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Johan de Jong
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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/change_preamble.sh new file: scripts/concat.sh new file: scripts/contents_html.sh new file: scripts/downloads_html.sh new file: scripts/stacks_html.sh new file: sets.tex new file: sites.tex new file: stacks-groupoids.tex new file: stacks.tex new file: template.tex
0 parents  commit 3d32323

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‎COPYING‎

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‎Makefile‎

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# Known suffixes.
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.SUFFIXES: .aux .bbl .bib .blg .dvi .html .log .out .pdf .ps .tex .toc .funny
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# Master list of stems of tex files in the project.
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# This should be in order.
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LIJST = introduction conventions sets categories sites flat etale injectives hypercovering stacks stacks-groupoids schemes algebraic desirables
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# Add fdl to get license latexed as well.
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LIJST_FDL = $(LIJST) fdl
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# Different extensions.
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PDFS = $(patsubst %,%.pdf,$(LIJST_FDL))
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DVIS = $(patsubst %,%.dvi,$(LIJST_FDL))
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PSS = $(patsubst %,%.ps,$(LIJST_FDL))
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FUNNYS = $(patsubst %,%.funny,$(LIJST_FDL))
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HTMLS = stacks.html contents.html downloads.html
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# Files in INSTALLDIR will be overwritten.
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INSTALLDIR=..
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# Make all the funny targets first so crossreferences work.
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.PHONY: all
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all: $(FUNNYS) $(PDFS) $(DVIS) $(PSS) $(HTMLS)
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# We need the following to cancel the built-in rule for
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# .dvi files (which uses tex not latex).
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%.dvi : %.tex
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# fld.funny is different because there is no bibliography
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# nor is there a table of contents...
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fdl.funny : fdl.tex
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echo "latex fdl.tex" >> logfile.log
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latex fdl.tex
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echo "touch fdl.funny" >> logfile.log
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touch fdl.funny
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# Other .pdf files do have bibliographies and
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# table of contents. But running make file.pdf
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# will not correctly insert external crossreferences
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# if the other .aux files aren't up to date.
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%.pdf : %.tex %.funny
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echo "2x pdflatex $<" >> logfile.log
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pdflatex $<
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pdflatex $<
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%.dvi : %.tex %.funny
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echo "2x latex $<" >> logfile.log
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latex $<
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latex $<
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# Funny target to prepare %.aux, %.toc and %.bbl.
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# The latex command creates %.aux and %.toc,
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# the bibtex command creates %.bbl, and finally
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# the touch command creates %.funny with a newer
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# modification time then any of %.dvi, %.aux, %.toc,
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# %.bbl, %.blg, %.log and %.out. Actually the modification
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# time resolution is not good enough so we remove %.dvi.
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%.funny: %.tex
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echo "latex $<" >> logfile.log
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latex $<
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echo "bibtex $*" >> logfile.log
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bibtex $*
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echo "rm $*.dvi" >> logfile.log
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rm $*.dvi
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echo "touch $@" >> logfile.log
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touch $@
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%.ps: %.dvi
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echo "dvips -o $@ $<" >> logfile.log
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dvips -o $@ $<
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.PHONY: clean
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clean:
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rm -f *.aux *.bbl *.blg *.dvi *.log *.pdf *.ps *.out *.toc *.html *.funny
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.PHONY: backup
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backup: clean
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cd .. ; tar -cjvf stacks-0.2.tar.bz2 src/
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# The script scripts/name_html.sh creates name.html in src directory.
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# We do not want an index.html in src! So we concatenate these into
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# $(INSTALLDIR)/index.html in the install target.
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#
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# FIXME: For contents.html We should really do some sanity checking to
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# see if the .toc files are up to date.
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%.html:
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echo "./scripts/$*_html.sh" >> logfile.log
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./scripts/$*_html.sh
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.PHONY: install
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install: all
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cp *.pdf *.ps *.dvi $(INSTALLDIR)
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cat stacks.html contents.html downloads.html > $(INSTALLDIR)/index.html

‎algebraic.tex‎

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\documentclass{amsart}
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% The following AMS packages are automatically loaded with amsart
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% documentclass:
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%\usepackage{amsmath}
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%\usepackage{amssymb}
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%\usepackage{amsthm}
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% For commutative diagrams you can use
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% \usepackage{amscd}
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% but Jason prefers xypic
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\usepackage[all]{xy}
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% To put source file link in headers.
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% Change "template.tex" to "this_filename.tex"
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\usepackage{fancyhdr}
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\pagestyle{fancy}
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\lhead{}
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\chead{}
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\rhead{Source file: \url{src/algebraic.tex}}
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\lfoot{}
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\cfoot{\thepage}
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\rfoot{}
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\renewcommand{\headrulewidth}{0pt}
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\renewcommand{\footrulewidth}{0pt}
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\renewcommand{\headheight}{12pt}
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% For cross-file-references
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\usepackage{xr-hyper}
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% Package for hypertext links:
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\usepackage[colorlinks=true]{hyperref}
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% For any local file, say "hello.tex" you want to refer to please use
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% \externaldocument[hello-]{hello}
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\externaldocument[conventions-]{conventions}
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\externaldocument[categories-]{categories}
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\externaldocument[hypercovering-]{hypercovering}
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\externaldocument[schemes-]{schemes}
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\externaldocument[desirables-]{desirables}
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\externaldocument[fdl-]{fdl}
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% The macro \autoref uses the macros \figurename, etc.
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% We list the default values and we change some of them
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% to start with a captial.
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% Figure \figurename
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% Table \tablename
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% Part \partname
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% Appendix \appendixname
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% Equation \equationname
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% item \Itemname
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% \renewcommand{\Itemname}{Item}
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\renewcommand{\Itemautorefname}{Item}
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% chapter \Chaptername
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% \renewcommand{\Chaptername}{Chapter}
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% \renewcommand{\Chapterautorefname}{Chapter}
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% section \sectionname
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\renewcommand{\sectionname}{Section}
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\renewcommand{\sectionautorefname}{Section}
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% subsection \subsectionname
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\renewcommand{\subsectionname}{Subsection}
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\renewcommand{\subsectionautorefname}{Subsection}
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% subsubsection \subsubsectionname
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\renewcommand{\subsubsectionname}{Subsubsection}
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\renewcommand{\subsubsectionautorefname}{Subsubsection}
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% paragraph \paragraphname
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\renewcommand{\paragraphname}{Paragraph}
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\renewcommand{\paragraphautorefname}{Paragraph}
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% footnote \Hfootnotename
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% \renewcommand{\Hfootnotename}{Footnote}
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\renewcommand{\Hfootnoteautorefname}{Footnote}
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% Equation \AMSname
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% Theorem \theoremname
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% Theorem environments.
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%
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\newtheorem{theorem}{Theorem}[subsection]
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\newtheorem{proposition}[theorem]{Proposition}
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\newtheorem{lemma}[theorem]{Lemma}
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\theoremstyle{definition}
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\newtheorem{definition}[theorem]{Definition}
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\newtheorem{example}[theorem]{Example}
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\newtheorem{exercise}[theorem]{Exercise}
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\newtheorem{situation}[theorem]{Situation}
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\theoremstyle{remark}
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\newtheorem{remark}[theorem]{Remark}
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\newtheorem{remarks}[theorem]{Remarks}
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\numberwithin{equation}{subsection}
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% OK, start here.
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%
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\begin{document}
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\title{Algebraic stacks}
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%\begin{abstract}
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%\end{abstract}
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\maketitle
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\thispagestyle{fancy}
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\tableofcontents
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\section{Introduction}
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\label{section-introduction}
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\noindent
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This is where we define algebraic stacks and make some very elementary
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observations. The general philosophy will be to have no separation
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conditions whatsoever and add those conditions necessary to make lemmas,
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propositions, theorems true/provable. Thus the notions discussed here
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differ slightly from those in other places in the literature, e.g.,
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\cite{LM-B}.
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\section{Definitions}
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\label{section-definitions}
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\subsection{Algebraic spaces}
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\label{subsection-algebraic-spaces}
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\noindent
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FIXME.
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\begin{definition}
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An algebraic space is a stack $\mathcal{S}$ over $\text{Aff}$ such that
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\begin{enumerate}
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\item every fibre category is setlike, see Categories,
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\autoref{categories-subsection-fibred-in-sets},
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\item the diagonal morphism
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$\Delta : \mathcal{S} \to \mathcal{S}\times\mathcal{S}$
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is representable by schemes, see Schemes,
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\autoref{schemes-subsection-definition-representable-by-schemes} and
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\item there exists a stack $\mathcal{X}$ representable by a scheme, see
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Schemes, \autoref{schemes-subsection-stack-representable-by-scheme}
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and an \'etale surjective morphism $\mathcal{X} \to \mathcal{S}$,
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see Schemes,
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\autoref{schemes-definition-property-morphism-representable-by-schemes}.
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\end{enumerate}
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\end{definition}
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\begin{remark}
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\label{remark-definition-correct}
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If you try to define some kind of more general algebraic space by requiring
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only that the diagonal is representable by algebraic spaces, and that there is
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a surjective etale morphism of an algebraic space onto $\mathcal{S}$, then
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you actually end up with the same notion.
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(FIXME: internal references, proofs.)
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\end{remark}
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\subsection{Morphisms representable by algebraic spaces}
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\label{subsection-morphism-representable-by-algebraic-spaces}
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\noindent
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Here is the formal definition. Please also see the informal discussion below.
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\begin{definition}
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\label{definition-representable-by-algebraic-spaces}
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Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of categories
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fibred in groupoids over $\text{Aff}$. We say $f$ is representable by
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algebraic spaces if for every stack $\mathcal{S}$ representable by a scheme
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(see Schemes, Definition \ref{schemes-definition-representable-by-scheme}),
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and every morphism $\mathcal{U} \to \mathcal{Y}$, the 2-fibre product
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$\mathcal{S}\times_\mathcal{Y}\mathcal{X}$ is an algebraic space.
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\end{definition}
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\noindent
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Informal discussion. Suppose that, with the notation of the definition,
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$S$ represents $\mathcal{S}$. Suppose that $W$ is a scheme and that
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$\text{Aff}/W \to \mathcal{S}\times_\mathcal{Y}\mathcal{X}$ is
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etale and surjective. According to
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Schemes, Lemma \ref{schemes-lemma-morphism-stacks-representable-by-schemes}
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we get a morphism of schemes $g : W \to S$ and a 2-commutative diagram
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of stacks
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$$
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\xymatrix{
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\text{Aff}/W \ar[d]^g \ar[r] &
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\mathcal{S}\times_\mathcal{X}\mathcal{Y} \ar[d] \ar[r] &
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\mathcal{Y} \ar[d] \\
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\text{Aff}/S &
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\mathcal{S} \ar[l]^j \ar[r] & \mathcal{X}
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}
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$$
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\begin{definition}
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\label{definition-property-morphism-representable-by-algebraic-spaces}
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Let $P$ be a property of morphisms of schemes, that is etale local
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on the source and such that if the morphism $f : X \to Y$ has property $P$,
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then so does every base change of $f$. (FIXME: introduce base change.)
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We say that a morphism of stacks $\mathcal{X}
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\to \mathcal{Y}$ representable by algebraic spaces has property
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$P$ if for every diagram as above the morphism of schemes
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$g : W \to S$ has property $P$.
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\end{definition}
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\noindent
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FIXME. Explain rationale behind this definition: what else could it be?
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\subsubsection{Algebraic stacks}
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\label{subsubsection-algebraic-stacks}
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\noindent
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FIXME.
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\begin{definition}
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An algebraic stack is a stack $\mathcal{S}$ over $\text{Aff}$ such that
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\begin{enumerate}
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\item the diagonal morphism
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$\Delta : \mathcal{S} \to \mathcal{S}\times\mathcal{S}$
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is representable by algebraic spaces, see Definition,
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\autoref{definition-representable-by-algebraic-spaces} and
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\item there exists a stack $\mathcal{X}$ representable by a scheme, see
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Schemes, \autoref{schemes-subsection-stack-representable-by-scheme}
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and a smooth surjective morphism $\mathcal{X} \to \mathcal{S}$,
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see Definition
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\ref{definition-property-morphism-representable-by-algebraic-spaces}.
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\end{enumerate}
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\end{definition}
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\smallskip\noindent
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To continue reading,
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\begin{enumerate}
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\item visit the next section: Algebraic stacks desirables,
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\autoref{desirables-section-foundational}, or
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\item go back to the
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table of contents: \url{index.html#contents}.
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\end{enumerate}
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\bibliography{my}
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\bibliographystyle{alpha}
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\end{document}

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