The Stacks project

LaTeX

Introduced a macro

\def\Ker{\text{Ker}}

and replace all occurrences of \text{Ker} with \Ker

New macro: \NL for naive cotangent complex

The naive cotangent complex is an important ingredient to several
topics discussed in the Stacks project. It deserves its own macro.

Fix error in artin.tex pointed out by David Rydh

The proof of Lemma Tag 07YM was wrong in two ways:

-- We were using the condition (RS*) in some cases where it didn't
   apply, namely for an extension A' of a Noetherian ring A by a
   module isomorphic to the residue field of a finite type point.
-- The second mistake was in some sense the same mistake thinking that
   the ring A' was Noetherian.

The solution, as pointed out by David Rydh, is twofold:

-- Strengthen assumption (RS*) to cover the first snafu
-- Only prove openess of versality at closed points

As far as I can see, this is harmless in all applications.

There are still some things left to fix. In particular, some of
the material concerning T_x(-) has to be fixed to deal with
non-finitely generated modules.

Move material on fibre products rings and add more

This is what happens when you initially aren't general enough.

Tags: Added new tags

Sharpen criterion openness versality

	Just having a single map with two properties is enough...

Some material about obstruction theories

	This includes the notion of a naive obstruction theory and a
	proof that such a thing is good enough to give openness of
	versality. This was introduced on the blog

	http://math.columbia.edu/~dejong/wordpress/?p=2472

	It is just working out what you would get if you actually had an
	algebraic stack. It seems likely that Artin was motivated in
	formulating his axioms by the idea that such a thing should
	exist.

	Hopefully we can use this to prove openness of versality for
	some nice cases such as families of proper flat algebraic
	spaces... We'll see.