diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/Makefile stacks-0.2/src/Makefile --- stacks-0.2.orig/src/Makefile 2006-03-05 03:48:21.000000000 +0000 +++ stacks-0.2/src/Makefile 2006-03-05 15:12:12.000000000 +0000 @@ -1,9 +1,9 @@ .SUFFIXES: .aux .bbl .bib .blg .dvi .html .log .out .pdf .ps .tex .toc -PDFS = conventions.pdf sites.pdf introduction.pdf categories.pdf hypercovering.pdf desirables.pdf injectives.pdf stacks-groupoids.pdf sets.pdf fdl.pdf stacks.pdf etale.pdf -DVIS = conventions.dvi sites.dvi introduction.dvi categories.dvi hypercovering.dvi desirables.dvi injectives.dvi stacks-groupoids.dvi sets.dvi fdl.dvi stacks.dvi etale.dvi -PSS = conventions.ps sites.ps introduction.ps categories.ps hypercovering.ps desirables.ps injectives.ps stacks-groupoids.ps sets.ps fdl.ps stacks.ps etale.ps -AUXS = conventions.aux sites.aux introduction.aux categories.aux hypercovering.aux desirables.aux injectives.aux stacks-groupoids.aux sets.aux stacks.aux etale.aux -TOCS = conventions.toc sites.toc introduction.toc categories.toc hypercovering.toc desirables.toc injectives.toc stacks-groupoids.toc sets.toc stacks.toc etale.toc +PDFS = conventions.pdf sites.pdf introduction.pdf categories.pdf hypercovering.pdf desirables.pdf injectives.pdf stacks-groupoids.pdf sets.pdf fdl.pdf stacks.pdf etale.pdf flat.pdf +DVIS = conventions.dvi sites.dvi introduction.dvi categories.dvi hypercovering.dvi desirables.dvi injectives.dvi stacks-groupoids.dvi sets.dvi fdl.dvi stacks.dvi etale.dvi flat.dvi +PSS = conventions.ps sites.ps introduction.ps categories.ps hypercovering.ps desirables.ps injectives.ps stacks-groupoids.ps sets.ps fdl.ps stacks.ps etale.ps flat.ps +AUXS = conventions.aux sites.aux introduction.aux categories.aux hypercovering.aux desirables.aux injectives.aux stacks-groupoids.aux sets.aux stacks.aux etale.aux flat.aux +TOCS = conventions.toc sites.toc introduction.toc categories.toc hypercovering.toc desirables.toc injectives.toc stacks-groupoids.toc sets.toc stacks.toc etale.toc flat.toc HTMLS = stacks.html contents.html downloads.html # Files in INSTALLDIR will be overwritten. diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/flat.tex stacks-0.2/src/flat.tex --- stacks-0.2.orig/src/flat.tex 1970-01-01 00:00:00.000000000 +0000 +++ stacks-0.2/src/flat.tex 2006-03-05 15:12:12.000000000 +0000 @@ -0,0 +1,487 @@ +\documentclass{amsart} +\usepackage[all]{xy} + +% To put source file link in headers. +% Change "template.tex" to "this_filename.tex" + +\usepackage{fancyhdr} +\pagestyle{fancy} +\lhead{} +\chead{} +\rhead{Source file: \url{src/flat.tex}} +\lfoot{} +\cfoot{\thepage} +\rfoot{} +\renewcommand{\headrulewidth}{0pt} +\renewcommand{\footrulewidth}{0pt} +\renewcommand{\headheight}{12pt} + +% For cross-file-references +\usepackage{xr-hyper} + +% Package for hypertext links: +\usepackage[colorlinks=true]{hyperref} +% For any local file, say "hello.tex" you want to refer to please use +% \externaldocument[hello-]{hello} +\externaldocument[conventions-]{conventions} +\externaldocument[hypercovering-]{hypercovering} +\externaldocument[sites-]{sites} +\externaldocument[categories-]{categories} +\externaldocument[stacks-groupoids-]{stacks-groupoids} +\externaldocument[desirables-]{desirables} + +% The macro \autoref uses the macros \figurename, etc. +% We list the default values and we change some of them +% to start with a captial. +% Figure \figurename +% Table \tablename +% Part \partname +% Appendix \appendixname +% Equation \equationname +% item \Itemname +% \renewcommand{\Itemname}{Item} +\renewcommand{\Itemautorefname}{Item} +% chapter \Chaptername +% \renewcommand{\Chaptername}{Chapter} +% \renewcommand{\Chapterautorefname}{Chapter} +% section \sectionname +\renewcommand{\sectionname}{Section} +\renewcommand{\sectionautorefname}{Section} +% subsection \subsectionname +\renewcommand{\subsectionname}{Subsection} +\renewcommand{\subsectionautorefname}{Subsection} +% subsubsection \subsubsectionname +\renewcommand{\subsubsectionname}{Subsubsection} +\renewcommand{\subsubsectionautorefname}{Subsubsection} +% paragraph \paragraphname +\renewcommand{\paragraphname}{Paragraph} +\renewcommand{\paragraphautorefname}{Paragraph} +% footnote \Hfootnotename +% \renewcommand{\Hfootnotename}{Footnote} +\renewcommand{\Hfootnoteautorefname}{Footnote} +% Equation \AMSname +% Theorem \theoremname + + +% Theorem environments. +% +\newtheorem{theorem}{Theorem}[subsection] +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{lemma}[theorem]{Lemma} + +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{example}[theorem]{Example} +\newtheorem{exercise}[theorem]{Exercise} +\newtheorem{situation}[theorem]{Situation} + +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\newtheorem{remarks}[theorem]{Remarks} + +\numberwithin{equation}{subsection} + +\begin{document} + +\title{Flat Descent for Quasi-Coherent sheaves} + +\maketitle +\thispagestyle{fancy} + +\tableofcontents + +\section{Introduction} +\label{section-introduction} + +\noindent +In this chapter we discuss the flat topology. + +\subsection{Descent for quasi-coherent sheaves} +\label{subsection-equivalence} + +\noindent +Let me state our goal now so we have a clear idea of what we are trying to +achieve. It is to show that the fibered category $(QCoh/S)$ of quasi-coherent +sheaves over a scheme $S$ over $(Sch/S)$ the category of schemes over $S$ is +a stack with respect to the flat topology. The idea is that we will be able +to exploit the fact that there is a standard equivalance of categories between +$QCoh (U)$ and $Mod_A$ where $U = \text{Spec}(A)$, and use the following lemma +which I will state without proof. I refer the reader to Vistoli's notes +\cite{Vis2}[Lemma 4.25, page 94] for the proof. + +\begin{lemma} +\label{lemma-zariski-flat} +Let $S$ be a scheme, $\mathcal F$ be a fibered category ovver the category +$(Sch/S)$. Suppose the following conditions are satisfied. +\begin{enumerate} +\item $\mathcal F$ is a stack with respect to the Zariski topology. +\item Whenever $V \rightarrow U$ is a flat surjective morphism of affine +$S$-schemes, the functor +$\mathcal F(U) \rightarrow \mathcal F(V \rightarrow U)$ is an equivalence of +categories. +\end{enumerate} +Then $\mathcal F$ is a stack with respect to the flat (fpqc) topology. +\end{lemma} + +\noindent +Note that the fpqc topology stands for "fidelment plat et quasi-compact" which +means faithfully flat and quasi-compact. This is a finer topology than the +fppf topology which is a finer topology than the etale topology. + +\smallskip\noindent +In the case of $(QCoh/S)$ over $(Sch/S)$ we can easily see that the first +condition is satisfied (i.e in the Zariski topology our definition for +quasi-coherent sheaves is exactly the conditions needed for the descent datum +to be effective). + +\smallskip\noindent +For the second condition, it will be necessary for me to define certain +notions and to prove certain algebraic results. + +\smallskip\noindent +First we begin with some definitions. + +\begin{definition} +\label{definition-faithfully-flat} +A morphism of schemes $f: X \rightarrow Y$ is faithfully flat if it is flat +and surjective. Let $B$ be an algebra over $A$, we say that $B$ is +faithfully flat if the associated morphism of schemes +$\text{Spec} B \rightarrow \text{Spec} A$ is. +\end{definition} + +\begin{proposition} +\label{proposition-faithfully-flat} +Let $B$ be an algebra over $A$, the following are equivalent. +\begin{enumerate} +\item $B$ is faithfully flat over $A$. +\item A sequence of $A$-modules $M' \rightarrow M \rightarrow M''$ is exact +if and only if the induced sequence of $B$-modules +$M' \otimes_A B \rightarrow M \otimes_A B \rightarrow M'' \otimes_A B$ +is exact. +\item A homomorphism of $A$ modules $M' \rightarrow M$ is injective if and +only if the associated homomorphism of $B$-modules +$M' \otimes_A B \rightarrow M \otimes_A B$ is injective. +\item $B$ is flat over $A$, and if $M$ is a module over $A$ with +$M \otimes_A B = 0$, we have $M = 0$. +\item $B$ is flat over $A$, and $mB \neq B$ for all maximal ideals $m$ +of $A$. +\end{enumerate} +\end{proposition} + +\begin{definition} +\label{definition-category-descent-datum} +Let $\mathcal C$ be a site. Let $\mathcal F$ be a category fibered over +$\mathcal C$. Let $\mathcal U = \{\sigma_i : U_i \rightarrow U\}$ be a +covering in $\mathcal C$. An object with descent data +$(\{\xi_i\}, \{\phi_{ij}\})$ on $\mathcal U$ is a collection of objects +$\xi_i \in \mathcal F(U_i)$, together with isomorphisms +$\phi_{ij}: pr_2^*\xi_j \simeq pr_1^*\xi_i$ in +$\mathcal F(U_i \times_U U_j)$, such that the following cocyle condition +is satisfied: for any triple of indicies $i, j, k$ we have the equality +$pr_{13}^*\phi_{ik} = pr_{12}^*\phi_{ij} \circ pr_{23}^*\phi_{jk}: +pr_3^*\xi_k \rightarrow pr_1^*\xi_i$. Where $pr_{ab}$ and $pr_a$ +projections onto the $a^{th}$ and $b^{th}$ factor, or the the $a^{th}$ +factor respectively. + +\smallskip\noindent +An arrow between objects with descent data +$\{\alpha_i\}: (\{\xi_i\},\{\phi_{ij}\}) \rightarrow +(\{\eta_i\},\{\psi_{ij}\})$ is a collection of arrows +$\alpha_i : \xi_i \rightarrow \eta_i$ in $\mathcal F(U_i)$ with the property +that for each pair of indicies $i,j$ the cocycle condition is satisfied. + +\smallskip\noindent +There is an obvious way of composing morphisms, which makes the objects with +descent data the objects of a cateogry which we will denote +$\mathcal F(\{U_i \rightarrow U\})$ + +\smallskip\noindent +Specifically in the case of the lemma we are only interested in the case where +$U = \text{Spec}(A)$ is affine. So it can be covered by one affine scheme +$V=\text{Spec}(B)$. In this case the above definition reduces to the +following: $\mathcal F(V \rightarrow U)$ is the category consisting of pairs +$(\xi, \phi)$ where $\xi \in \mathcal F(V)$ and +$\phi: p_1^*\xi \simeq p_2^*\xi$ is an isomorphism which satisfies the +cocycle condition. +\end{definition} + +\noindent +To continue we need to set some conventions. Let $A$ be a commutative ring, +and denote $Mod_A$ as the category of modules over $A$. We also have a ring +homomorphism $f: A \rightarrow B$. We define a cateogry +$Mod_{A\rightarrow B}$ as follows. Let the objects be pairs $(N,\phi)$ +where $N$ is a $B$-module and $\phi: N \otimes_A B \simeq B \otimes_A N$ +is an isomorphism of $B^{\otimes 2}$-modules such that the following cocycle +condition is satisfied: +\begin{eqnarray} +\phi_1: B \otimes_A N \otimes_A B & \rightarrow & B \otimes_A B \otimes_A N, +\nonumber\\ +\phi_2: N \otimes_A B \otimes_A B & \rightarrow & B \otimes_A B \otimes_A B, +\nonumber\\ +\phi_3: N \otimes_A B \otimes_A B & \rightarrow & B \otimes_A N \otimes_A B, +\nonumber +\end{eqnarray} +where $\phi_1 = id_B \otimes \phi$, $\phi_3 = \phi \otimes id_B$, and +$\phi_2 = \phi_1\phi_3$. + +\smallskip\noindent +A morphism $\beta: (N,\phi) \rightarrow (N', \phi')$ is a homomorphism of +$B$-modules $\beta: N \rightarrow N'$ making the following diagram commute: +\begin{center} +\( +\begin{array}{ccc} +N \otimes_A B & \stackrel{\phi}{\longrightarrow} & B \otimes_A N \\ +\beta \otimes id_B \downarrow & & \downarrow id_B \otimes \beta \\ +N' \otimes_A B & \stackrel{\phi'}{\longrightarrow} & B \otimes_A N' +\end{array} +\) +\end{center} + +\smallskip\noindent +Given a functor $F: Mod_A \rightarrow Mod_{A\rightarrow B}$ which takes an +$A$-module $M$ to the pair $(B\otimes_A M, \phi_M)$ where +$\phi_M: (B\otimes_A M)\otimes_A B \rightarrow B \otimes_A (B \otimes_A M)$ +maps $b \otimes m \otimes b'$ to $b \otimes b' \otimes m$ (and satisfies the +cocyle condition). + +\begin{theorem} +\label{flat-descent-rings} +If $B$ is faithfully flat over $A$, the functor +$F: Mod_A \rightarrow Mod_{A\rightarrow B}$ as defined above is an +equivalence of categories. +\end{theorem} + +\noindent +To prove this we first need the following lemma + +\begin{lemma} +\label{lemma-exactness} +Let $M$ be an $A$-module. Then the following sequence +$$ +0 \rightarrow M \stackrel{\alpha_M}{\longrightarrow} +B \otimes_A M \stackrel{(e_1 - e_2)\otimes id_M}{\longrightarrow} +B^{\otimes 2}\otimes_A M +\stackrel{(e_1-e_2+e_3)\otimes id_M}{\longrightarrow} +B^{\otimes 3} \otimes_A M \rightarrow \cdots +$$ +is exact. Where $e_i: B^{\otimes n} \rightarrow B^{\otimes n+1}$ is the +map that puts a $1$ into the $i^{th}$ place of the tensor product. +\end{lemma} + +\begin{proof} +It is easy to see that $\alpha_M$ is an injective map, and that images of +all of the maps will be contained in the appropriate kernels. So what we +need to show is that we have reverse containment (i.e. kernels contained +in images) at each step of the sequence. If there existed a section +$B\otimes_A M \rightarrow M$ then this will be easy (for reasons we will see +soon). However, there does not always exist such a map. + +\smallskip\noindent +Luckily we can use the fact that $B$ is flat over $A$, and that a sequence +of $A$-modules is exact if and only if the sequence tensored with $B$ is +exact. This is fortunate because once we tensor our sequence with $B$ we get +the following sequence (with the same maps as in the statement of the lemma +just with tensored with a $id_B$ on the left): +$$ +0 \rightarrow B \otimes_A M \rightarrow B^{\otimes_2} \otimes_A M +\rightarrow B^{\otimes 3}\otimes_A M \rightarrow +B^{\otimes 4} \otimes_A M \rightarrow \cdots. +$$ + +\smallskip\noindent +And there is a natural map $B^{\otimes_2} \otimes_A M +\stackrel{mult}{\rightarrow} B \otimes_A M$ which just takes +$b\otimes b' \otimes m$ to the element $bb' \otimes m$. In other words it +is just the multipication map $B\otimes_A B \rightarrow B$ composed with the +identity on $M$. + +Now, to prove that the sequence is exact at $B^{\otimes_2} \otimes_A M$, we +pick an element $\Sigma b_i \otimes b'_i \otimes m_i$ in the kernel of the +map $id_B \circ (e_1-e_2) \circ id_M$. This means that in +$B^{\otimes_3} \otimes_A M$ we have the following relation: +$\Sigma b_i\otimes b'_i \otimes 1 \otimes m_i = +\Sigma b_i \otimes 1 \otimes b'_i \otimes m_i$. We then apply the map +$mult \circ id_B \circ id_M$ to the equality and get the +$\Sigma b_ib'_i \otimes 1 \otimes m_i = b_i \otimes b'_i \otimes m_i$ +in $B^{\otimes_2} \otimes_A M$. So we are getting +$id_B \circ \alpha_M (\Sigma b_ib'_i \otimes m_i) = +\Sigma b_i\otimes b'_i \otimes m_i$ which was the element from the kernel +that we chose. Thus every element in the kernel is also in the image of +the appropriate map hence the sequence is exact there. The same argument +can be made at each step of the sequence to show that the appropriate +kernels are contained in the appropriate images. A choice for sections +that work is simply multipication composed with whatever number of identity +maps are necessary. +\end{proof} + +\begin{proof}[Proof of the theorem] +So to prove the theorem we need to first consider the functor +$F: Mod_A \rightarrow Mod_{A\rightarrow B}$ which takes an $A$-module $M$ +to the pair $(B\otimes_A M, \phi_M)$ where +$\phi_M: (B\otimes_A M)\otimes_A B \rightarrow B \otimes_A (B \otimes_A M)$ +maps $b \otimes m \otimes b'$ to $b \otimes b' \otimes m$. + +\smallskip\noindent +To show that $F$ is an equivalence of categories, we need to show that there +is a functor $G: Mod_{A\rightarrow B} \rightarrow Mod_A$ such that $GF$ and +$FG$ are isomorphic to the identity. + +\smallskip\noindent +So let us define a functor $G$ to take pairs $(N, \phi)$ to elements +$GN = \{n \in N | 1\otimes n = \phi(n\otimes 1)\}$ and given a morphism +$\beta: (N, \phi) \rightarrow (N', \phi')$ in $Mod_{A\rightarrow B}$ we +get a morphism $\beta_G: GN \rightarrow GN'$. + +\smallskip\noindent +So first let us check that $GF$ is isomorphic to the identity. Notice +that +$$ +((e_1-e_2)\otimes id_M)(b \otimes m) = +b \otimes 1 \otimes m - 1 \otimes b \otimes m = +\phi_M(b\otimes m \otimes 1) - 1 \otimes b \otimes m +$$ +for all $m$ and $b$. For simplicity we can rewrite this as +$((e_1 - e_2)\otimes id_M)(x) = \phi_M(x) - 1\otimes x$ for all +$x \in B \otimes_A M$. So then by our definition of the functor $G$ we get +that $G(B \otimes_A M, \phi_M) = ker ((e_1-e_2)\otimes id_M)$. However, due +to our lemma, we know that the sequence +$$ +0 \rightarrow M \rightarrow^{\alpha_M} B \otimes_A M +\rightarrow B^{\otimes 2}\otimes_A M +\rightarrow B^{\otimes 3} \otimes_A M \rightarrow \cdots +$$ +is exact, thus $ker ((e_1-e_2)\otimes id_M) = im(\alpha_M) \simeq M$. +So $M \simeq G(B \otimes_A M) = GF(M)$ as needed. + +\smallskip\noindent +Now we will show that $FG$ is isomorphic to the identity. So we take +$(N, \phi)$ in $Mod_{A\rightarrow B}$ and we set +$M = G(N,\phi) = \{n \in N | 1 \otimes n = \phi(n \otimes 1)\}$. Since +$M$ is an $A$-submodule of the $B$-submodule $N$ we get a homomorphism of +$B$-modules $\theta: B \otimes_A M \rightarrow N$ which takes $b\otimes m$ +to $bm$. It is easy to check that this is a morphism in +$Mod_{A\rightarrow B}$. So notice that we can also think of $\theta$ as a +map $F(M) \rightarrow N$, thus we can see that $\theta$ defines a natural +transformation $id \rightarrow FG$. So to complete the proof we need to +show that $\theta$ is an isomorphism. + +\smallskip\noindent +First we will need to define some maps $i$ will just be inclusion, +$\iota_M: M \otimes B \rightarrow B \otimes M$ is the map taking +$m \otimes b$ to $b \otimes m$, and +$\alpha, \beta: N \rightarrow B \otimes_A M$ are defined by +$\alpha (n) = 1 \otimes n$ and $\beta (n) = \phi(n \otimes 1)$. So by +definition, $M = ker(\alpha - \beta)$. We have the following diagram, +where the rows are exact: +\begin{center} +\( +\begin{array}{cccccccc} +0 & \longrightarrow & M \otimes_A B & +\stackrel{i \otimes id_B}{\longrightarrow} & N \otimes_A B & +\stackrel{(\alpha-\beta)\otimes id_B}{\longrightarrow }& +B \otimes_A N \otimes_A B \\ + & & \downarrow \theta \circ \iota_M & + & \downarrow \phi & & +\downarrow \phi_1\\ +0 & \longrightarrow & N & \stackrel{\alpha_N}{\longrightarrow} & +B \otimes_A N & \stackrel{(e_1-e_2)\otimes id_N}{\longrightarrow} & +B \otimes_A B \otimes_A N +\end{array} +\) +\end{center} + +\smallskip\noindent +So by showing that the diagram commutes, using the fact that both $\phi$ +and $\phi_1$ are isomorphisms we are able to get that $\theta$ is an +isomorphism. To show the diagram commutes let us focus on one square at a +time. For the first square we want +$\phi(i \otimes id_B)(m\otimes b) = \alpha_M \theta \iota_M (m\otimes b)$. +We know that $\alpha_M\theta \iota_M(m\otimes b) = 1 \otimes bm$. So we +just need to show that we get the same thing for +$(\phi(i \otimes id_B))(m\otimes b)$. We have +\begin{eqnarray*} +(\phi (i \otimes id_B))(b \otimes m) & = & \phi(m \otimes b) \\ +& = & \phi((1\otimes b)(m \otimes 1))\\ +& = & (1 \otimes b) \phi (m \otimes 1)\\ +& = & (1 \otimes b)(1 \otimes m)\\ +& = & 1 \otimes bm +\end{eqnarray*} +as needed. Now we just need to show the second square commutes. For the +second square it should be clear that +$\phi_1(\alpha \otimes id_B) = (e_2 \otimes id_N) \circ \phi$. So we just +need to check that +$\phi_1(\beta \otimes id_B) = (e_1 \otimes id_N)\circ \phi$. We have: +\begin{eqnarray*} +\phi_1(\beta \otimes id_B)(n\otimes b) & = & +\phi_1(\phi(n\otimes 1) \otimes b) \\ +& = & \phi_1 \phi_3(n \otimes 1 \otimes b) \\ +& = & \phi_2 (n \otimes 1 \otimes b)\\ +& = & (e_1 \otimes id_N) \phi( n \otimes b) +\end{eqnarray*} + +\smallskip\noindent +So, by the argument above $\theta$ is an isomorphism and thus +$FG(N, \phi) \simeq (N, \phi)$. +\end{proof} + +\noindent +And now we can restate and sketch the proof of our desired result + +\begin{theorem} +\label{theorem-quasi-coherent-stack} +Let $S$ be a scheme. The fibered category $(QCoh/S)$ over $(Sch/S)$ is +stack with respect to the flat (fpqc) topology. +\end{theorem} + +\begin{proof} FIXME: Sketch of proof. + +\smallskip\noindent +Let me remind you that we just need to check the second condition of the +lemma. So for a flat and surjective morphism $V \rightarrow U$ +(corresponding to a faithfully flat ring homomorphism $f: A \rightarrow B$). +We need to show that there is an equivalence of categories between $QCoh(U)$ +and $QCoh(V\rightarrow U)$. We will do this using the previous theorem which +states that there is an equivalence of categories between $Mod_A$ and +$Mod_{A\rightarrow B}$. + +\smallskip\noindent +There is a standard equivalence of categories between $QCoh(U)$ and $Mod_A$. +So we just need to show that there is an equivalence of categories between +$QCoh(V\rightarrow U)$ and $Mod_{A\rightarrow B}$. To do this let us look at +$\mathcal N$ an object in $QCoh(V)$ which corresponds to an $B$-module $N$. +Looking at $p_1^* \mathcal N$ and $p_2^* \mathcal N$ in +$V \times_U V = \text{Spec} (B \otimes_A B)$ we get $N \otimes_A B$ and +$B \otimes_A N$ respectively. So the descent datum +$\psi: p_1^* \mathcal N \simeq p_2^* \mathcal N$ will correspond to the +descent data $\phi: N \otimes_A B \simeq B \otimes_A N$ in +$Mod_{A \rightarrow B}$. So $(\mathcal N, \psi)$ is an object of +$QCoh(V\rightarrow U)$ if and only if $\phi$ satisfies the cocycle condition, +thus giving us an equivalence of categories between $QCoh(V\rightarrow U)$ +and $Mod_{A\rightarrow B}$. Thus the functor +$QCoh(U) \rightarrow QCoh(V\rightarrow U)$ corresponds to the functor +$Mod_A \rightarrow Mod_{A\rightarrow B}$! So since, the later is an +equivalence we get the equivalence of categories that we need. Thus +finishing the proof. +\end{proof} + +\noindent +FIXME: put in example with descent of projective schemes and put in Galois descent example. + +\smallskip\noindent +To continue reading, +\begin{enumerate} + +\item visit the next section: Algebraic Stacks Desirables, +\autoref{desirables-section-foundational}, or + +\item go back to the +table of contents: \url{index.html#contents}. + +\end{enumerate} + + +\bibliography{my} +\bibliographystyle{alpha} + + +\end{document} diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/scripts/contents_html.sh stacks-0.2/src/scripts/contents_html.sh --- stacks-0.2.orig/src/scripts/contents_html.sh 2006-03-05 03:48:21.000000000 +0000 +++ stacks-0.2/src/scripts/contents_html.sh 2006-03-05 15:12:12.000000000 +0000 @@ -33,7 +33,7 @@ # LIJST is the list of STEMS of .toc files in the order in which you want it # to appear on the web-site. Do not include fdl. -LIJST="introduction conventions sets categories sites etale injectives hypercovering stacks stacks-groupoids desirables" +LIJST="introduction conventions sets categories sites etale injectives hypercovering stacks stacks-groupoids flat desirables" TELLER=0 for STAM in $LIJST; do diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/scripts/downloads_html.sh stacks-0.2/src/scripts/downloads_html.sh --- stacks-0.2.orig/src/scripts/downloads_html.sh 2006-03-05 03:48:21.000000000 +0000 +++ stacks-0.2/src/scripts/downloads_html.sh 2006-03-05 15:12:12.000000000 +0000 @@ -3,7 +3,7 @@ # Write a downloads section to downloads.html. # Same list as in contents_html.sh. -LIJST="introduction conventions sets categories sites etale injectives hypercovering stacks stacks-groupoids desirables" +LIJST="introduction conventions sets categories sites etale injectives hypercovering stacks stacks-groupoids flat desirables" cat > downloads.html << "EOF"

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diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/stacks-groupoids.tex stacks-0.2/src/stacks-groupoids.tex --- stacks-0.2.orig/src/stacks-groupoids.tex 2005-09-26 19:48:15.000000000 +0000 +++ stacks-0.2/src/stacks-groupoids.tex 2006-03-05 15:12:12.000000000 +0000 @@ -46,6 +46,7 @@ % \externaldocument[hello-]{hello} \externaldocument[conventions-]{conventions} \externaldocument[desirables-]{desirables} +\externaldocument[flat-]{flat} % The macro \autoref uses the macros \figurename, etc. % We list the default values and we change some of them @@ -223,8 +224,8 @@ To continue reading, \begin{enumerate} -\item visit the next section: Algebraic Stacks Desirables, -\autoref{desirables-section-foundational}, or +\item visit the next section: Flat descent for quasi-coherent sheaves, +\autoref{flat-section-introduction}, or \item go back to the table of contents: \url{index.html#contents}.