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# Rules and indications

First of all, we want a logical source file, that any user can easily adapt to his wishes (use of other page size, other fonts, just to name some examples). The layout at this moment is not so important. Hence one should try to keep the typesetting commands as simple and as standard as possible, and preferably no spacing commands at all should be done by hand'.

We want the participants to understand what they are typing. A good reason for this is that one has to know what types of objects one is dealing with, in order to choose the proper font. For example, an underlined F that represents a sheaf should be typeset as \mathcal{F}, and a handwritten X that denotes a formal scheme should be typeset as \goth{X}.

## Typesetting rules

Do not edit outside your part unless you have a good reason. It won't have any effect anyway, as I will copy-paste only your parts into the master file. So, in fact, you are free to edit the header, but keep in mind that what you type has to work with the header of the skeleton.

Do not add macros (i.e., \newcommands etc.). If you want one, ask me (email) to put one in the header of the skeleton. See here for the list of macros in the skeleton. Of course, you can also just read the header of the skeleton. Most notations in math mode that are based on abbreviations (to be typeset in roman) such as an' (for analytification), Gal' (for Galois group) are in the list of macros (\an and \Gal, for example). In such cases the name of the macro is the piece of text that should appear, unless this is impossible (e.g., \divisor for div' for divisor of a function). If such a notation is not available as a macro, use \mathrm{...} or ask me to make a macro.

Text that is emphasized, either by underlining inside a context that is not underlined, or not underlined in a context that is underlined, should be typeset using \emph{}. In the last case, the doubly emphasized text will appear upright in a slanted context. For environments such as theorems and such, the emphasizing is automatic. Italic corrections are automatic, using \emph{}.

Formulas. Underlined symbols in formulas have to be typeset in a suitable font. As a general rule, this means bold, script (i.e., calligraphic), gothic, or blackboard bold, depending on what type of object is concerned. See here for a list of types of objects and how they should be typeset, and here for a dvi file explaining how to typeset the notations from the index des notations' of SGA 1.

Use \begin{enumerate}\item[...] ..... \end{enumerate} for lists within theorems and such, when appropriate.

For spacing in math mode, if necessary, use \,', \:', \;', \quad' or \qquad'. Example: in a displayed equation, with some text on the same line, you can have ...., \qquad\text{avec $S'=...$)'.

For the colon in the notation f: X --> Y' use \colon (e.g., f\colon X\to Y).

(Co)Homology. Use the macro \H. Example: \H^1(X,\mathcal{F}). Similarly, for right derived functors, use \R.

For footnotes (they appear as (*) in the text) use the \footnote{} command. Footnotes are then numbered within each exposé.

Inverse and direct limits: \varinjlim and \varprojlim can be used, together with \underset. For example: \underset{i}{\varinjlim}.

For the squiggly arrow (\leadsto in TeX) that indicates what a map does with elements of a set: use \mapsto. Example: $f\colon X \to Y$, $x\mapsto f(x)$.

Spectral sequence notation: use \Rightarrow, or \Longrightarrow, or their left versions.

Internal references in the book: use \ref{}. For example: \ref{I.3.7}, which produces 3.7'. Note that all numbered sections and theorems and such have been labeled in this format: the number of the exposé in Roman, followed by the numbers given in the book, separated by dots. However, note that the number produced by \ref does not contain that of the chapter. So if you want to get I.3.7', then you type I.\ref{I.3.7}'.

References to the bibliography of the exposé: type by hand: [1] for example.

Opposite category: use ^\circ. Example: \cal{C}^\circ.

Use \ell' for handwritten or script l's, and l' for ordinary typed l's.

For blackboard bold mu (used for roots of unity), use \Bbb{\mu} (this gives the same result as \mu, but that might improve some day).

Use \No for N^o' (N with superscript o) used for Numéro'. Example: \No~2.

Use ties to prevent line breaks at all inappropriate places (remember that a user should be able to modify the page width and still get a reasonable result). Example: ......de $B$ sur~$A$.' will not be broken between sur' and $A$'.

For left superscripts there is a macro \leftexp (see below).

If you have a question, send me an email (or first ask somebody that you have around). For example, I can imagine that one hesitates with the dual of \mathit{\Omega}_{X/S} on page 73, which I think should be \goth{g}_{X/S}.

## Typesetting indications

A word about documentation for TeX and LaTeX. The main source is the CTAN archive. Each reasonable TeX distribution should come with a lot of documentation, directly available on your computer. For example, on my computer, the documentation of the teTeX distribution is accessible from /usr/share/texmf/doc/index.html. For your convenience, here is a link to a full reference manual of LaTeX2e commands in dvi format.

Accented characters. Since we are typing a french text, it is easier to type the accented characters as such. Even when using a keyboard without accented characters, you can type them in emacs using the iso-accents-mode (the use of the package isolatin1 makes it possible to run latex on such a source file). You can switch back and forth between between TeX format ({\'e}) and iso-format (é) with the commands iso-tex2iso and iso-iso2tex.

• babel, with option francais. This selects the french style of typesetting. For example, a space is generated automatically between the end of a word and a punctuation character; hence one should type question?' and not question ?' or (the more robust) question~?'.
• isolatin1. This makes it possible to use accented characters (iso-8859-1 encoded) in the source file. In case your TeX distribution does not have this, here it is.
• amsmath. For documentation, see Chapter 8 of The LaTeX Companion of which a revised version is available online in pdf form. Another reference is the User's Guide. For a piece of text in math mode one can use \text{}.
• amssymb. This package contains the amsfonts package. See here for a table of many available mathematical symbols in dvi format.
• amscd. To make simple commutative diagrams. For documentation, see section 8 of the User's Guide mentioned just above.
• XY-pic. To make more complicated diagrams. See the XY-pic User's Guide or the XYpic reference manual in ps format for documentation. Here are two examples, with double and triple arrows between two objects (useful for diagrams involving descent):
      \xymatrix{S \ar[r] & X \ar@<2pt>[r]\ar@<-2pt>[r] & Y
\ar@<4pt>[r] \ar[r] \ar@<-4pt>[r] & Z}

\xymatrix{X\times Y\times
Z\ar@<8pt>[r]^-{f}\ar@<-2pt>[r]^-{g}\ar@<-6pt>[r]_-{h} &S}

• a4wide. To use the paper more efficiently. If your TeX distribution does not have this, here it is.
• mathrsfs and times. For fonts. If your TeX distribution does not have this, please look at the CTAN archive how to install it. And complain to your distributor, especially if you had to pay!

A few words on fonts in LaTeX2e. A font consists of three attributes: a family (rm (roman), ssf (sans serif) or tt (typewriter)), a shape (up (up), it (italic), sl (slanted), sc (small caps)), and a series (md (medium), bf (bold face)). Each command such as \textrm, \textup, \textmd changes just one of the three attributes. For example, the \textup{} command in $\Hom_\text{\textup{$k$-alg}}(A,B)$ ensures that alg' is typeset in upshape, even if the surrounding environment is slanted (as in a theorem).

In math mode, the commands that change the font attributes are of the form \mathrm, \mathit, etc. For convenience, the old LaTeX 2.09 commands \rm, \it, \bf and \cal have been redefined to be \mathrm, \mathit, etc. Note that they have to be used as \rm{Hom}' and not as {\rm Hom}' (in the last case, only the H of Hom will be in roman). Their use in textmode will now lead to an error message.

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## List of macros

### Macros based on abbreviations


### Other macros

 \leftexp \newcommand{\leftexp}[2]{{\vphantom{#2}}^{#1}{#2}} \leftexp{e}{F} produces a left superscript eF

### Some extra symbols


### Symbols in blackboard bold

The macro \Bbb is defined as follows: \renewcommand{\Bbb}{\mathbb}.


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## List of types of objects, their corresponding font (command), and examples

Here are rules for choosing the font for underlined or handwritten symbols in formulas. If a symbol is not underlined, then it is to be typeset in standard math mode (i.e., do nothing to change the font).

 formal scheme \goth \goth{X} ideals in rings, in lower case \goth \goth{n}, \goth{p} sheaf \cal \cal{F} greek' sheaf \it \it{\Omega} category \cal \cal{E}, unless standard such as \Ens, \Sch stack (champ) \cal \cal{F} analytic space \goth \goth{X}

## A few special remarks.

Notation for fundamental groups. Most handwritten pi's in the text should become \pi', but some, also handwritten, wider and straighter (see page 132) should become \Pi'. The difference is that \pi is a topological group, whereas \Pi is a pro-group in a galois category.

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