PlanetPhysics/Index of Algebraic Geometry

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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.

Disciplines in algebraic geometry

 * 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
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
 * xyz

Seminars on Algebraic Geometry and Topos Theory (SGA)

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

Algebraic varieties and the GAGA principle

 * 1) new1x
 * 2) new2y
 * 3) new3z

Cohomology theory

 * 1) Cohomology group
 * 2) Cohomology sequence
 * 3) DeRham cohomology
 * 4) new4

Homology theory

 * 1) homology group #Homology sequence
 * 2) Homology complex
 * 3) new4

Duality in algebraic topology and category theory

 * 1) Tanaka-Krein duality
 * 2) Grothendieck duality
 * 3) categorical duality #tangled duality #DA5
 * DA6
 * DA7

Category theory applications

 * 1) abelian categories
 * 2) topological category #fundamental groupoid functor #Categorical Galois theory
 * 3) non-Abelian algebraic topology #Group category
 * 4) groupoid category #$$\mathcal{T}op$$ 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
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
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
 * NT2
 * NT3

Index of natural transformations
The following is a contributed listing of natural transformations:


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

Grothendieck proposals
\item Pursuing Stacks
 * 1) Esquisse d'un Programme
 * S2
 * S3

Descent theory

 * D1
 * D2
 * D3

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 #weak n-categories of algebraic varieties
 * 9) Bi-dimensional Algebraic Geometry
 * 10) Anabelian Geometry
 * 11) Noncommutative geometry
 * 12) Higher-homology/cohomology theories
 * H1
 * H2
 * H3
 * H4

Axioms of cohomology theory

 * A1
 * A2
 * A3

Axioms of homology theory

 * A1


 * A2
 * A3

Quantum algebraic topology (QAT)
'''(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}

, JB- and JL- algebras, $$C^*$$ - or C*- algebras,
 * 1) quantum operator algebras (such as: involution, *-algebras, or $$*$$-algebras, von Neumann 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 $$C^*$$-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, $$\L{}-M_n$$- 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

 * 1) Quantum Geometry overview
 * 2) Quantum non-commutative geometry

2x

 * 1) new1x
 * 2) new2y

13

 * 1) new1x
 * 2) new2y

Textbooks and bibliograpies
Bibliography on Category theory, AT and QAT

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