Talk:PlanetPhysics/Index of Algebraic Geometry

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 This is a contributed entry in progress

\section{Index of Algebraic Geometry}


Algebraic Geometry (AG), and \htmladdnormallink{non-commutative geometry/}{http://planetphysics.us/encyclopedia/NAQAT2.html}. On the other hand, there are also close ties between \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} geometry and number theory.


\subsection{Outline}

\subsection{Disciplines in algebraic geometry}

\begin{enumerate}
\item \emph{Birational geometry, Dedekind \htmladdnormallink{domains}{http://planetphysics.us/encyclopedia/Bijective.html} and Riemann-Roch \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html}}
\item Homology and \htmladdnormallink{cohomology theories}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}
\item Algebraic \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}: \htmladdnormallink{Lie groups}{http://planetphysics.us/encyclopedia/BilinearMap.html}, \htmladdnormallink{matrix}{http://planetphysics.us/encyclopedia/Matrix.html} group schemes,group machines, linear groups, generalizing Lie groups, \htmladdnormallink{representation}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html} theory
\item {\em Abelian varieties}
\item {\em Arithmetic algebraic geometry}
\item \htmladdnormallink{duality}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item \htmladdnormallink{category theory applications}{http://planetphysics.us/encyclopedia/CategoricalOntology.html} in algebraic geometry
\item \htmladdnormallink{indexes of category}{http://planetphysics.us/encyclopedia/IndexOfCategories.html}, \htmladdnormallink{functors}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} and \htmladdnormallink{natural transformations}{http://planetphysics.us/encyclopedia/VariableCategory2.html}
\item \htmladdnormallink{Grothendieck's Descent theory}{http://www.uclouvain.be/17501.html}
\item `\htmladdnormallink{Anabelian Geometry}{http://planetphysics.us/encyclopedia/IsomorphismClass.html}' \item Categorical Galois theory
\item \htmladdnormallink{higher dimensional algebra}{http://planetphysics.us/encyclopedia/2Groupoid2.html} (\htmladdnormallink{HDA}{http://planetphysics.us/encyclopedia/2Groupoid2.html})
\item \htmladdnormallink{Quantum Algebraic Topology}{http://planetphysics.us/encyclopedia/TriangulationMethodsForQuantizedSpacetimes2.html} (\htmladdnormallink{QAT}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html})
\item Quantum Geometry
\item \htmladdnormallink{computer}{http://planetphysics.us/encyclopedia/Program3.html} algebra \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html}; an example is: explicit projective resolutions for finitely-generated \htmladdnormallink{modules}{http://planetphysics.us/encyclopedia/RModule.html} over suitable rings
\end{enumerate}

\subsection{Cohomology}
Cohomology is an essential theory in the study of complex \htmladdnormallink{manifolds}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}.
\htmladdnormallink{computations}{http://planetphysics.us/encyclopedia/LQG2.html} in cohomology studies of complex manifolds in algebraic geometry utilize similar computations to those of cohomology theory in \htmladdnormallink{algebraic topology}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html}: spectral sequences, excision, the Mayer-Vietoris sequence, etc.
\begin{enumerate}
\item \htmladdnormallink{cohomology groups}{http://planetphysics.us/encyclopedia/CohomologyTheoryOnCWComplexes.html} are defined and then cohomology functors associate \htmladdnormallink{Abelian groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} to sheaves on a scheme; one may view such Abelian groups them as cohomology with coefficients in a scheme.
\item Cohomology functors
\item \htmladdnormallink{fundamental cohomology theorems}{http://planetphysics.us/encyclopedia/NaturalIsomorphism.html}
\item A basic \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} of cohomology for schemes is the sheaf cohomology
\item Whitehead groups, torsion and towers
\item xyz
\end{enumerate}

\subsection{Seminars on Algebraic Geometry and Topos Theory (SGA)}
\begin{enumerate}
\item \htmladdnormallink{SGA1}{http://planetphysics.org/?op=getobj&from=books&id=171}
\item \htmladdnormallink{SGA2}{http://planetphysics.org/?op=getobj&from=books&id=193}
\item \htmladdnormallink{SGA3}{http://planetphysics.org/?op=edit&from=books&id=201}
\item \htmladdnormallink{SGA4}{http://planetphysics.org/?op=getobj&from=lec&id=199}
\item \htmladdnormallink{SGA5}{http://planetphysics.org/?op=getobj&from=lec&id=196}
\item \htmladdnormallink{SGA6}{http://planetphysics.org/?op=getobj&from=books&id=200}
\item \htmladdnormallink{SGA7}{http://planetphysics.org/?op=getobj&from=lec&id=198}
\end{enumerate}

\subsection{Algebraic varieties and the GAGA principle}
\begin{enumerate}
\item new1x
\item new2y
\item new3z
\end{enumerate}


\subsection{Number theory applications}

\subsection{Cohomology theory}
\begin{enumerate}
\item Cohomology group
\item Cohomology sequence
\item DeRham cohomology
\item new4
\end{enumerate}

\subsection{Homology theory}
\begin{enumerate}
\item \htmladdnormallink{homology group}{http://planetphysics.us/encyclopedia/ExtendedHurewiczFundamentalTheorem.html} \item Homology sequence
\item Homology complex
\item new4

\end{enumerate}

\subsection{Duality in algebraic topology and category theory}
\begin{enumerate}

\item Tanaka-Krein duality
\item Grothendieck duality
\item \htmladdnormallink{categorical duality}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \item \htmladdnormallink{tangled duality}{http://planetphysics.us/encyclopedia/DualityAndTriality.html} \item DA5
\item DA6
\item DA7

\end{enumerate}

\subsection{Category theory applications}
\begin{enumerate}
\item \htmladdnormallink{abelian categories}{http://planetphysics.us/encyclopedia/AbelianCategory2.html}
\item \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} \item \htmladdnormallink{fundamental groupoid functor}{http://planetphysics.us/encyclopedia/QuantumFundamentalGroupoid.html} \item Categorical Galois theory
\item \htmladdnormallink{non-Abelian algebraic topology}{http://planetphysics.us/encyclopedia/ModuleAlgebraic.html} \item Group category
\item \htmladdnormallink{groupoid category}{http://planetphysics.us/encyclopedia/GroupoidCategory3.html} \item $\mathcal{T}op$ category
\item \htmladdnormallink{topos}{http://planetphysics.us/encyclopedia/GrothendieckTopos.html} and topoi axioms
\item \htmladdnormallink{generalized toposes}{http://planetphysics.us/encyclopedia/ManyValuedLogicSubobjectClassifiers.html} \item Categorical logic and algebraic topology
\item \htmladdnormallink{meta-theorems}{http://planetphysics.us/encyclopedia/MetaTheorems.html} \item Duality between spaces and algebras

\end{enumerate}


\subsection{Examples of Categories}
The following is a listing of categories relevant to algebraic topology:

\begin{enumerate}
\item \htmladdnormallink{Algebraic categories}{http://www.uclouvain.be/17501.html}
\item Topological category
\item Category of sets, Set
\item Category of topological spaces
\item \htmladdnormallink{category of Riemannian manifolds}{http://planetphysics.us/encyclopedia/CategoryOfRiemannianManifolds.html} \item Category of CW-complexes
\item Category of Hausdorff spaces
\item \htmladdnormallink{category of Borel spaces}{http://planetphysics.us/encyclopedia/CategoryOfBorelSpaces.html} \item Category of CR-complexes
\item Category of \htmladdnormallink{graphs}{http://planetphysics.us/encyclopedia/Cod.html} \item Category of \htmladdnormallink{spin networks}{http://planetphysics.us/encyclopedia/SimplicialCWComplex.html} \item Category of groups
\item Galois category
\item Category of \htmladdnormallink{fundamental groups}{http://planetphysics.us/encyclopedia/HomotopyCategory.html} \item Category of \htmladdnormallink{Polish groups}{http://planetphysics.us/encyclopedia/InvariantBorelSet.html}
\item Groupoid category
\item \htmladdnormallink{category of groupoids}{http://planetphysics.us/encyclopedia/GroupoidCategory.html} (or groupoid category)
\item \htmladdnormallink{category of Borel groupoids}{http://planetphysics.us/encyclopedia/CategoryOfBorelGroupoids.html} \item Category of \htmladdnormallink{fundamental groupoids}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html}
\item Category of functors (or \htmladdnormallink{functor category}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html})
\item \htmladdnormallink{double groupoid}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} category
\item \htmladdnormallink{double category}{http://planetphysics.us/encyclopedia/HorizontalIdentities.html} \item \htmladdnormallink{category of Hilbert spaces}{http://planetphysics.us/encyclopedia/CategoryOfHilbertSpaces.html} \item \htmladdnormallink{category of quantum automata}{http://planetphysics.us/encyclopedia/CategoryOfQuantumAutomata.html} \item \htmladdnormallink{R-category}{http://planetphysics.us/encyclopedia/RCategory.html} \item Category of \htmladdnormallink{algebroids}{http://planetphysics.us/encyclopedia/Algebroids.html} \item Category of \htmladdnormallink{double algebroids}{http://planetphysics.us/encyclopedia/GeneralizedSuperalgebras.html}
\item Category of \htmladdnormallink{dynamical systems}{http://planetphysics.us/encyclopedia/ContinuousGroupoidHomomorphism.html}
\end{enumerate}

\subsection{Index of functors}
\emph{The following is a contributed listing of functors:}

\begin{enumerate}
\item Covariant functors
\item Contravariant functors
\item \htmladdnormallink{adjoint functors}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html}
\item \htmladdnormallink{preadditive functors}{http://planetphysics.us/encyclopedia/PreadditiveFunctor.html}
\item Additive functor
\item \htmladdnormallink{representable functors}{http://planetphysics.us/encyclopedia/CategoryOfLogicAlgebras.html}
\item Fundamental groupoid functor
\item Forgetful functors
\item Grothendieck group functor
\item Exact functor
\item Multi-functor
\item \htmladdnormallink{section functors}{http://planetphysics.us/encyclopedia/RightAdjointFunctor.html}
\item NT2
\item NT3
\end{enumerate}


\subsection{Index of natural transformations}
\emph{The following is a contributed listing of natural transformations:}

\begin{enumerate}
\item \htmladdnormallink{natural equivalence}{http://planetphysics.us/encyclopedia/IsomorphismClass.html} \item Natural transformations in a \htmladdnormallink{2-category}{http://planetphysics.us/encyclopedia/2Category.html} \item NT3
\item NT1

\end{enumerate}

\subsection{Grothendieck proposals}
\begin{enumerate}
\item Esquisse d'un Programme
\item
\htmladdnormallink{Pursuing Stacks}{http://www.math.jussieu.fr/~leila/grothendieckcircle/stacks.ps}
\item S2
\item S3


\end{enumerate}

\subsection{Descent theory}
\begin{enumerate}
\item D1
\item D2
\item D3


\end{enumerate}

\subsection{Higher Dimensional Algebraic Geometry (HDAG)}

\begin{enumerate}
\item Categorical groups and \htmladdnormallink{supergroup}{http://planetphysics.us/encyclopedia/Paragroups.html} algebras
\item Double groupoid varieties
\item Double algebroids
\item Bi-algebroids
\item $R$-algebroid
\item $2$-category
\item $n$-category
\item \htmladdnormallink{super-category}{http://planetphysics.us/encyclopedia/SuperCategory6.html} \item weak \htmladdnormallink{n-categories}{http://planetphysics.us/encyclopedia/InfinityGroupoid.html} of \htmladdnormallink{algebraic varieties}{http://planetphysics.us/encyclopedia/IsomorphismClass.html}
\item Bi-dimensional Algebraic Geometry
\item Anabelian Geometry
\item \htmladdnormallink{Noncommutative geometry}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry.html}
\item Higher-homology/cohomology theories
\item H1
\item H2
\item H3
\item H4

\end{enumerate}



\subsubsection{Axioms of cohomology theory}
\begin{enumerate}

\item A1
\item A2
\item A3

\end{enumerate}

\subsubsection{Axioms of homology theory}
\begin{enumerate}

\item A1

\item A2
\item A3

\end{enumerate}

\subsection{Quantum algebraic topology (QAT)}

\textbf{(a). Quantum algebraic topology} is described as \emph{the mathematical and physical study of \htmladdnormallink{general theories}{http://planetphysics.us/encyclopedia/GeneralTheory.html} of quantum \htmladdnormallink{algebraic structures}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} from the standpoint of algebraic topology, \htmladdnormallink{category theory}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} and
their \htmladdnormallink{non-Abelian}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} extensions in higher dimensional algebra and \htmladdnormallink{supercategories}{http://planetphysics.us/encyclopedia/SuperCategory6.html}}
\begin{enumerate}
\item \htmladdnormallink{quantum operator algebras}{http://planetphysics.us/encyclopedia/Groupoid.html} (such as: involution, *-algebras, or $*$-algebras, \htmladdnormallink{von Neumann algebras}{http://planetphysics.us/encyclopedia/CoordinateSpace.html},
, JB- and JL- algebras, $C^*$ - or C*- algebras,
\item Quantum von Neumann algebra and subfactors; Jone's towers and subfactors
\item Kac-Moody and K-algebras
\item categorical groups
\item \htmladdnormallink{Hopf algebras}{http://planetphysics.us/encyclopedia/QuantumGroup4.html}, quantum Groups and \htmladdnormallink{quantum group}{http://planetphysics.us/encyclopedia/QuantumGroup4.html} algebras
\item \htmladdnormallink{quantum groupoids}{http://planetphysics.us/encyclopedia/WeakHopfAlgebra.html} and weak Hopf $C^*$-algebras
\item \htmladdnormallink{groupoid C*-convolution algebras}{http://planetphysics.us/encyclopedia/GroupoidCConvolutionAlgebra.html} and *-convolution algebroids
\item \htmladdnormallink{quantum spacetimes}{http://planetphysics.us/encyclopedia/NonAbelianQuantumAlgebraicTopology3.html} and \htmladdnormallink{quantum fundamental groupoids}{http://planetphysics.us/encyclopedia/QuantumFundamentalGroupoid4.html}
\item Quantum double Algebras
\item \htmladdnormallink{quantum gravity}{http://planetphysics.us/encyclopedia/LQG2.html}, \htmladdnormallink{supersymmetries}{http://planetphysics.us/encyclopedia/Supersymmetry.html}, \htmladdnormallink{supergravity}{http://planetphysics.us/encyclopedia/AntiCommutationRelations.html}, \htmladdnormallink{superalgebras}{http://planetphysics.us/encyclopedia/NewtonianMechanics.html} and graded `\htmladdnormallink{Lie' algebras}{http://planetphysics.us/encyclopedia/BilinearMap.html} \item Quantum \htmladdnormallink{categorical algebra}{http://planetphysics.us/encyclopedia/CategoryOfLogicAlgebras.html} and higher--dimensional, $\L{}-M_n$- Toposes
\item Quantum R-categories, \htmladdnormallink{R-supercategories}{http://planetphysics.us/encyclopedia/RDiagram.html} and \htmladdnormallink{spontaneous symmetry breaking}{http://planetphysics.us/encyclopedia/LongRangeCoupling.html} \item \htmladdnormallink{Non-Abelian Quantum Algebraic Topology}{http://planetphysics.us/encyclopedia/NonAbelianQuantumAlgebraicTopology3.html} (NA-QAT): closely related to NAAT and HDA.
\end{enumerate}

\subsection{Quantum Geometry}
\begin{enumerate}
\item \htmladdnormallink{Quantum Geometry overview}{http://planetphysics.us/encyclopedia/QuantumGeometry2.html}
\item Quantum non-commutative geometry
\end{enumerate}


\subsection{2x}


\begin{enumerate}

\item new1x
\item new2y


\end{enumerate}

\subsection{13}
\begin{enumerate}

\item new1x
\item new2y


\end{enumerate}

\subsection{14}


\subsection{Textbooks and bibliograpies}

\htmladdnormallink{Bibliography on Category theory, AT and QAT}{http://planetmath.org/?op=getobj&from=objects&id=10746}


\subsubsection{Textbooks and Expositions:}
\begin{enumerate}
\item A \htmladdnormallink{Textbook1}{http://planetmath.org/?op=getobj&from=books&id=172}
\item A \htmladdnormallink{Textbook2}{http://planetmath.org/?op=getobj&from=books&id=156}
\item A \htmladdnormallink{Textbook3}{http://planetmath.org/?op=getobj&from=books&id=159}
\item A \htmladdnormallink{Textbook4}{http://planetmath.org/?op=getobj&from=books&id=160}
\item A \htmladdnormallink{Textbook5}{http://planetmath.org/?op=getobj&from=books&id=153}
\item A \htmladdnormallink{Textbook6}{http://planetmath.org/?op=getobj&from=lec&id=68}
\item A \htmladdnormallink{Textbook7}{http://planetmath.org/?op=getobj&from=books&id=158}
\item A \htmladdnormallink{Textbook8}{http://planetmath.org/?op=getobj&from=lec&id=75}
\item A \htmladdnormallink{Textbook9}{http://planetmath.org/?op=getobj&from=lec&id=73}
\item A \htmladdnormallink{Textbook10}{http://planetmath.org/?op=getobj&from=books&id=174}
\item A \htmladdnormallink{Textbook11}{http://planetmath.org/?op=getobj&from=books&id=169}
\item A \htmladdnormallink{Textbook12}{http://planetmath.org/?op=getobj&from=books&id=178}
\item A \htmladdnormallink{Textbook13}{http://www.math.cornell.edu/~hatcher/VBKT/VB.pdf}
\item new1x
\end{enumerate}

\begin{thebibliography}{99}
\bibitem{AG-JD60}
Alexander Grothendieck and J. Dieudonn\'{e}.: 1960, El\'{e}ments de geometrie alg\'{e}brique., \emph{Publ. Inst. des Hautes Etudes de Science}, \textbf{4}.

\bibitem{Alex4}
Alexander Grothendieck. \emph{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 \htmladdnormallink{freely available}{http://modular.fas.harvard.edu/sga/sga/pdf/index.html} in French;
also available here is an extensive
\htmladdnormallink{Abstract in English}{http://planetmath.org/?op=getobj&from=books&id=158}.

\bibitem{Alex62}
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).,
\htmladdnormallink{Typewritten manuscript available in French}{http://modular.fas.harvard.edu/sga/sga/2/index.html};
\htmladdnormallink{see also a brief summary in English}{http://planetmath.org/?op=getobj&from=books&id=78} . Available for free downloads at  \htmladdnormallink{on the web}{http://www.numdam.org/numdam-bin/recherche?au=Grothendieck&format=short}.

\bibitem{Alex84}
Alexander Grothendieck, 1984. ``Esquisse d'un Programme'', (1984 manuscript), {\em finally published in ``Geometric Galois Actions''}, L. Schneps, P. Lochak, eds., {\em London Math. Soc. Lecture Notes} {\bf 242}, Cambridge University Press, 1997, pp.5-48; English transl., ibid., pp. 243-283. MR 99c:14034 .

\bibitem{Liu2k2}
Qing Liu.2002. \emph{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).

\bibitem{Shafarevich76}
Igor Shafarevich, \emph{Basic Algebraic Geometry} Vols. 1 and 2;
Vol.2:  {\em 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.

\bibitem{Milne}
James Milne, \emph{Elliptic Curves}, online course notes. \htmladdnormallink{Available at his website}{http://www.jmilne.org/math/CourseNotes/math679.html}.

\bibitem{Silverman86}
Joseph H. Silverman, \emph{The Arithmetic of Elliptic Curves}. Springer-Verlag, New York, 1986.

\bibitem{Silverman94}
Joseph H. Silverman, \emph{Advanced Topics in the Arithmetic of Elliptic Curves}. Springer-Verlag, New York, 1994.

\bibitem{Shimura71}
Goro Shimura, \emph{Introduction to the Arithmetic Theory of Automorphic Functions}. Princeton University Press, Princeton, New Jersey, 1971.

\bibitem{Mumford70}
David Mumford, \emph{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.''

\end{thebibliography} 

\end{document}