The Stacks project

Spell check: words starting with a or A

zerodivisor and nonzerodivisor

	Seems better this way.

Whitespace changes

LaTeX: \Spec

	Introduced the macro

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

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

More variants of splitting lemma

	Sometimes you get affine schemes...

Typos

	Thanks to David Rydh.

More Groupoids: Split out slicing lemmas

	We split the argument in the proof of Bootstrap, Lemma Tag 0489
	into separated pieces and put them in More on Groupoids. This
	will let us recycle those pieces in the (far) future.

More on Groupoids: Slicing lemma fixed

	After a bit of a detour we have finally fixed the lemma.

More Groupoids: Small changes

Groupoids: Put advanced material on groupoids in separated chapter

	We will rewrite the technical lemmas, the slicing lemma, and
	etale localization lemmas in order to fix errors and for
	clarity.

Groupoids: Put advanced material on groupoids in separated chapter

	We will rewrite the technical lemmas, the slicing lemma, and
	etale localization lemmas in order to fix errors and for
	clarity.

Groupoids: Error in proof slicing lemma

	This causes another lemma in bootstrap to be wrong. The proof
	can likely be fixed by adding some hypotheses, but a better
	replacement may be found as well.

Algebra: Grothendieck's lemma for finite presentation case

	We added the non-Noetherian case of what we like to call
	Grothendieck's lemma. It says that if R --> S is a flat and
	essentially of finite presentation local map of local rings, and
	if f is in the maximal ideal such that f maps to a nonzero
	divisor on S/m_RS, then f is a nonzero divisor on S and S/fS is
	flat over R.

	There is also a more general statement for modules.

	See EGA IV 11.3.7.

	For some reason we did not have this version of Grothendieck's
	lemma and it may be that some of the arguments in the algebra
	chapter may be simplified by using this lemma.

Tags: added new tags

Groupoid schemes: slicing lemma

	Here is the lemma promised in commit
	79c56068dee8636e2ae061dd70e3af6d8f9bd30d. The proof is really
	straightforward once you formulate it in the setting where you
	already know that s,t are Cohen-Macaulay.