The Stacks project

Invisible changes

Limit the number of different commands used in math mode

Explain what e_i is

Thanks to Matthieu Romagny
http://stacks.math.columbia.edu/tag/00HK#comment-1936

details on the definition of a map

Minor corrections on algebra matters.

algebra.tex:355: The "this" in the "Informally this means..." paragraph
refers to the notion of "finitely presented", and so belongs inside part (2) of the
definition, rather than after it. (Otherwise it should be more explicit
about what it means by "this".)

algebra.tex:718: I have never seen transitive relations referred to as
"associative", so I take this for a typo.

algebra.tex:1863: There are various R-modules around, so I figured a bit more
explicitness wouldn't hurt.

algebra.tex:2021: It is certainly not the only iso $f : S^{-1}R \otimes_R M
\to S^{-1}M$.

algebra.tex:8067: The "(faithfully)" alternatives were added to the second
sentence to help prove the first sentence. It wouldn't hurt splitting the
lemma in two, though, as it now makes four different claims.

more-algebra.tex:4556: Fixed to the best of my knowledge. I am not 100% sure
of this since I don't know your notations.

Typos in algebra.tex

Thanks to Minseon Shin

LaTeX

Introduced a macro

\def\Ker{\text{Ker}}

and replace all occurrences of \text{Ker} with \Ker

Small changes

Thanks to Pieter Belmans

An application of criterion flatness per fibre

	Namely, if S ---> S' is a map of finitely presented algebras
	over R and I is a locally nilpotent ideal of R such that S/IS
	---> S'/IS' is an isomorphism, and S' is flat over R, then S
	---> S' is an isomorphism.

Whitespace changes

Code changes only

	Changes to algebra,tex which have no effect on the produced pdf
	files.

Started tags infrastructure

	new file:   scripts/add_tags.py
	modified:   scripts/functions.py
	new file:   tags/initial_tags
	new file:   tags/tags

More neurotic changes

	modified:   algebra.tex