The Stacks project

Try to use L/K notation for field extensions

We could also try to consistenly use "field extension" and not just
"extension" and consistently use "ring extension", etc.

Remove 'f.f.'

Sad IMHO.

Thanks to BCnrd, Dario Weissmann, and sdf
https://stacks.math.columbia.edu/tag/02JQ#comment-2762
https://stacks.math.columbia.edu/tag/02JQ#comment-2765
https://stacks.math.columbia.edu/tag/02JQ#comment-2766
https://stacks.math.columbia.edu/tag/02JQ#comment-2767

valuation ---> valuative

Thanks toKestutis Cesnavicius
http://stacks.math.columbia.edu/tag/03K8#comment-2101

Valuative criteria

Thanks to Brian Conrad

Here is a part of his email concerning the topic of this commit:

"Here is a more direct way to say what is going on in the case of alg.
spaces, it case it might be of some use to include a Remark along such
lines in the Stacks Project.  Let f:X ---> Y be a quasi-compact
separated map between quasi-separated alg. spaces.  Let R be a valuation
ring with fraction field k, and suppose we are given y in Y(R) and x_k
in X(k) over the associated y_k in Y(k).   We want to consider the
problem of whether x_k extends uniquely to an x in X(R) over y, and
possibly after some local extension on R to a bigger valuation ring.  We
can at least pull back along y so that we may rename Y as Spec(R).  That
is, we're give X = qc separated algebraic space over Spec(R), and x_k in
X(k).  We wonder if it extends to X(R), possibly after some local
extension on R to a bigger valuation ring.  Since X_k is separated, so
x_k is a closed immersion into X_k, there is no harm in replacing X with
the "schematic closure" of x_k.

This reduces our study to when X_k = Spec(k) and X is R-flat (as
flatness over val. ring is the same as being torsion-free).  In such a
situation, the key thing is to show that X is univ. closed over Spec(R)
iff X = Spec(R).  The implication "<==" is obvious, and for the converse
it suffices to show X is quasi-finite over Spec(R) (as then X is a
*scheme*, so we can apply the usual thing).  To check being quasi-finite
it is harmless to make a local extension on R to a bigger valuation ring
since that is an fpqc base change (and such base change preserves the
hypotheses we have arranged to hold).  But if we can make such a base
change to acquire a section then the section is a closed immersion (as X
is separated) and its defining ideal must vanish (since by R-flatness
this can be checked at the generic point, where all is clear)."

Fix for xyjax error on Tag 089G

The fix is to make sure there is no &\Spec in xymatrix
However, having this doesn't trigger the bug consistently.

Anyway, the easiest solution for now is to make sure every
& is followed by a space in the LaTeX files.

LaTeX: \Spec

	Introduced the macro

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

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

End conversion of etale to \'etale.

Morphisms of Spaces: Relative conditions

	Trying to understand the relative versions of the local
	conditions found in Properties of Spaces

	Todo:
		Fix currently unfinished discussion of the above
		Add lemma about algebraic spaces etale over fields
		When does an algebraic space satisfy the sheaf
			condition for fpqc-coverings? This is missing in
			the discussion of algebraic space in the
			introductory chapter on algebraic spaces, but it
			doesn't have a high priority.
		Add remark discussing informally the relative conditions
			and what to do with them.

Morphisms of Spaces: Valuative criterion + finite separable extensions

	In the existence part of the valuative criterion for algebraic
	spaces it is enough to take a finite separable extension of the
	fraction field of the dvr. It is sometimes also necessary. It
	seems that if the morphism is separated, then it shouldn't be
	necessary, but we'll return to that later.