diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/categories.tex stacks-0.2/src/categories.tex --- stacks-0.2.orig/src/categories.tex 2006-02-08 14:12:07.000000000 +0000 +++ stacks-0.2/src/categories.tex 2006-02-10 13:42:42.000000000 +0000 @@ -587,7 +587,8 @@ $$ \xymatrix{ B' \ar[r]^{g'} & T' & & B \ar[r]^g & T & \\ -A' \ar@{-->}[u]^{??} \ar[ru]_{h'} & & \ar@{}[u]^{above} & A \ar[u]^f \ar[ru]_{gf = h} & \\ +A' \ar@{-->}[u]^{??} \ar[ru]_{h'} & & \ar@{}[u]^{above} & +A \ar[u]^f \ar[ru]_{gf = h} & \\ } $$ \end{example} @@ -702,7 +703,8 @@ \smallskip\noindent Finally, we can define pullback functors on $\mathcal{S}'$ by setting $g^\ast(x,f) = (x, f \circ g)$ on objects if $g : V' \to V$ and -$f : V \to U$. On morphisms $(\varphi,\text{id}_V) : (x_1, f_1) \to (x_2,f_2)$ between morphisms in $\mathcal{S}'_V$ we set $g^\ast(\varphi,\text{id}_V) = +$f : V \to U$. On morphisms $(\varphi,\text{id}_V) : (x_1, f_1) \to (x_2,f_2)$ +between morphisms in $\mathcal{S}'_V$ we set $g^\ast(\varphi,\text{id}_V) = (g^\ast\varphi, \text{id}_{V'})$ where we use the unique identifications $g^\ast f_i^\ast x_i = (f_i \circ g)^\ast x_i$ from Lemma \ref{lemma-fibred-groupoids} to think of $g^\ast\varphi$ as a morphism from diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/sites.tex stacks-0.2/src/sites.tex --- stacks-0.2.orig/src/sites.tex 2006-02-08 05:14:07.000000000 +0000 +++ stacks-0.2/src/sites.tex 2006-02-13 23:42:57.000000000 +0000 @@ -386,36 +386,59 @@ functorial morphism $\mathcal{F} \rightarrow \mathcal{F}^\#$ and such that for any morphism from $\mathcal{F}$ to an abelian sheaf $\mathcal{G}$ there is a unique factorization $\mathcal{F} \rightarrow -\mathcal{F}^\# \rightarrow \mathcal{G}$. $\mathcal{F}^\#$ will be +\mathcal{F}^\# \rightarrow \mathcal{G}$. The sheaf $\mathcal{F}^\#$ will be called the sheafification of $\mathcal{F}$. +\smallskip\noindent +FIXME: Move the following stuff on limits to another file, and make it +more general. Find nice lim symbol anyone? + \begin{definition} A directed set is a set $S$ together with a relation $\geq$ which is transitive and reflexive such that for $a, b \in S$ there exists another element $c \in S$ such that $c \geq a$ and $c \geq b$. \end{definition} + \noindent -Let $\mathcal{J}_U$ be the set of all coverings of $U$. It is not hard to check -that $\mathcal{J}_U$, along with all possible refinements of coverings, -is a category. It is also clear that given any two coverings of $U$, $\mathcal{U}_1$ -and $\mathcal{U}_2$, there is another covering refining them both. -That is, the covering $\{U_{1i} \times_U U_{2j} \rightarrow U\}$ is a cover of $U$ and the -natural projection maps give the refinements. -(Note: it is exactly conditions 2 and 3 in \ref{definition-site} that allow us to -know that this will be a cover). -Now if $\mathcal{F}$ is a functor from $\mathcal{C} \rightarrow \mathcal{A}b$ then we can -define the direct limit relative to a directed set $I$ to be $\stackrel{\lim}{\rightarrow}\mathcal{F}(U_i)$ to be -$\bigoplus \mathcal{F}(U_i) / \Box$ where $\Box$ is generated by the relations $m_i = \sigma_{ij}(m_i)$ -where $m_i \in \mathcal{F}(U_i)$ and $\sigma_{ij}: \mathcal{F}(U_i) \rightarrow \mathcal{F}(U_j)$. -Now, by the above remarks, we see that $\mathcal{J}_U$ is a directed set, and so we can take -the direct limit over the set of coverings of $U$. -We let $\mathcal{F}^\dagger(U) := \stackrel{\lim}{\rightarrow}\mathcal{F}(\mathcal{U})$ +A directed system over a directed set $S$, is given by a set $M_s$ for +every $s\in S$ and a map $M_a \to M_b$ for every pair $b\geq a$ such +that all the composition $M_a \to M_b \to M_c$ equals the map +$M_a \to M_c$ whenever $c \geq b \geq a$. The limit of the directed system +is the set $\lim_{s\in S} M_s = \big(\coprod_{a\in S} M_a\big)/\sim$. Here, +if $m\in M_a$ and $m'\in M_{a'}$, then $m \sim m'$ if and only if $m$ and $m'$ +map to the same element in some $M_b$ for some $b$ with $b\geq a$ and +$b \geq a'$. If the system is in the category of abelian groups then the +limit has the structure of an abelian group. + +\noindent +Let $\mathcal{J}_U$ be the set of all coverings of $U$. It is not hard to +check that $\mathcal{J}_U$ with morphisms being morphisms of coverings +over $U$, is a category. It is also clear that given any two coverings of $U$, +$\mathcal{U}_1$ and $\mathcal{U}_2$, there is another covering refining them +both. That is, the covering $\{U_{1i} \times_U U_{2j} \rightarrow U\}$ is a +cover of $U$ and the natural projection maps give the refinements: it is +exactly conditions (2) and (3) in \ref{definition-site} that allow us to conclude +that this is a cover. Now, by the above remarks, we see that $\mathcal{J}_U$ +is a directed set, where we say that $\{U_i \to U\} \geq \{V_j \to U\}$ if +and only if $\{U_i \to U\}$ is a refinement of $\{V_j \to U\}$. Lemma +\ref{lemma-indepent-refinement} tells us that $\mathcal{U} \mapsto +\mathcal{F}(\mathcal{U})$ is a directed system over $\mathcal{J}(U)$, if +we define, for $\mathcal{U} \geq \mathcal{V}$ the map +$\mathcal{F}(\mathcal{U}) \to \mathcal{F}(\mathcal{V})$ to be induced from any +morphism of coverings of $U$. Hence we can take the direct limit over the set +of coverings of $U$. Thus we define +$$ +\mathcal{F}^\dagger(U) = \lim_{\mathcal{U}}\mathcal{F}(\mathcal{U}) +$$ where the limit is over the directed set of coverings $\mathcal{J}_U$. -\\Note that it is exactly \ref{lemma-indepent-refinement} which makes this definition make sense. -\\Remark: This is also denoted $\check{H}^o(U, F)$. It corresponds to taking the directed limit of global sections -over all coverings of $U$, ie, the 0th \v{C}ech cohomology group. -\\We say that $\mathcal{F}$ is separated if, for all coverings of $U, \{U_i \rightarrow U$ the map +This is sometimes denoted $\check{H}^0(U, F)$, ie, the $0$th \v{C}ech +cohomology group. + +\smallskip\noindent +Finally, we say that $\mathcal{F}$ is separated if, for all coverings +of $U$, $\{U_i \rightarrow U$ the map $\mathcal(F) \rightarrow \prod \mathcal(F)(U_i)$ is injective. + \begin{theorem} With $\mathcal{F}$ as above \begin{enumerate} @@ -424,10 +447,14 @@ \item $\mathcal{F}^{\dagger\dagger}$ is always a sheaf. \end{enumerate} \end{theorem} + +\begin{proof} +FIXME. +\end{proof} + \noindent -FIXME. PROOF. -\\\noindent -FIXME. Discuss the more general case when $\mathcal{F}$ may not be a sheaf with values in $\mathcal{A}b$. +FIXME. Discuss the more general case when $\mathcal{F}$ may not be a sheaf with +values in $\mathcal{A}b$. \section{Representable sheaves} \label{section-representable-sheaves} @@ -529,8 +556,6 @@ \end{enumerate} - - \bibliography{my} \bibliographystyle{alpha} diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/stacks.tex stacks-0.2/src/stacks.tex --- stacks-0.2.orig/src/stacks.tex 2005-10-15 22:23:19.000000000 +0000 +++ stacks-0.2/src/stacks.tex 2006-02-11 16:45:26.000000000 +0000 @@ -125,11 +125,18 @@ \item $p : \mathcal{S} \to \mathcal{C}$ is a category fibred in groupoids, see Categories, \autoref{categories-definition-fibred-groupoids}, -\item descend for morphisms holds, and +\item descent for morphisms holds, and \item descent data for objects are effective. \end{enumerate} + +\subsection{Explanation and examples} +\label{subsection-defintion-explanation} + +\noindent To explain this, we choose a collection of pullback functors as in Categories, \hyperref[categories-lemma-fibred-groupoids]% +{Lemma~\ref*{categories-lemma-fibred-groupoids}}. Another approach is to use +Categories, \hyperref[categories-lemma-fibred-groupoids]% {Lemma~\ref*{categories-lemma-fibred-groupoids}}. \smallskip\noindent @@ -172,9 +179,35 @@ \end{lemma} \begin{proof} -FIXME: A fun way to do this is to use Categories, -\hyperref[categories-lemma-fibred-strict]% -{Lemma~\ref*{categories-lemma-fibred-strict}} +Let $(g',\text{id}_U) : (f' : V' \to U) \to (f : V\to U)$ and +$(g'',\text{id}_U) : (f'' : V'' \to U) \to (f' : V' \to U)$ be morphisms in +$\mathcal{C}/U$, and let $\phi \in {Mor}_{\mathcal{S}_V}(f^\ast x, f^\ast y)$. +It suffices to show that +$$ +[(\text{Isom}(x,y))(g' \circ g'')](\phi) = +[(\text{Isom}(x,y))(g'')]([(\text{Isom}(x,y))(g')](\phi)) +$$ +By \hyperref[categories-lemma-fibred-groupoids]% +{Lemma~\ref*{categories-lemma-fibred-groupoids}} there are pullback functors +$r: (g' \circ g'')^\ast f'^\ast \longrightarrow f''^\ast$, +$t': g'^\ast f^\ast \longrightarrow f'^\ast$, +$t'': g''^\ast f'^\ast \longrightarrow f''^\ast$ and +$u: g''^\ast g'^\ast \longrightarrow (g' \circ g'')^\ast$. It follows from the +uniqueness part of the lemma that +$t''_{y} \circ g''^\ast t'_{y} = r_{y} \circ u_{f^\ast y}$ and +$g''^\ast t'^{-1}_{x} \circ t''^{-1}_{x} = u^{-1}_{f^\ast x} \circ r^{-1}_{x}$. +We now have that +\begin{eqnarray*} +[(\text{Isom}(x,y))(g' \circ g'')](\phi) & = & +r_{y} \circ (g' \circ g'')^\ast \phi \circ r^{-1}_{x}\\ +& = & r_{y} \circ u_{f^\ast y} \circ +g''^\ast g'^\ast \phi \circ u^{-1}_{f^\ast x} \circ r^{-1}_{x} \\ +& = & t''_{y} \circ g''^\ast t'_{y} \circ +g''^\ast g'^\ast \phi \circ g''^\ast t'^{-1}_{x} \circ t''^{-1}_{x} \\ +& = & t''_{y} \circ g''^\ast (t'_{y} \circ +g'^\ast \phi \circ t'^{-1}_{x}) \circ t''^{-1}_{x} \\ +& = & [(\text{Isom}(x,y))(g'')]([(\text{Isom}(x,y))(g')](\phi)). +\end{eqnarray*} \end{proof} \noindent @@ -186,24 +219,41 @@ $\text{Isom}(x,y)$ is a sheaf on the site $\mathcal{C}/U$! \smallskip\noindent -We still have to explain the meaning of the third condition. -For this we have to explain what a descent datum is. We -introduce some notation. Suppose that $\{f_i : U_i \to U\}_{i\in I}$ -is a convering in the site $\mathcal{C}$. We will be looking at -fibre products $U_{ij} = U_i \times_U U_j$ and fibre products -$U_{ijk} = U_i \times_U U_j \times_U U_k$. We will denote -$\text{pr}_1 : U_{ij} \to U_i$, resp.\ $\text{pr}_1 : -U_{ijk} \to U_i$ the projection onto the first factor, -$\text{pr}_{12} : U_{ijk} \to U_{ij}$ -the projection onto the first and second factor, etc. Let -$x_i \in \text{Ob}(\mathcal{S}_{U_i})$, $i\in I$ be a collection of +In order to explain the meaning of the third condition, we must define a +descent datum. First, we introduce some notation. +Suppose that $\{f_i : U_i \to U\}_{i\in I}$ +is a covering in the site $\mathcal{C}$. We will be looking at +fibre products $U_{ij} = U_i \times_U U_j$ and +$U_{ijk} = U_i \times_U U_j \times_U U_k$. +When three indices, say $i,j$, and $k$, are in play, the projection maps from +the twofold fibre products will be denoted as follows: $\text{pr}_{1/2}$ is +the projection $U_{ij} \to U_i$, $\text{pr}_{3/1}$ is the projection +$U_{ik} \to U_k$, and so on. If only two indices, say $i$ and $j$, are in +play, the projection maps $U_{ij} \to U_i$ and $U_{ij} \to U_j$ will be +denoted by $\text{pr}_{1}$ and $\text{pr}_{2}$, resp.; this should cause no +confusion. The projection maps from $U_{ijk}$ to the twofold fibre products +are $\text{pr}_{12} : U_{ijk} \to U_{ij}$, +$\text{pr}_{13} : U_{ijk} \to U_{ik}$, and +$\text{pr}_{23} : U_{ijk} \to U_{jk}$. Finally, the projection maps from +$U_{ijk}$ to $U_{i}$, $U_{j}$, and $U_{k}$ are ${\widetilde{pr}}_1$, +${\widetilde{pr}}_2$, and ${\widetilde{pr}}_3$, respectively. +(Note the relations ${\widetilde{pr}}_1 = \text{pr}_{1/2} \circ \text{pr}_{12} += \text{pr}_{1/3} \circ \text{pr}_{13}$, +${\widetilde{pr}}_2 = \text{pr}_{2/1} \circ \text{pr}_{12} = +\text{pr}_{2/3} \circ \text{pr}_{23}$, and +${\widetilde{pr}}_3 = \text{pr}_{3/1} \circ \text{pr}_{13} = +\text{pr}_{3/2} \circ \text{pr}_{23}$.) +Let $x_i \in \text{Ob}(\mathcal{S}_{U_i})$, $i\in I$ be a collection of objects, and let $$ -\phi_{ij} : \text{pr}_1^\ast x_i \longrightarrow -\text{pr}_2^\ast x_j , +\phi_{ij} : \text{pr}_{1}^\ast x_i \longrightarrow +\text{pr}_{2}^\ast x_j \qquad (i,j \in I) $$ -$i,j\in I$ be a collection of morphisms in the fibre categories -$\mathcal{S}_{U_{ij}}$. +be a collection of morphisms in the fibre categories +$\mathcal{S}_{U_{ij}}$. In the definition that follows we will abuse notation +by identifying, for instance, ${\widetilde{pr}}_1^\ast x_i$ with both +$\text{pr}_{12}^\ast \text{pr}_{1/2}^\ast x_i$ and +$\text{pr}_{13}^\ast \text{pr}_{1/3}^\ast x_i$. \begin{definition} \label{definition-descent-data} @@ -212,37 +262,44 @@ triple $(i,j,k)\in I^3$ the diagram $$ \xymatrix{ -\text{pr}_1^\ast x_i +{\widetilde{pr}}_1^\ast x_i \ar[rr]^{\text{pr}_{13}^\ast \phi_{ik}} \ar[rd]_{\text{pr}_{12}^\ast \phi_{ij}} & & -\text{pr}_3^\ast x_k \\ -& \text{pr}_2^\ast x_j \ar[ru]_{\text{pr}_{23}^\ast \phi_{jk}} +{\widetilde{pr}}_3^\ast x_k \\ +& {\widetilde{pr}}_2^\ast x_j \ar[ru]_{\text{pr}_{23}^\ast \phi_{jk}} } $$ -is commutative. +in $\mathcal{S}_{U_{ijk}}$ is commutative. \end{definition} \noindent -Let us make a small remark about this definition. There is a -diagonal morphism $\Delta_i : U_i \to U_{ii}$. We can pull back +Two remarks about this definition are in order. + +\begin{itemize} +\item{There is a diagonal morphism $\Delta_i : U_i \to U_{ii}$. We can pull back $\phi_{ii}$ via this morphism to get an automorphism -$\Delta^\ast \phi_{ii} \in \text{Aut}_{U_i}(x_i)$. +${\Delta_i}^\ast \phi_{ii} \in \text{Aut}_{U_i}(x_i)$. On pulling back the cocycle condition for the triple $(i,i,i)$ by $\Delta_{123} : U_i \to U_{iii}$ we deduce that -$\Delta^\ast \phi_{ii} \circ \Delta^\ast \phi_{ii} = -\Delta^\ast \phi_{ii}$, in other words $\Delta^\ast \phi_{ii} = -\text{id}_{x_i}$. Furthermore, there is a morphism +${\Delta_i}^\ast \phi_{ii} \circ {\Delta_i}^\ast \phi_{ii} = +{\Delta_i}^\ast \phi_{ii}$; thus ${\Delta_i}^\ast \phi_{ii} = +\text{id}_{x_i}$.} + +\medskip + +\item{There is a morphism $\Delta_{13}: U_{ij} \to U_{iji}$ and we can pull back the cocycle condition for the triple $(i,j,i)$ to get the -identity $\text{switch}^\ast \phi_{ji} \circ \phi_{ij} = -\text{id}_{x_i}$. Here $\text{switch}$ is the obvious -morphism $U_{ij} \to U_{ji}$. +identity $(\sigma^\ast \phi_{ji}) \circ \phi_{ij} = +\text{id}_{\text{pr}_{1/2}^\ast x_i}$, where $\sigma: U_{ij} \to U_{ji}$ is the +switching morphism.} +\end{itemize} \smallskip\noindent -A morphism of descent data $\{x_i, \phi_{ij}\}$ to -$\{y_i, \psi_{ij}\}$ is given by a collection of -morphisms $\alpha_i : x_i \to y_i$ in the fibre category +A morphism of descent data +$\alpha : \{x_i, \phi_{ij}\} \rightarrow \{y_i, \psi_{ij}\}$ is given by a +collection of morphisms $\alpha_i : x_i \to y_i$ in the fibre category over $U_i$ such that the following diagrams $$ \xymatrix{ @@ -262,17 +319,16 @@ commute. Note that every morphism of descent data is an isomorphism. \smallskip\noindent -An object $x \in \text{Ob}(\mathcal{S}_U)$ gives rise to a descend +An object $x \in \text{Ob}(\mathcal{S}_U)$ gives rise to a descent datum in the following manner. First we set $x_i = f_i^\ast x$. -Second we let $\phi_{ij}$ be the canonical isomorphism -$$ -\text{pr}_1^\ast x_i = \text{pr}_1^\ast f_i^\ast x -= (f_i \circ \text{pr}_1)^\ast x = -(f_j \circ \text{pr}_2)\ast x = -\text{pr}_2^\ast f_j^\ast x = \text{pr}_2^\ast x_j. -$$ +Then we set $\phi_{ij} = {t_{j}}^{-1} \circ t_{i}$, where +$t_{i}: \text{pr}_1^\ast f_i^\ast x \to (f_i \circ \text{pr}_1)^\ast x$ +and $t_{j}: \text{pr}_2^\ast f_j^\ast x \to (f_j \circ \text{pr}_2)^\ast x$ +are the canonical isomorphisms guaranteed by Categories, +\hyperref[categories-lemma-fibred-groupoids]% +{Lemma~\ref*{categories-lemma-fibred-groupoids}}. The lemma below shows this is a descent datum; we will call -this the canonical descent datum associated to $x$. +this the {\it canonical descent datum} associated to $x$. \begin{lemma} \label{lemma-trivial-cocycle} @@ -280,9 +336,12 @@ \end{lemma} \begin{proof} -FIXME: Use Categories, -\hyperref[categories-lemma-fibred-strict]% -{Lemma~\ref*{categories-lemma-fibred-strict}} +First, note that $f_i \circ {\widetilde{pr}}_1 = f_j \circ {\widetilde{pr}}_2= +f_k\circ {\widetilde{pr}}_3$. Then note that ${\text{pr}_{13}^\ast \phi_{ik}}$, +${\text{pr}_{23}^\ast \phi_{jk}}$ and ${\text{pr}_{12}^\ast \phi_{ij}}$ factor +uniquely through $(f_i \circ {\widetilde{pr}}_1)^\ast x = +(f_j \circ {\widetilde{pr}}_2)^\ast x = (f_k \circ {\widetilde{pr}}_3)^\ast x$ +by Lemma 3.1.3. \end{proof} \begin{definition}