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	new file: COPYING
	new file: Makefile
	new file: algebraic.tex
	new file: categories.tex
	new file: conventions.tex
	new file: desirables.tex
	new file: documentation/dontdiff
	new file: documentation/submitting-patches
	new file: etale.tex
	new file: fdl.tex
	new file: flat.tex
	new file: hypercovering.tex
	new file: injectives.tex
	new file: introduction.tex
	new file: my.bib
	new file: schemes.tex
	new file: scripts/
	new file: scripts/
	new file: scripts/
	new file: scripts/
	new file: scripts/
	new file: sets.tex
	new file: sites.tex
	new file: stacks-groupoids.tex
	new file: stacks.tex
	new file: template.tex
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Johan de Jong committed May 20, 2008
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397 changes: 397 additions & 0 deletions COPYING

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93 changes: 93 additions & 0 deletions Makefile
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# Known suffixes.
.SUFFIXES: .aux .bbl .bib .blg .dvi .html .log .out .pdf .ps .tex .toc .funny

# Master list of stems of tex files in the project.
# This should be in order.
LIJST = introduction conventions sets categories sites flat etale injectives hypercovering stacks stacks-groupoids schemes algebraic desirables

# Add fdl to get license latexed as well.

# Different extensions.
PDFS = $(patsubst %,%.pdf,$(LIJST_FDL))
DVIS = $(patsubst %,%.dvi,$(LIJST_FDL))
PSS = $(patsubst %,,$(LIJST_FDL))
FUNNYS = $(patsubst %,%.funny,$(LIJST_FDL))
HTMLS = stacks.html contents.html downloads.html

# Files in INSTALLDIR will be overwritten.

# Make all the funny targets first so crossreferences work.
.PHONY: all
all: $(FUNNYS) $(PDFS) $(DVIS) $(PSS) $(HTMLS)

# We need the following to cancel the built-in rule for
# .dvi files (which uses tex not latex).
%.dvi : %.tex

# fld.funny is different because there is no bibliography
# nor is there a table of contents...
fdl.funny : fdl.tex
echo "latex fdl.tex" >> logfile.log
latex fdl.tex
echo "touch fdl.funny" >> logfile.log
touch fdl.funny

# Other .pdf files do have bibliographies and
# table of contents. But running make file.pdf
# will not correctly insert external crossreferences
# if the other .aux files aren't up to date.
%.pdf : %.tex %.funny
echo "2x pdflatex $<" >> logfile.log
pdflatex $<
pdflatex $<

%.dvi : %.tex %.funny
echo "2x latex $<" >> logfile.log
latex $<
latex $<

# Funny target to prepare %.aux, %.toc and %.bbl.
# The latex command creates %.aux and %.toc,
# the bibtex command creates %.bbl, and finally
# the touch command creates %.funny with a newer
# modification time then any of %.dvi, %.aux, %.toc,
# %.bbl, %.blg, %.log and %.out. Actually the modification
# time resolution is not good enough so we remove %.dvi.
%.funny: %.tex
echo "latex $<" >> logfile.log
latex $<
echo "bibtex $*" >> logfile.log
bibtex $*
echo "rm $*.dvi" >> logfile.log
rm $*.dvi
echo "touch $@" >> logfile.log
touch $@ %.dvi
echo "dvips -o $@ $<" >> logfile.log
dvips -o $@ $<

.PHONY: clean
rm -f *.aux *.bbl *.blg *.dvi *.log *.pdf *.ps *.out *.toc *.html *.funny

.PHONY: backup
backup: clean
cd .. ; tar -cjvf stacks-0.2.tar.bz2 src/

# The script scripts/ creates name.html in src directory.
# We do not want an index.html in src! So we concatenate these into
# $(INSTALLDIR)/index.html in the install target.
# FIXME: For contents.html We should really do some sanity checking to
# see if the .toc files are up to date.
echo "./scripts/$*" >> logfile.log

.PHONY: install
install: all
cp *.pdf *.ps *.dvi $(INSTALLDIR)
cat stacks.html contents.html downloads.html > $(INSTALLDIR)/index.html
241 changes: 241 additions & 0 deletions algebraic.tex
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% The following AMS packages are automatically loaded with amsart
% documentclass:

% For commutative diagrams you can use
% \usepackage{amscd}
% but Jason prefers xypic

% To put source file link in headers.
% Change "template.tex" to "this_filename.tex"
\rhead{Source file: \url{src/algebraic.tex}}

% For cross-file-references

% Package for hypertext links:
% For any local file, say "hello.tex" you want to refer to please use
% \externaldocument[hello-]{hello}

% 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}
% chapter \Chaptername
% \renewcommand{\Chaptername}{Chapter}
% \renewcommand{\Chapterautorefname}{Chapter}
% section \sectionname
% subsection \subsectionname
% subsubsection \subsubsectionname
% paragraph \paragraphname
% footnote \Hfootnotename
% \renewcommand{\Hfootnotename}{Footnote}
% Equation \AMSname
% Theorem \theoremname

% Theorem environments.




% OK, start here.

\title{Algebraic stacks}





This is where we define algebraic stacks and make some very elementary
observations. The general philosophy will be to have no separation
conditions whatsoever and add those conditions necessary to make lemmas,
propositions, theorems true/provable. Thus the notions discussed here
differ slightly from those in other places in the literature, e.g.,


\subsection{Algebraic spaces}


An algebraic space is a stack $\mathcal{S}$ over $\text{Aff}$ such that
\item every fibre category is setlike, see Categories,
\item the diagonal morphism
$\Delta : \mathcal{S} \to \mathcal{S}\times\mathcal{S}$
is representable by schemes, see Schemes,
\autoref{schemes-subsection-definition-representable-by-schemes} and
\item there exists a stack $\mathcal{X}$ representable by a scheme, see
Schemes, \autoref{schemes-subsection-stack-representable-by-scheme}
and an \'etale surjective morphism $\mathcal{X} \to \mathcal{S}$,
see Schemes,

If you try to define some kind of more general algebraic space by requiring
only that the diagonal is representable by algebraic spaces, and that there is
a surjective etale morphism of an algebraic space onto $\mathcal{S}$, then
you actually end up with the same notion.
(FIXME: internal references, proofs.)

\subsection{Morphisms representable by algebraic spaces}

Here is the formal definition. Please also see the informal discussion below.

Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of categories
fibred in groupoids over $\text{Aff}$. We say $f$ is representable by
algebraic spaces if for every stack $\mathcal{S}$ representable by a scheme
(see Schemes, Definition \ref{schemes-definition-representable-by-scheme}),
and every morphism $\mathcal{U} \to \mathcal{Y}$, the 2-fibre product
$\mathcal{S}\times_\mathcal{Y}\mathcal{X}$ is an algebraic space.

Informal discussion. Suppose that, with the notation of the definition,
$S$ represents $\mathcal{S}$. Suppose that $W$ is a scheme and that
$\text{Aff}/W \to \mathcal{S}\times_\mathcal{Y}\mathcal{X}$ is
etale and surjective. According to
Schemes, Lemma \ref{schemes-lemma-morphism-stacks-representable-by-schemes}
we get a morphism of schemes $g : W \to S$ and a 2-commutative diagram
of stacks
\text{Aff}/W \ar[d]^g \ar[r] &
\mathcal{S}\times_\mathcal{X}\mathcal{Y} \ar[d] \ar[r] &
\mathcal{Y} \ar[d] \\
\text{Aff}/S &
\mathcal{S} \ar[l]^j \ar[r] & \mathcal{X}

Let $P$ be a property of morphisms of schemes, that is etale local
on the source and such that if the morphism $f : X \to Y$ has property $P$,
then so does every base change of $f$. (FIXME: introduce base change.)
We say that a morphism of stacks $\mathcal{X}
\to \mathcal{Y}$ representable by algebraic spaces has property
$P$ if for every diagram as above the morphism of schemes
$g : W \to S$ has property $P$.

FIXME. Explain rationale behind this definition: what else could it be?

\subsubsection{Algebraic stacks}


An algebraic stack is a stack $\mathcal{S}$ over $\text{Aff}$ such that
\item the diagonal morphism
$\Delta : \mathcal{S} \to \mathcal{S}\times\mathcal{S}$
is representable by algebraic spaces, see Definition,
\autoref{definition-representable-by-algebraic-spaces} and
\item there exists a stack $\mathcal{X}$ representable by a scheme, see
Schemes, \autoref{schemes-subsection-stack-representable-by-scheme}
and a smooth surjective morphism $\mathcal{X} \to \mathcal{S}$,
see Definition

To continue reading,

\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}.




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