The Stacks project

Fix snafu in flat.tex

The proof of a small part of lemma-weak-bourbaki-pre
was erroneous... The problem was in the sentence

"Let $z \in M \otimes_R N$ be an element whose annihilator in $S$
has radical $\mathfrak q$."

In general weakly associated primes don't automatically have
this property, only after you localize does it become true.
To work around it we invented a trick where you choose a z
such that \mathfrak q is minimal over its annihilator and
then you look at what happens after you map z up into a
suitably chosen overring (using the machinery of RG).

If you look at the proof of the corresponding fact in RG
then you don't find any explanation of how to deal with this
whatsoever, so this could mean there is a much simpler way
to deal with it. Anyway, I am quite convinced the current
argument is correct, so I will leave it like this for now.

Get rid of list environments

Fix 'the the ' occurences

Thanks to Burt Totaro

pell check: words starting with n, o, p, q, r, N, O, P, Q, or R

Typos

Thanks to Bhargav Bhatt for the typo in limits.tex

Whitespace changes

LaTeX: \Spec

	Introduced the macro

	\def\Spec{\mathop{\rm Spec}}

	and changed all the occurences of \text{Spec} into \Spec.

Fixing references

	This time we're fixing the type of the reference

More local freeness

	As opposed to generic freeness. These lemmas are all trivial
	consequences of the proposition in the finitely presented case
	combined with earlier results on finite type flat modules over
	local rings.

Tags: Added new tags

Impurities...

	A bunch of lemmas on impurities. Enough to show that pureness is
	preserved under base change if we start with a noetherian base
	scheme. To be continued.

Theorem on when ft + flat => fp

	plus a bunch of easy consequences.

Bourbaki lemma in finite type case

	Finally!