The Stacks project

Typos in dpa.tex.

Invisible changes

Limit the number of different commands used in math mode

Revised the material in commits Darij Grinberg

Removed the PBW like lemma in commit
b28cd605026f9194c57022c3b301843e52a66f50
Although we should have lemmas like this in the future, it wasn't
necessary for the proof of lemma-finite-finitely-presented-extension.
Instead we moved this lemma later so we can make use of (very
elementary) theory of finite ring maps, thereby reducing to the one
variable case which should be clear to anyone with an undergraduate
course in modern algebra,

Of course this means we *should* add a discussion of the Grobner
basis algorithm and at that point the lemma will be obvious.

dpa.tex, proof of {lemma-gamma-extends}: added alternative to Lazard's theorem

Two small changes

1. Removed the comment on line 8218 of algebra.tex because just in the
paragraph preceding it, it was stated that it suffices

2. Try to keep definitions clean.

Comment on exactness: We make the link between exactness of functors
between abelian categories and general categories in Section Tag 010M of
the chapter on homology. As the commutative algebra chapter precedes the
homology chapter, we cannot use this in the commutative algebra chapter.

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.

oops, nothing was wrong about the principal-ideal case

minor corrections and clarifications; comment on unclear argument

New chapter on divided power algebra

Contains the rudiments of divided powers as well as the material on Tate
resolutions and its application to Avramov's theorem we just added.

Tags: Added new tags

Some more lemmas on divided power structures

Divided power algebras...