The Stacks project

Flip components for consistency

THanks to Jia Jia
https://stacks.math.columbia.edu/tag/0264#comment-6292

Typo in spaces

Thanks to Yu-Liang Huang
https://stacks.math.columbia.edu/tag/02WU#comment-2964

Fix typos

Thanks to Minseon Shin (email)

R/U ---> U/R

Thanks to Yogesh More
http://stacks.math.columbia.edu/tag/0264#comment-613

LaTeX

Added a new macro

\def\Im{\text{Im}}

and replaced all occurrences of \text{Im} by \Im

Fix a few typos

Pointed out by Rex and David Holmes the comments (too many to list
here).

LaTeX: get rid of useless brackets

Groupoids: Fix dumb error.

	The error was not realizing that ``relation generated by'' is
	a kind of tricky operation...

	This basically has no further ramifications since we only used
	the result in situations where the the pre-relation is actually
	a pre-equivalence relation (which would be better named a
	equivalence pre-relation).

Groupoid schemes: slicing lemma

	Here is the lemma promised in commit
	79c56068dee8636e2ae061dd70e3af6d8f9bd30d. The proof is really
	straightforward once you formulate it in the setting where you
	already know that s,t are Cohen-Macaulay.

Restriction and equivalences

	We inserted a couple of lemmas describing when the restriction
	of a groupoid scheme gives rise to an isomorphic quotient sheaf
	or when it gives rise to an equivalent quotient stack.

Renamed fpqc-descent.tex descent.tex

	modified:   Makefile
	modified:   algebraic.tex
	modified:   browse.html
	modified:   chapters.tex
	renamed:    fpqc-descent.tex -> descent.tex
	modified:   groupoids.tex
	modified:   more-morphisms.tex
	modified:   morphisms.tex
	modified:   preamble.tex
	modified:   scripts/functions.py
	modified:   spaces.tex
	modified:   stacks.tex
	modified:   tags/Makefile
	modified:   tags/tags

Added new tags to stacks project

	modified:   tags/tags

Towards a better descent

	modified:   fpqc-descent.tex
	modified:   groupoids.tex
	modified:   spaces.tex

Towards a better descent

	modified:   fpqc-descent.tex
	modified:   groupoids.tex
	modified:   spaces.tex

Small confused changes

	modified:   spaces.tex

Beginning work on algebraic spaces

	modified:   algebra.tex
	modified:   fpqc-descent.tex
	modified:   groupoids.tex
	modified:   morphisms.tex
	modified:   sites.tex
	modified:   spaces.tex