The Stacks project

algebra.tex:8210--8223: Clarifying injectivity argument.

You are right -- it is a straightforward argument; I was mostly confused
by the misconception that I had to use something about tensor products.
It might become clearer if a part of this proof is factored out into a
general criterion for when right-exact functors preserving colimits are
exact; do you see any other applications for such a criterion?

Forgot to save the file before commit
538c1a727d5976222806a4f4b1054c67bc79dbaf

algebra.tex:8218: Something I don't understand.

algebra.tex:8179: Clarifying the WLOG finiteness argument for my understanding. Probably more complicated than necessary (I haven't seen enough commutative algebra to do these things slickly).

algebra.tex:8151: I found this argument rather nontrivial, so I added it in. Probably it is more complicated than necessary, though; also, the injectivity of the second arrow is not used.

There is one more thing I don't understand. You define exact functors in
categories.tex:2876 using limits and colimits. In algebra.tex, you seem to be
using a different definition: A functor between abelian(?) categories is
left, resp. right, exact if it preserves exactness of sequences of the form 0
-> A -> B -> C, resp. A -> B -> C -> 0. Is the equivalence between these two
definitions clear? Wouldn't it be helpful to make it explicit and emphasize
its use?

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.

Some more spell checking

Typos in algebra.tex

Thanks to Minseon Shin

LaTeX

Added a new macro

\def\Im{\text{Im}}

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

LaTeX

Introduced a macro

\def\Ker{\text{Ker}}

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

Fixes for material flat modules

Thanks to remarks by Fan on the site, especially
http://stacks.math.columbia.edu/tag/00HD#comment-396

Normalize use of item labels

	Also check for this in parse.py

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

Clean up labels and references
Add more stringent reference checking to parse.py

	modified:   algebra.tex
	modified:   algebraic.tex
	modified:   categories.tex
	modified:   constructions.tex
	modified:   desirables.tex
	modified:   flat.tex
	modified:   fpqc-descent.tex
	modified:   morphisms.tex
	modified:   schemes.tex
	modified:   scripts/functions.py
	modified:   scripts/make_index.py
	modified:   scripts/parse.py
	modified:   sets.tex
	modified:   sheaves.tex
	modified:   simplicial.tex
	modified:   sites.tex
	modified:   stacks.tex
	modified:   topology.tex

	modified:   algebra.tex
	modified:   topology.tex

	modified:   algebra.tex

	modified:   algebra.tex