diff -urN -X dontdiff stacks-0.2.orig/src/desirables.tex stacks-0.2/src/desirables.tex --- stacks-0.2.orig/src/desirables.tex Tue Sep 27 09:40:53 2005 +++ stacks-0.2/src/desirables.tex Mon Oct 17 10:59:55 2005 @@ -96,7 +96,8 @@ OK, here are some ideas about how to write a text about stacks, what should go into it, and fixing ideas for some of the definitions. Please email comments to the -\href{http://www.math.columbia.edu/mailman/listinfo/algebraic_geometry}{mailing list}. +\href{http://www.math.columbia.edu/mailman/listinfo/algebraic_geometry}% +{mailing list}. \end{abstract} \maketitle diff -urN -X dontdiff stacks-0.2.orig/src/documentation/dontdiff stacks-0.2/src/documentation/dontdiff --- stacks-0.2.orig/src/documentation/dontdiff Wed Dec 31 19:00:00 1969 +++ stacks-0.2/src/documentation/dontdiff Mon Oct 17 10:59:55 2005 @@ -0,0 +1,15 @@ +*.aux +*.bbl +*.blg +*.dvi +*.log +*.pdf +*.ps +*.out +*.toc +*.html +*~ +.* +*.patch +*.gz +*.bz2 diff -urN -X dontdiff stacks-0.2.orig/src/documentation/submitting-patches stacks-0.2/src/documentation/submitting-patches --- stacks-0.2.orig/src/documentation/submitting-patches Wed Dec 31 19:00:00 1969 +++ stacks-0.2/src/documentation/submitting-patches Mon Oct 17 10:59:55 2005 @@ -0,0 +1,45 @@ +How to get your changes into the project +---------------------------------------- + + +Certainly you can simply edit any of the TeX files and email the new version to +the mailing list algebraic_geometry@math.columbia.edu (subscribe first please). + + +On the other hand it is sometimes convenient to send only the difference +between the new version and the old one. Please use "diff -u" in this case to +produce the patch. It will help if you follow the suggestions below, or do +something with equivalent endproduct. + + +To create a patch for a single file, say stacks.tex, you can do the following. +First download stacks.tex. Make a copy: + + cp stacks.tex stacks.tex.orig + +and then edit stacks.tex as you like. (Or, if you've already edited the file, +you download and save the version from the web directly as stacks.tex.orig.) To +make the patch use + + diff -u stacks.tex.orig stacks.tex > patchfile + +Submit the patchfile to the mailing list; please include PATCH in the subject. + + +To create a patch for the whole project directory, you can do the following. +Download the file stacks-0.2.tar.bz2 to ~ say. To set up a working directory +and a local copy of the original: + + cd ~ + mkdir stacks-0.2 + tar --directory stacks-0.2 -xjf stacks-0.2.tar.bz2 + cp -a stacks-0.2 stacks-0.2.orig + +At this point you edit the files in stacks-0.2/src, run make, etc. When you are +done, produce a patch as follows: + + cd ~ + diff -ruN -X stacks-0.2.orig/src/documentation/dontdiff \ + stacks-0.2.orig stacks-0.2 > patchfile + +Submit the patchfile to the mailing list; please include PATCH in the subject. diff -urN -X dontdiff stacks-0.2.orig/src/introduction.tex stacks-0.2/src/introduction.tex --- stacks-0.2.orig/src/introduction.tex Mon Sep 26 15:24:22 2005 +++ stacks-0.2/src/introduction.tex Mon Oct 17 10:59:55 2005 @@ -121,7 +121,8 @@ \smallskip\noindent To read about this project and its goals please visit the -\href{http://www.math.columbia.edu/mailman/listinfo/algebraic_geometry}{mailing list}. This is also the place to submit patches, etc. +\href{http://www.math.columbia.edu/mailman/listinfo/algebraic_geometry}% +{mailing list}. This is also the place to submit patches, etc. \smallskip\noindent To continue reading, diff -urN -X dontdiff stacks-0.2.orig/src/sites.tex stacks-0.2/src/sites.tex --- stacks-0.2.orig/src/sites.tex Sat Oct 15 15:14:24 2005 +++ stacks-0.2/src/sites.tex Mon Oct 17 10:59:55 2005 @@ -220,44 +220,154 @@ is exact as before. If the products in $(*)$ exist then this condition just means that the first arrow is the equalizer of the other two. -\begin{remark} -A morphism of coverings $\{U_i \to U\}_{i\in I} \to -\{V_j \to V\}_{j\in J}$ is given by -a morphism $U \to V$, a map of sets $\alpha : I \to J$ and +\subsection{More about coverings} +\label{subsection-coverings} + +\noindent +Let $\mathcal{C}$ be a site. A morphism of coverings of $\mathcal{C}$ from +$\mathcal{U}=\{U_i \to U\}_{i\in I}$ to $\mathcal{V}=\{V_j \to V\}_{j\in J}$ +is given by a morphism $U \to V$, a map of sets $\alpha : I \to J$ and for each $i\in I$ a morphism $U_i \to V_{\alpha(i)}$ such that -the diagram +the diagram $$ \xymatrix{ -U_i \ar[r] \ar[d] & V_{\alpha(I)} \ar[d] \\ +U_i \ar[r] \ar[d] & V_{\alpha(i)} \ar[d] \\ U \ar[r] & V } $$ -is commutative. -\end{remark} +is commutative. In the special case that $U=V$ and $U\to V$ is the identity +we call $\mathcal{U}$ a refinement of the covering $\mathcal{V}$. +\smallskip\noindent +Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$, and let +$\mathcal{U}$ be a covering in $\mathcal{C}$ as above. Let us use the +notation $\mathcal{F}(\mathcal{U})$ to indicate the equalizer +$$ +\mathcal{F}(\mathcal{U}) = \{ (s_i)_i \in \prod_i \mathcal{F}(U_i) +\mid \text{pr}_1^\ast s_i = \text{pr}_2^\ast s_j \forall i,j \in I\}. +$$ +There is a canonical map $\mathcal{F}(U) \to \mathcal{F}(\mathcal{U})$. +It is clear that a morphism of coverings $\mathcal{U} \to \mathcal{V}$ +induces commutative diagrams +$$ +\xymatrix{ +& U_i \ar[rr] & & V_{\alpha(i)} \\ +U_i \times_U U_j \ar[rr] \ar[ur] \ar[dr] & & +V_{\alpha(i)}\times_V V_{\alpha(j)} \ar[ur] \ar[dr] & \\ +& U_j \ar[rr] & & V_{\alpha(j)} +}. +$$ +This in turn produces a map $\mathcal{F}(\mathcal{V}) \to +\mathcal{F}(\mathcal{U})$, compatible with the map $\mathcal{F}(V) +\to \mathcal{F}(U)$. + +\begin{lemma} +\label{lemma-indepent-refinement} +Any two morphisms $f,f': \mathcal{U} \to \mathcal{V}$ of coverings +inducing the same morphism $U \to V$ induce the same +map $\mathcal{F}(\mathcal{U}) \to \mathcal{F}(\mathcal{V})$. +\end{lemma} + +\begin{proof} +FIXME. +\end{proof} \noindent -FIXME. Explain about sheafification. - +FIXME. Use the lemma to explain about sheafification. \section{Representable sheaves} \label{section-representable-sheaves} \noindent -FIXME. Talk about representable presheaves, canonical topology and +FIXME. Talk about representable presheaves, canonical topology and representable sheaves. +\section{Morphisms of sites} +\label{section-morphism-sites} + +\noindent +FIXME. Talk about continuous functors, and explain the condition that leads to +the correct functoriality on sheaves (i.e., exactness of the pullback functor). +It makes sense to not always assume this holds. + \section{Topoi} \noindent -FIXME. What are topoi? Do we need them? (Yes.) +The topos associated to a site $\mathcal{C}$ is its ``category'' of sheaves of +sets. Conversely, any topos is equivalent to such a ``category'' of sheaves. +Our conventions do not allow us to talk about topoi. Of course we can choose a +large cardinal $\alpha$ and consider the category of sheaves of sets +$\text{Sh}_\alpha(\mathcal{C})$ contained in $\alpha$, but this does not have +the same flavor. + +\smallskip\noindent +FIXME. What are topoi? What are morphisms of topoi? Do we need them? (Yes, in a +way.) + +\smallskip\noindent +As a result some of the discussion in this project uses sites in places where +it might be more convenient to use the language of topoi. We discuss a few +of these ``inconveniences'' in this section. + +\subsection{Sites and points} +\label{subsection-points} + +\noindent +A point of a topos $\mathcal{S}$ is a morphism of topoi from $\text{Sets}$ to +$\mathcal{S}$. As discussed above we do not use this definition. In stead, we +somewhat akwardly define a point as follows. A point is a functor +$p : \mathcal{C} \to \text{Sets}$ such that +\begin{enumerate} +\item if $V\times_U W$ exists then $p(V\times_U W)=p(V)\times_{p(U)}p(W)$, +\item if $\{U_i \to U\}$ is a covering, then $\coprod_i p(U_i) \to p(U)$ is +surjective, +\item for any $x\in p(U)$ and $y\in p(V)$ there exists a $z\in p(W)$ and +morphisms $\alpha:W \to U$, $\beta:W \to V$ such that $p(\alpha)(z)=x$, +and $p(\beta)(z)=y$, and +\item for any pair of morphisms $f,g : V \to U$, and $y\in p(V)$ such +that $p(f)(x)=p(g)(x)$, there exists a $h: W \to V$, $z\in p(W)$ such that +$p(h)(z)=y$ and $g\circ h = f \circ h$. +\end{enumerate} +Once we have this, then we can define the stalk of a (pre)sheaf $\mathcal{F}$ +at $p$ as follows +$$ +\mathcal{F}_p = \lim_{(U,x)} \mathcal{F}, +$$ +where the limit is over the category of pairs +$\{(U,x) \mid U \in \text{Ob}(\mathcal{C}), x\in p(U)$. The conditions +above imply this is a FIXME filtered limit. This implies that taking +stalks is an exact functor. FIXME: Need a section on limits. + +\begin{lemma} +\label{lemma-points-recover} +In the situation above we have $p(U) = (U^{++})_p$. FIXME: notation. +\end{lemma} + +\begin{proof} +FIXME. +\end{proof} + +\noindent +We say that a site $\mathcal{C}$ has enough points if the following equivalence +is true for every morphism of sheaves of sets +$\phi : \mathcal{F} \to \mathcal{G}$: +$$ +\phi\ \text{is}\ \text{injective} +\Leftrightarrow +\forall p, \phi_p\ \text{is}\ \text{injective} +$$ +This will then imply the same thing for ``bijective'' and ``surjective'', and +it allows you to check exactness of sequences of sheaves of abelian groups +on stalks. (FIXME: explain?) Often sites that we work with have enough points +and it is easier to work with them, e.g., it is fairly easy to construct +injective sheaves of abelian groups on such a site. \smallskip\noindent -To continue reading, +To continue reading, \begin{enumerate} \item visit the next section: Injectives, -\autoref{injectives-section-introduction}, or +\autoref{injectives-section-introduction}, or \item go back to the table of contents: \url{index.html#contents}. diff -urN -X dontdiff stacks-0.2.orig/src/template.tex stacks-0.2/src/template.tex --- stacks-0.2.orig/src/template.tex Sat Jul 30 13:21:58 2005 +++ stacks-0.2/src/template.tex Mon Oct 17 10:59:55 2005 @@ -113,8 +113,9 @@ \noindent This is an example reference to \autoref{conventions-section-categories} of the chapter Conventions, and here is another: Hypercoverings, -\hyperref[hypercovering-lemma-construct-new-covers]{Lemma~\ref*{hypercovering-lemma-construct-new-covers}}. Yes, if you look at the TeX file you see this is -cumbersome to type but it works. +\hyperref[hypercovering-lemma-construct-new-covers]% +{Lemma~\ref*{hypercovering-lemma-construct-new-covers}}. Yes, if you look at +the TeX file you see this is cumbersome to type but it works. \subsubsection{Even more nonsense} \label{subsubsection-nonsense}