The stack of spaces

Consider the fibred category p : Spaces —> (Sch) where an object of Spaces over the scheme U is an algebraic space X over U. A morphism (f, g) : X/U \to Y/V is given by morphisms f : X —> Y and g : U —> V fitting into an obvious commutative diagram.

Theorem: This is a stack over (Sch)_{fppf}.

In essence the thing you have to prove here is that any descent data for spaces relative to an fppf covering of a scheme is effective. This follows immediately from the results discussed in this post, see Lemma Tag 04SK. You can find a detailed discussion in the chapter Examples of Stacks of the stacks project (in the stacks project we have only formulated this exact statement for the full subcategory of pairs X/U whose structure morphism X —> U is of finite type; this is due to our insistence to be honest about set theoretical issues).

Note how absurdly general this is! There are no assumptions on the morphisms X —> U at all. Now we can use this to show that suitable full subcategories of Spaces form stacks. For example, if we want to construct the stack parametrizing flat families of d-dimensional proper algebraic spaces, all we have to do is show that given an fppf covering {U_i —> U} of schemes and an algebraic space X —> U over U such that for each i the base change U_i \times_U X —> U_i is flat, proper with d-dimensional fibres, then also the morphism X —> U is flat, proper and has d-dimensional fibres. This is peanuts (compared to what goes into the theorem above).

Of course, to show that (under additional hypotheses on the families) we sometimes obtain an algebraic stack is quite a bit more work! For example you likely will have to add the hypothesis that X —> U is locally of finite presentation, which I intentionally omitted above, to make this work.

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  1. Pingback: Families of algebraic spaces « Stacks Project Blog

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