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\section*{EGA}

This entry is about the text

\begin{itemize}%
\item [[Alexander Grothendieck]], \emph{\'E{}l\'e{}ments de g\'e{}om\'e{}trie alg\'e{}brique} (\href{http://www.numdam.org/numdam-bin/recherche?au=Grothendieck&format=short}{numdam})

\end{itemize}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{ega}{EGA}\dotfill \pageref*{ega} \linebreak
\noindent\hyperlink{list_of_volumes_of_ega}{List of volumes of EGA}\dotfill \pageref*{list_of_volumes_of_ega} \linebreak
\noindent\hyperlink{ega_v_and_beyond}{EGA V and beyond}\dotfill \pageref*{ega_v_and_beyond} \linebreak
\noindent\hyperlink{the_other_series}{The other series:}\dotfill \pageref*{the_other_series} \linebreak
\noindent\hyperlink{fga_fondements_de_la_gometrie_algbrique}{FGA (Fondements de la G\'e{}ometrie Alg\'e{}brique)}\dotfill \pageref*{fga_fondements_de_la_gometrie_algbrique} \linebreak
\noindent\hyperlink{sga_sminaire_de_gomtrie_algbrique_du_bois_marie}{SGA (S\'e{}minaire de G\'e{}om\'e{}trie Alg\'e{}brique du Bois Marie)}\dotfill \pageref*{sga_sminaire_de_gomtrie_algbrique_du_bois_marie} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{ega}{}\subsection*{{EGA}}\label{ega}

EGA stands for \emph{\'E{}l\'e{}ments de g\'e{}om\'e{}trie alg\'e{}brique} (Elements of algebraic geometry), which was written by [[Alexandre Grothendieck]] and co-edited with [[Jean Dieudonne|Jean Dieudonné]]. These volumes (and a list is given below) were among his many works attempting to build the foundations for [[algebraic geometry]] in the language of [[schemes]]. They were preceded by FGA and were followed up by the SGA, more details of these can be found below.

The published part of \emph{EGA} is in Publ. IH\'E{}S, now free online at \href{http://www.numdam.org/numdam-bin/recherche?au=Grothendieck&format=short}{numdam} (detailed links to chapters, and their contents will be added here later, see also below for a list of the volumes in the series). We plan to list here the grand plan and some remarks and links.

\hypertarget{list_of_volumes_of_ega}{}\subsubsection*{{List of volumes of EGA}}\label{list_of_volumes_of_ega}

Note that there is a chapter 0 that continues at the beginning of each chapter, establishing preliminaries from [[topology]], [[category theory]], [[commutative algebra]], [[homological algebra]], etc.

\begin{itemize}%
\item I. Le langage des sch\'e{}mas, Inst. Hautes \'E{}tudes Sci. Publ. Math. 4 (1960), 5--228. Ch. 0.\S{}1--7 ; Ch I.\S{}1--10.


\item II. \'E{}tude globale \'e{}l\'e{}mentaire de quelques classes de morphismes, Inst. Hautes \'E{}tudes Sci. Publ. Math. 8 (1961), 5--222. Ch. II.\S{}1--8.


\item III\_1. \'E{}tude cohomologique des faisceaux coh\'e{}rents (Premi\`e{}re partie) Inst. Hautes \'E{}tudes Sci. Publ. Math. 11 (1961), 5--167. Ch. 0.\S{}8--13 ; Ch.III.\S{}1--5.


\item III\_2. \'E{}tude cohomologique des faisceaux coh\'e{}rents (Seconde partie) Inst. Hautes \'E{}tudes Sci. Publ. Math. 17 (1963), 5--91. Ch. III .\S{}6--7.


\item IV\_1. \'E{}tude locale des sch\'e{}mas et des morphismes de sch\'e{}mas (Premi\`e{}re partie) Inst. Hautes \'E{}tudes Sci. Publ. Math. 20 (1964), 5--259. Ch.0.\S{}14--23 ; Ch. IV.\S{}1.


\item IV\_2. \'E{}tude locale des sch\'e{}mas et des morphismes de sch\'e{}mas (Seconde partie) Inst. Hautes \'E{}tudes Sci. Publ. Math. 24 (1965), 5--231. Ch. IV2.\S{}2--7 .


\item IV\_3. \'E{}tude locale des sch\'e{}mas et des morphismes de sch\'e{}mas (Troisi\`e{}me partie) Inst. Hautes \'E{}tudes Sci. Publ. Math. 28 (1966), 5--255. Ch. IV.\S{}8--15.


\item IV\_4. \'E{}tude locale des sch\'e{}mas et des morphismes de sch\'e{}mas (Quatri\`e{}me partie) Inst. Hautes \'E{}tudes Sci. Publ. Math. 32 (1967), 5--361. Ch.IV.\S{}16--21.



\end{itemize}
There was a second edition of EGA I:

\begin{itemize}%
\item I. Le langage des sch\'e{}mas. (2nd ed.) Grundlehren der Mathematischen Wissenschaften (1971). 166. Springer-Verlag.

\end{itemize}
In this edition, the term ``prescheme'' (now outdated but used in the first edition for what we now call schemes) was replaced by ``scheme'', and the term ``scheme'' became [[separated scheme]]. Also, there was a transition to the [[functor of points]] approach to scheme theory.

\hypertarget{ega_v_and_beyond}{}\subsubsection*{{EGA V and beyond}}\label{ega_v_and_beyond}

EGA was never completed. The listed volumes I-IV are just a part of the original plan. Grothendieck outlined what was meant to be in chapters V-VII, at least, and some handwritten prenotes existed for a small part of those. See e.g. these prenotes for some parts of \href{http://www.jmilne.org/math/Documents/EGA-V.pdf}{EGA V}.

\href{http://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique}{Wikipedia} lists the titles of planned chapters I-XIII.

\hypertarget{the_other_series}{}\subsection*{{The other series:}}\label{the_other_series}

\hypertarget{fga_fondements_de_la_gometrie_algbrique}{}\subsubsection*{{FGA (Fondements de la G\'e{}ometrie Alg\'e{}brique)}}\label{fga_fondements_de_la_gometrie_algbrique}

The earliest of the series is \emph{FGA} written alone at the end of the 1950s; it is sort of quick outline of the theory and some deep results with proofs at the beginning of the period of the development of [[scheme theory]]. A modern version of the material in FGA appeared as a book [[FGA explained]] (ICTP, Trieste 2003--2005). The original is available \href{http://www.math.jussieu.fr/~leila/grothendieckcircle/FGA.pdf}{here}.

\hypertarget{sga_sminaire_de_gomtrie_algbrique_du_bois_marie}{}\subsubsection*{{SGA (S\'e{}minaire de G\'e{}om\'e{}trie Alg\'e{}brique du Bois Marie)}}\label{sga_sminaire_de_gomtrie_algbrique_du_bois_marie}

(Bois Marie is the name of an area of woodland at Bures-sur-Yvette, where the [[math institutions|IHÉS]] is situated.)

\emph{SGA} consists of the writeups of seminars at IH\'E{}S lead by Grothendieck; the work of the members of the seminar under the guidance of Grothendieck are included. In $n$lab we have pages on [[SGA1]], and [[SGA4]].

The \href{http://en.wikipedia.org/wiki/S%C3%A9minaire_de_G%C3%A9om%C3%A9trie_Alg%C3%A9brique_du_Bois_Marie}{Wikipedia entry} lists all of the seminars.

[[Pierre Deligne]] edited a supplementary volume \emph{SGA $4\frac{1}{2}$} comprisings parts of [[SGA5]], his own articles proving new results and some substantially rewritten and expanded material exposed in less satisfactory form in SGA5 (these results were used for his proof of the [[Weil conjecture]]s which appeared before SGA5).

An ongoing project aims to retype the SGA volumes in TeX. So far the following volumes have been retyped:

\begin{itemize}%
\item \href{http://arxiv.org/abs/math/0206203}{SGA 1} - Edited by B. Edixhoven, 2004
\item \href{http://arxiv.org/abs/math/0511279}{SGA 2} - Edited by Y. Laszlo, 2004
\item \href{http://webusers.imj-prg.fr/~patrick.polo/SGA3/}{SGA 3} - Edited by P. Gilles and P. Polo, 2011
\item SGA 4 - Edited by Y. Laszlo. \href{http://www.math.polytechnique.fr/~laszlo/sga4/SGA4-1/sga41.pdf}{Part 1}, \href{http://www.math.polytechnique.fr/~laszlo/sga4/SGA4-2/sga42.pdf}{Part 2}, \href{http://www.math.polytechnique.fr/~laszlo/sga4/SGA4-3/sga43.pdf}{Part 3}.
\item \href{https://github.com/dkmiller/sga4.5}{SGA $4{1\over2}$} - Edited by Daniel Miller.

\end{itemize}
See also:

\begin{itemize}%
\item \href{http://fabrice.orgogozo.perso.math.cnrs.fr/SGA4/index.html}{SGA 4} Edited by F. Orgogozo

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

There are Wikipedia entries

\begin{itemize}%
\item Wikipedia, \href{http://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique}{\'E{}l\'e{}ments de g\'e{}om\'e{}trie alg\'e{}brique}


\item Wikipedia, \href{http://en.wikipedia.org/wiki/S%C3%A9minaire_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique}{S\'e{}minaire de G\'e{}om\'e{}trie Alg\'e{}brique du Bois Marie}


\item Wikipedia, \href{http://en.wikipedia.org/wiki/Fondements_de_la_G%C3%A9ometrie_Alg%C3%A9brique}{Fondements de la G\'e{}ometrie Alg\'e{}brique}



\end{itemize}
EGA and the other published texts of Grothendieck are available from

\begin{itemize}%
\item [[Grothendieck Circle]], \href{http://www.math.jussieu.fr/~leila/grothendieckcircle/pubtexts.php}{Scans of published texts}.

\end{itemize}
and some of the EGA and SGA links are at this \href{http://www.cmls.polytechnique.fr/perso/chambert/DJVU/index.html}{html}.

David Madore has prepared a detailed table of contents of EGA:

\begin{itemize}%
\item \href{https://math.berkeley.edu/~mhaiman/math256-fall13-spring14/EGA-contents.pdf}{Detailed table of contents of EGA}

\end{itemize}
A translated table of contents has been prepared by Mark Haiman, available at

\begin{itemize}%
\item [[Mark Haiman]], \href{http://math.berkeley.edu/~mhaiman/math256/EGA-contents.pdf}{English translation of table of contents of EGA}

\end{itemize}
along with synopsises, in English, of many sections at

\begin{itemize}%
\item [[Mark Haiman]], \href{https://math.berkeley.edu/~mhaiman/math256-fall18-spring19/#ega}{English synopsises of most of EGA I and II}

\end{itemize}
Daniel Murfet has written notes for parts of EGA, see

\begin{itemize}%
\item [[Daniel Murfet]], \href{http://therisingsea.org/?page_id=3}{The Rising Sea}

\end{itemize}
[[!redirects SGA]] [[!redirects FGA]] [[!redirects Éléments de géométrie algébrique]] [[!redirects Elements de geometrie algebrique]] [[!redirects Elements of algebraic geometry]]

category: reference



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