ega1.tex

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\title{The language of schemes (EGA I)}
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\label{section:ega1}

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\section*{Summary}

\begin{longtable}{ll}
  \textsection\hyperref[section:I.1]{1}.   & Affine schemes.\\
  \textsection\hyperref[section:I.2]{2}.   & Preschemes and morphisms of preschemes.\\
  \textsection\hyperref[section:I.3]{3}.   & Products of preschemes.\\
  \textsection\hyperref[section:I.4]{4}.   & Subpreschemes and immersion morphisms.\\
  \textsection\hyperref[section:I.5]{5}.   & Reduced preschemes; the separation condition.\\
  \textsection\hyperref[section:I.6]{6}.   & Finiteness conditions.\\
  \textsection\hyperref[section:I.7]{7}.   & Rational maps.\\
  \textsection\hyperref[section:I.8]{8}.   & Chevalley schemes.\\
  \textsection\hyperref[section:I.9]{9}.   & Supplement on quasi-coherent sheaves.\\
  \textsection\hyperref[section:I.10]{10}. & Formal schemes.
\end{longtable}
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\oldpage[I]{79}
In \textsection\textsection1--8 we do little more than develop a language to be used in what follows.
It should be noted, however, that, in accordance with the general spirit of this treatise, \textsection\textsection7--8 will be used less than the others, and in a less essential way; we speak of Chevalley schemes only for the purpose of relating to the language of Chevalley \cite{I-1} and Nagata \cite{I-9}.
Then, in \textsection9, we give definitions and results concerning quasi-coherent sheaves, some of which are no longer simply a translation of known notions of commutative algebra into a ``geometric'' language, but are instead already of a global nature; they will be indispensable, in the following chapters, when it comes to the global study of morphisms.
Finally, in \textsection10, we introduce a generalization of the notion of a scheme, which will be used as an intermediary in Chapter~III to conveniently formulate and prove the fundamental results of the cohomological study of proper morphisms;
moreover, it should be noted that the notion of formal schemes seems indispensable in expressing certain facts about the ``theory of modules'' (classification problems of algebraic varieties).
The results of \textsection10 will not be used before \textsection3 of Chapter~III, and it is recommended to skip their reading until then.
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