index of algebraic geometry
IndexOfAlgebraicGeometry
2009-05-02 16:09:46
2009-05-10 15:07:39
Topic
index of algebraic geometry
This is a contributed entry in progress
\section{Index of Algebraic Geometry}
\PMlinkname{Algebraic Geometry (AG)}{AlgebraicGeometry}, and Non-commutative Geometry/NAAT. On the other hand, there are also close ties between algebraic geometry and number theory.
\subsection{Outline}
\subsection{Disciplines in algebraic geometry}
\begin{enumerate}
\item \emph{Birational geometry, Dedekind domains and Riemann-Roch theorem}
\item Homology and cohomology theories
\item Algebraic groups: Lie groups, matrix group schemes,group machines, linear groups, generalizing Lie groups, representation theory
\item {\em Abelian varieties}
\item {\em Arithmetic algebraic geometry}
\item Duality
\item Category theory applications in algebraic geometry
\item Index of categories, functors and natural transformations
\item \PMlinkexternal{Grothendieck's Descent theory}{http://www.uclouvain.be/17501.html}
\item `Anabelian geometry'
\item Categorical Galois theory
\item Higher dimensional algebra (HDA)
\item Quantum algebraic topology (QAT)
\item Quantum Geometry
\item Computer algebra systems; an example is: explicit projective resolutions for finitely-generated modules over suitable rings
\end{enumerate}
\subsection{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.
\begin{enumerate}
\item 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.
\item Cohomology functors
\item Fundamental cohomology theorems
\item A basic type 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}
\begin{enumerate}
\item \PMlinkexternal{SGA1}{http://planetphysics.org/?op=getobj&from=books&id=171}
\item \PMlinkexternal{SGA2}{http://planetphysics.org/?op=getobj&from=books&id=193}
\item \PMlinkexternal{SGA3}{http://planetphysics.org/?op=getobj&from=lec&id=199}
\item \PMlinkexternal{SGA4}{http://planetphysics.org/?op=getobj&from=lec&id=199}
\item \PMlinkexternal{SGA5}{http://planetphysics.org/?op=getobj&from=lec&id=196}
\item \PMlinkexternal{SGA6}{http://planetphysics.org/?op=getobj&from=books&id=200}
\item \PMlinkexternal{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 Homology group
\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 Categorical duality
\item Tangled duality
\item DA5
\item DA6
\item DA7
\end{enumerate}
\subsection{Category theory applications}
\begin{enumerate}
\item Abelian categories
\item Topological category
\item Fundamental groupoid functor
\item Categorical Galois theory
\item Non-Abelian algebraic topology
\item Group category
\item Groupoid category
\item $\mathcal{T}op$ category
\item Topos and topoi axioms
\item Generalized toposes
\item Categorical logic and algebraic topology
\item Meta-theorems
\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 \PMlinkexternal{Algebraic categories}{http://www.uclouvain.be/17501.html}
\item Topological category
\item Category of sets, Set
\item Category of topological spaces
\item Category of Riemannian manifolds
\item Category of CW-complexes
\item Category of Hausdorff spaces
\item Category of Borel spaces
\item Category of CR-complexes
\item Category of graphs
\item Category of spin networks
\item Category of groups
\item Galois category
\item Category of fundamental groups
\item Category of Polish groups
\item Groupoid category
\item Category of groupoids (or groupoid category)
\item Category of Borel groupoids
\item Category of fundamental groupoids
\item Category of functors (or functor category)
\item Double groupoid category
\item Double category
\item Category of Hilbert spaces
\item Category of quantum automata
\item R-category
\item Category of algebroids
\item Category of double algebroids
\item Category of dynamical systems
\end{enumerate}
\subsection{Index of functors}
\emph{The following is a contributed listing of functors:}
\begin{enumerate}
\item Covariant functors
\item Contravariant functors
\item Adjoint functors
\item Preadditive functors
\item Additive functor
\item Representable functors
\item Fundamental groupoid functor
\item Forgetful functors
\item Grothendieck group functor
\item Exact functor
\item Multi-functor
\item Section functors
\item NT2
\item NT3
\end{enumerate}
\subsection{Index of natural transformations}
\emph{The following is a contributed listing of natural transformations:}
\begin{enumerate}
\item Natural equivalence
\item Natural transformations in a 2-category
\item NT3
\item NT1
\end{enumerate}
\subsection{Grothendieck proposals}
\begin{enumerate}
\item \PMlinkname{Esquisse d'un Programme}{AlexSMathematicalHeritageEsquisseDunProgramme}
\item
\PMlinkexternal{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 supergroup algebras
\item Double groupoid varieties
\item Double algebroids
\item Bi-algebroids
\item $R$-algebroid
\item $2$-category
\item $n$-category
\item Super-category
\item weak n-categories of algebraic varieties
\item Bi-dimensional Algebraic Geometry
\item Anabelian Geometry
\item \PMlinkname{Noncommutative geometry}{NoncommutativeGeometry}
\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 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}
\begin{enumerate}
\item Quantum operator algebras (such as: involution, *-algebras, or $*$-algebras, von Neumann algebras,
, 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 Hopf algebras, quantum Groups and quantum group algebras
\item Quantum groupoids and weak Hopf $C^*$-algebras
\item Groupoid C*-convolution algebras and *-convolution algebroids
\item Quantum spacetimes and quantum fundamental groupoids
\item Quantum double Algebras
\item Quantum gravity, supersymmetries, supergravity, superalgebras and graded `Lie' algebras
\item Quantum categorical algebra and higher--dimensional, $\L{}-M_n$- Toposes
\item Quantum R-categories, R-supercategories and spontaneous symmetry breaking
\item Non-Abelian Quantum algebraic topology (NA-QAT): closely related to NAAT and HDA.
\end{enumerate}
\subsection{Quantum Geometry}
\begin{enumerate}
\item \PMlinkname{Quantum Geometry overview}{QuantumGeometry2}
\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}
\PMlinkexternal{Bibliography on Category theory, AT and QAT}{http://planetmath.org/?op=getobj&from=objects&id=10746}
\subsubsection{Textbooks and Expositions:}
\begin{enumerate}
\item A \PMlinkexternal{Textbook1}{http://planetmath.org/?op=getobj&from=books&id=172}
\item A \PMlinkexternal{Textbook2}{http://planetmath.org/?op=getobj&from=books&id=156}
\item A \PMlinkexternal{Textbook3}{http://planetmath.org/?op=getobj&from=books&id=159}
\item A \PMlinkexternal{Textbook4}{http://planetmath.org/?op=getobj&from=books&id=160}
\item A \PMlinkexternal{Textbook5}{http://planetmath.org/?op=getobj&from=books&id=153}
\item A \PMlinkexternal{Textbook6}{http://planetmath.org/?op=getobj&from=lec&id=68}
\item A \PMlinkexternal{Textbook7}{http://planetmath.org/?op=getobj&from=books&id=158}
\item A \PMlinkexternal{Textbook8}{http://planetmath.org/?op=getobj&from=lec&id=75}
\item A \PMlinkexternal{Textbook9}{http://planetmath.org/?op=getobj&from=lec&id=73}
\item A \PMlinkexternal{Textbook10}{http://planetmath.org/?op=getobj&from=books&id=174}
\item A \PMlinkexternal{Textbook11}{http://planetmath.org/?op=getobj&from=books&id=169}
\item A \PMlinkexternal{Textbook12}{http://planetmath.org/?op=getobj&from=books&id=178}
\item A \PMlinkexternal{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 \PMlinkexternal{freely available}{http://modular.fas.harvard.edu/sga/sga/pdf/index.html} in French;
also available here is an extensive
\PMlinkexternal{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).,
\PMlinkexternal{Typewritten manuscript available in French}{http://modular.fas.harvard.edu/sga/sga/2/index.html};
\PMlinkexternal{see also a brief summary in English}{http://planetmath.org/?op=getobj&from=books&id=78} . Available for free downloads at \PMlinkexternal{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. \PMlinkexternal{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}