The Stacks project

Isolate a general argument

Thanks to Ryan Chen who writes

"A few comments about Lemma 03LS in the Stacks project.

--

In the lemma statement, the expression g^{-1}_{small} c_f \circ c_g should
instead be c_g \circ g^{-1}_{small} c_f (the arrows are going the wrong way).
Indeed, note that c_g has domain g^{-1}_{small} \mathcal{F} _{U'}, while the
arrow c_{f \circ g} has domain (f \circ g)^{-1}_{small} \mathcal{F} _U.

This same typo also appears in Lemma 03LY and Lemma 03LZ.

--

I don't think the result and proof of Lemma 03LS are special to the small
étale site of an algebraic space. It should be a general site-theoretic fact.
Perhaps it would be good to put this site-theoretic fact in the chapter
on sites, e.g. Section 04TP, and then just reference this fact for the
proof of Lemma 03LS.

I have attached a pdf and TeX file of the corresponding fact about sites.
I tried to match the writing style in Section 04TP (where this lemma could
go). The reference appearing Lemma ?? in the .pdf is a reference to
Lemma 03EH.

--

If you do add the general site-theoretic fact to the Stacks project, perhaps
the proof of Lemma 03LS could just be rewritten as:

We may interpret g^{-1}_{small} as in Lemma \ref{lemma-etale-morphism-topoi}.
Then, the lemma follows from a general fact about sites, see Lemma
\ref{lemma-glue-sheaves-absolute}."

LaTeX: \etale

Introduced the macro

\def\etale{{\acute{e}tale}}

and replaced all occurences of \acute{e}tale by \etale

LaTeX: \Ob

	Introduced a macro

	\def\Ob{\mathop{\rm Ob}\nolimits}

	and replaced any occurence of \text{Ob}( with \Ob(. There are
	still some occurences of \text{Ob} but these are sets, not the
	operator that takes the set of objects of a category.

Neurotic changes

	Fix (almost all) complaints of parse.py

End conversion of etale to \'etale.

Tags: Added new tags

	Also commented obsolete tags corresponding to sections of
	desirables.tex that have been removed.

Properties of Spaces: Quasi-coherent sheaves in terms of a presentation

	In addition we introduced the abstract notion of a
	quasi-coherent sheaf on a groupoid, as well as a G-equivariant
	sheaf on a scheme endowed with an action of a group scheme.