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PlanetPhysics/Index of Algebraic Geometry

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Index of Algebraic Geometry

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

Outline

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Disciplines in algebraic geometry

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  1. Birational geometry, Dedekind \htmladdnormallink{domains {http://planetphysics.us/encyclopedia/Bijective.html} 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 #category theory applications in algebraic geometry
  7. indexes of category, functors and natural transformations
  8. Grothendieck's Descent theory
  9. `Anabelian Geometry' #Categorical Galois theory
  10. higher dimensional algebra (HDA)
  11. Quantum Algebraic Topology (QAT)
  12. Quantum Geometry
  13. computer algebra systems; an example is: explicit projective resolutions for finitely-generated modules over suitable rings

Cohomology

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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

Seminars on Algebraic Geometry and Topos Theory (SGA)

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

Algebraic varieties and the GAGA principle

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  1. new1x
  2. new2y
  3. new3z

Number theory applications

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Cohomology theory

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

Homology theory

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

Duality in algebraic topology and category theory

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

Category theory applications

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  1. abelian categories
  2. topological category #fundamental groupoid functor #Categorical Galois theory
  3. non-Abelian algebraic topology #Group category
  4. groupoid category #  category
  5. topos and topoi axioms
  6. generalized toposes #Categorical logic and algebraic topology
  7. meta-theorems #Duality between spaces and algebras

Examples of Categories

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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 #Category of CW-complexes
  6. Category of Hausdorff spaces
  7. category of Borel spaces #Category of CR-complexes
  8. Category of graphs #Category of spin networks #Category of groups
  9. Galois category
  10. Category of fundamental groups #Category of Polish groups
  11. Groupoid category
  12. category of groupoids (or groupoid category)
  13. category of Borel groupoids #Category of fundamental groupoids
  14. Category of functors (or functor category)
  15. double groupoid category
  16. double category #category of Hilbert spaces #category of quantum automata #R-category #Category of algebroids #Category of double algebroids
  17. Category of dynamical systems

Index of functors

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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

Index of natural transformations

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The following is a contributed listing of natural transformations:

  1. natural equivalence #Natural transformations in a 2-category #NT3
  2. NT1

Grothendieck proposals

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  1. Esquisse d'un Programme

\item Pursuing Stacks

  1. S2
  2. S3

Descent theory

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  1. D1
  2. D2
  3. D3

Higher Dimensional Algebraic Geometry (HDAG)

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

Axioms of cohomology theory

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  1. A1
  2. A2
  3. A3

Axioms of homology theory

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  1. A1
  1. A2
  2. A3

Quantum algebraic topology (QAT)

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(a). Quantum algebraic topology is described as the mathematical and physical study of \htmladdnormallink{general theories {http://planetphysics.us/encyclopedia/GeneralTheory.html} 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,   - or C*- algebras,

  1. Quantum von Neumann algebra and subfactors; Jone's towers and subfactors
  2. Kac-Moody and K-algebras
  3. categorical groups
  4. Hopf algebras, quantum Groups and quantum group algebras
  5. quantum groupoids and weak Hopf  -algebras
  6. groupoid C*-convolution algebras and *-convolution algebroids
  7. quantum spacetimes and quantum fundamental groupoids
  8. Quantum double Algebras
  9. quantum gravity, supersymmetries, supergravity, superalgebras and graded `Lie' algebras #Quantum categorical algebra and higher--dimensional, Failed to parse (unknown function "\L"): {\displaystyle \L{}-M_n} - Toposes
  10. Quantum R-categories, R-supercategories and spontaneous symmetry breaking #Non-Abelian Quantum Algebraic Topology (NA-QAT): closely related to NAAT and HDA.

Quantum Geometry

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

2x

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  1. new1x
  2. new2y

13

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  1. new1x
  2. new2y

14

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Textbooks and bibliograpies

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Bibliography on Category theory, AT and QAT

Textbooks and Expositions:

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  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

All Sources

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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

References

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  1. ↑ Alexander Grothendieck and J. Dieudonn\'{e}.: 1960, El\'{e}ments de geometrie alg\'{e}brique., Publ. Inst. des Hautes Etudes de Science , 4 .
  2. ↑ Alexander Grothendieck. S\'eminaires en G\'eometrie Alg\`ebrique- 4 , Tome 1, Expos\'e 1 (or the Appendix to Expos\'ee 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. S\'eminaires en G\'eom\'etrie Alg\'ebrique du Bois-Marie, Vol. 2 - Cohomologie Locale des Faisceaux Coh\`erents et Th\'eor\`emes de Lefschetz Locaux et Globaux. , pp.287. (with an additional contributed expos\'e 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 {\mathbf 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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