This post is a response to Brian Conrad asking the following question: “How come the stacks project includes a base scheme S in the definition of algebraic spaces? Namely, we could think of an algebraic space over S as just an algebraic space over Spec(Z) equipped with a morphism to S.”<\/p>\n
The short answer is that everywhere in the stacks project you can just think of X as an algebraic space over Z endowed with a morphism to S whenever you see the statement “let X be an algebraic space over S”. If you do this, then in many statements mentioning S is indeed completely superfluous.<\/p>\n
A longer answer is that it is related to the setup in the stacks project, including our choices regarding set-theory.<\/p>\n
When you see “Let S be a scheme” at the beginning of a lemma\/proposition\/theorem about algebraic spaces then this really means “Choose a partial universe of schemes to work with which contains S”. I can quantify exactly what I mean with “partial universe” and we prove using ZFC that partial universes exist containing any given set of schemes. (See Lemma Tag 000J<\/a>.)<\/p>\n For the stacks project an algebraic space is a functor defined on the comma category C\/S where C is this partial universe. So an algebraic space is<\/strong> a functor F : (C\/S)^{opp} —> Sets. If you want to get an algebraic space over Spec(Z) you have to apply “Change of base scheme” (Section Tag 03I3<\/a> of the chapter “Algebraic Spaces”). Of course this is a completely trivial operation, but to get all the details right this is what you have to do.<\/p>\n A consequence is that an algebraic space over Spec(Z) doesn’t (a priori) have a value on all schemes, only on the schemes in the partial universe C. But you can apply “Change of big site” (Section Tag 03FO<\/a> of the chapter “Algebraic Spaces”) to enlarge your partial universe to contain any given set of schemes.<\/p>\n A similar story goes for algebraic stacks. But… what we’ve done for algebraic stacks in Properties of Stacks, Section Tag 04XA<\/a> is introduce the customary abuse of language which forgets about all of this set-theoretical nonsense. This language is also less precise.<\/p>\n We could (and maybe should) do the same thing for algebraic spaces. On the other hand, it mostly doesn’t hurt; it just looks a bit funny here and there.<\/p>\n [Post edited on May 30, 2012.]<\/p>\n","protected":false},"excerpt":{"rendered":" This post is a response to Brian Conrad asking the following question: “How come the stacks project includes a base scheme S in the definition of algebraic spaces? Namely, we could think of an algebraic space over S as just … Continue reading