The Stacks project

fix small typos

Spell check: words starting with b, c, B, or C

LaTeX: Remove useless brackets

LaTeX: \Mor

	Introduced a macro

	\def\Mor{\mathop{\rm Mor}\nolimits}

	and replaced all the occurences of \text{Mor} with \Mor.

Reduced algebraic stacks

Check representability by AS on flat and fp cover

	The statement is that a morphism of algebraic stacks X ---> Y is
	representable by algebraic spaces if there exists an algebraic
	space W and a surjective, flat and locally finite presentation
	morphism W ---> Y such that W \times_Y X is an algebraic space.

Tags: added new tags

Morphisms representable by algebraic spaces and presentations

	This is what one would call a trivial lemma, were it not for the
	fact that we haven't put enough work into earlier chapters. What
	we need to make the proof of this lemma easier is a clearer
	formulation of the cocartesian property of the diagram

		R ---> U
		|      |
		v      v
		U -> [U/R]

	This should be done by describing the category

		Mor([U/R], \mathcal{X})

	as follows: Objects are pairs (x, \beta) where x is an object of
	\mathcal{X} over U and \beta : s^*x ---> t^*x is a morphism such
	that c^*\beta = pr_0^*\beta \circ pr_1^*\beta. Morphisms are
	going to be maps of x's with some property regarding the
	\beta's.

	The only slight problem with this is that U, R may not be
	schemes, in which case we have strictly speaking not defined
	what it means to take an objects of \mathcal{X} over them.