloads at on the web.

[4]   Alexander Grothendieck, 1984. “Esquisse d’un Programme”, (1984 manuscript), finally published in “Geometric Galois Actions”, L. Schneps, P. Lochak, eds., London Math. Soc. Lecture Notes 242, Cambridge University Press, 1997, pp.5-48; English transl., ibid., pp. 243-283. MR 99c:14034 .

[5]   Qing Liu.2002. Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics 6, 2002. 300 pages on schemes followed by geometry and arithmetic surfaces. (Serre duality is approached via Grothendieck duality).

[6]   Igor Shafarevich, Basic Algebraic Geometry Vols. 1 and 2; Vol.2: Schemes and Complex Manifolds., Second Revised and Expanded Edition. Springer-Verlag; scheme theory, varieties as schemes, varieties and schemes over the complex numbers, and complex manifolds.

[7]   James Milne, Elliptic Curves, online course notes. Available at his website.

[8]   Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.

[9]   Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.

[10]   Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.

[11]   David Mumford, Abelian Varieties, Oxford University Press, London, 1970. This book is a canonical reference on the subject. “It is written in the language of modern algebraic geometry, and provides a thorough grounding in the theory of abelian varieties.”


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Also defines:  algebraic geometry index, SGA1, SGA2, SGA3, SGA4, SGA5, SGA6, SGA7, SGA
Keywords:  index of algebraic geometry, related: algebraic topology index

Cross-references: non-Abelian Quantum Algebraic Topology, spontaneous symmetry breaking, R-supercategories, categorical algebra, Lie algebras, supergravity, supersymmetries, quantum gravity, quantum fundamental groupoids, quantum spacetimes, groupoid C*-convolution algebras, quantum groupoids, quantum group, Hopf algebras, von Neumann algebras, quantum operator algebras, supercategories, non-Abelian, category theory, general theories, n-categories, super-category, supergroup, 2-category, section functors, representable functors, preadditive functors, dynamical systems, double algebroids, algebroids, R-category, category of quantum automata, category of Hilbert spaces, double category, double groupoid, functor category, groupoids, category of Borel groupoids, category of groupoids, Polish groups, fundamental groups, spin networks, graphs, category of Borel spaces, category of Riemannian manifolds, meta-theorems, generalized toposes, topos, groupoid category, fundamental groupoid functor, category, topological, Abelian categories, tangled duality, categorical duality, homology group, type, fundamental cohomology theorems, cohomology groups, algebraic topology, computations, manifolds, modules, systems, computer, QAT, Quantum Algebraic Topology, HDA, higher dimensional algebra, natural transformations, functors, indexes of category, category theory applications, duality, representation, matrix, Lie groups, cohomology theories, theorem, domains, algebraic, non-commutative geometry
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