The Stacks project

Improved characterization of open immersions

zerodivisor and nonzerodivisor

	Seems better this way.

LaTeX: \Spec

	Introduced the macro

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

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

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.

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

Lemmas on universally injective maps

	will be continued...

Inverting elements

	sometimes gives universally injective maps