The Stacks project

Fixed references to short titles

	using the script we will add in the next commit

More examples of pure modules

	In particular it turns out that if we have R ---> S and N an
	S-module which is countably generated and Mittag-Leffler as an
	R-module, then it is pure relative to S/R. This finally puts a
	direct link between purity and the Mittag-Leffler condition
	which I was looking for.

	Note that there is an example that shows that one cannot go the
	other way. Namely, there exist pure modules over S/R (even when
	everything of finite type and Noetherian) which are not
	Mittag-Leffler.

Remove finite type hypothesis

	In some sense this is a simplification.

Fix an example

	Actually the example was correct but the thing it was a counter
	example against was formulated incorrectly. Now we added it as
	an actual lemma in more-morphisms.tex

Explain purity

	We added the following fun lemma: Suppose R ---> S is a ring map
	with R local and M an R-module. Assume that
		S/m_RS is Noetherian
		M/m_RM is finite over S/m_RS
		M is projective as an R-module
	Then any prime q of S which is an associated prime of M \otimes
	k(p) where p = q \cap R is contained in a prime of S lying over
	m_R. This lemma in some sense explains the notion of purity
	introduced in Raynaud-Gruson...

Tags: Added new tags

First projectivity result

	as stated this lemma can be used to proceed by Noetherian
	induction on the base in certain situations.