index of algebraic geometry
This is a contributed entry in progress
Algebraic Geometry (AG), and non-commutative geometry/. On the other hand, there are also close ties between algebraic
geometry and number theory.
- Birational geometry, Dedekind domains
and Riemann-Roch theorem
- Homology and cohomology theories
- Algebraic groups: Lie groups, matrix
group schemes,group machines, linear groups, generalizing Lie groups, representation
theory
- Abelian varieties
- Arithmetic algebraic geometry
- duality
- category theory applications
in algebraic geometry
- indexes of category, functors
and natural transformations
- Grothendieck's Descent theory
- `Anabelian geometry'
- Categorical Galois theory
- higher dimensional algebra
(HDA)
- Quantum Algebraic Topology
(QAT)
- Quantum Geometry
- computer
algebra systems; an example is: explicit projective resolutions for finitely-generated modules
over suitable rings
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.
- 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.
- Cohomology functors
- fundamental cohomology theorems
- A basic type
of cohomology for schemes is the sheaf cohomology
- Whitehead groups, torsion and towers
- xyz
- SGA1
- SGA2
- SGA3
- SGA4
- SGA5
- SGA6
- SGA7
- new1x
- new2y
- new3z
- Cohomology group
- Cohomology sequence
- DeRham cohomology
- new4
- homology group
- Homology sequence
- Homology complex
- new4
- Tanaka-Krein duality
- Grothendieck duality
- categorical duality
- tangled duality
- DA5
- DA6
- DA7
- abelian categories
- topological
category
- fundamental groupoid functor
- Categorical Galois theory
- non-Abelian algebraic topology
- Group category
- groupoid category
category
- topos
and topoi axioms
- generalized toposes
- Categorical logic and algebraic topology
- meta-theorems
- Duality between spaces and algebras
The following is a listing of categories relevant to algebraic topology:
- Algebraic categories
- Topological category
- Category of sets, Set
- Category of topological spaces
- category of Riemannian manifolds
- Category of CW-complexes
- Category of Hausdorff spaces
- category of Borel spaces
- Category of CR-complexes
- Category of graphs
- Category of spin networks
- Category of groups
- Galois category
- Category of fundamental groups
- Category of Polish groups
- Groupoid category
- category of groupoids
(or groupoid category)
- category of Borel groupoids
- Category of fundamental groupoids
- Category of functors (or functor category)
- double groupoid
category
- double category
- category of Hilbert spaces
- category of quantum automata
- R-category
- Category of algebroids
- Category of double algebroids
- Category of dynamical systems
The following is a contributed listing of functors:
- Covariant functors
- Contravariant functors
- adjoint functors
- preadditive functors
- Additive functor
- representable functors
- Fundamental groupoid functor
- Forgetful functors
- Grothendieck group functor
- Exact functor
- Multi-functor
- section functors
- NT2
- NT3
The following is a contributed listing of natural transformations:
- Natural equivalence
- Natural transformations in a 2-category
- NT3
- NT1
- Esquisse d'un Programme
- Pursuing Stacks
- S2
- S3
- D1
- D2
- D3
- Categorical groups and supergroup
algebras
- Double groupoid varieties
- Double algebroids
- Bi-algebroids
-algebroid
-category
-category
- super-category
- weak n-categories
of algebraic varieties
- Bi-dimensional Algebraic Geometry
- Anabelian Geometry
- Noncommutative geometry
- Higher-homology/cohomology theories
- H1
- H2
- H3
- H4
- A1
- A2
- A3
- A1
- A2
- A3
(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
- quantum operator algebras
(such as: involution, *-algebras, or
-algebras, von Neumann algebras,
, JB- and JL- algebras,
- or C*- algebras,
- Quantum von Neumann algebra and subfactors; Jone's towers and subfactors
- Kac-Moody and K-algebras
- categorical groups
- Hopf algebras, quantum Groups and quantum group
algebras
- quantum groupoids
and weak Hopf
-algebras
- groupoid C*-convolution algebras
and *-convolution algebroids
- quantum spacetimes
and quantum fundamental groupoids
- Quantum double Algebras
- quantum gravity, supersymmetries, supergravity, superalgebras
and graded `Lie' algebras
- Quantum categorical algebra
and higher-dimensional,
- Toposes
- Quantum R-categories, R-supercategories
and spontaneous symmetry breaking
- Non-Abelian Quantum Algebraic Topology
(NA-QAT): closely related to NAAT and HDA.
- Quantum Geometry overview
- Quantum non-commutative geometry
- new1x
- new2y
- new1x
- new2y
Bibliography on Category theory, AT and QAT
- A Textbook1
- A Textbook2
- A Textbook3
- A Textbook4
- A Textbook5
- A Textbook6
- A Textbook7
- A Textbook8
- A Textbook9
- A Textbook10
- A Textbook11
- A Textbook12
- A Textbook13
- new1x
- 1
-
Alexander Grothendieck and J. Dieudonné.: 1960, Eléments de geometrie algébrique., Publ. Inst. des Hautes Etudes de Science, 4.
- 2
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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
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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
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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
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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
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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
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James Milne, Elliptic Curves, online course notes. Available at his website.
- 8
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Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
- 9
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Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.
- 10
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Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, Princeton, New Jersey, 1971.
- 11
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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.''
Contributors to this entry (in most recent order):
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