Functorial point of view

If you take a look at the current version of the stacks project you will see that essentially the last chapter is the one on algebraic stacks, and moreover that it is practically empty. Why?

One of the decisions made early on was to build up the material along historical lines in the usual manner. Namely:

  1. We develop some commutative algebra and some theory of sheaves on topological spaces.
  2. We define schemes as locally ringed spaces which locally look like the spectrum of a ring.
  3. We develop the theory of sheaves on a site and we discuss schemes and morphisms of schemes. We define the big fppf/etale site of a scheme. We also discuss in some detail the notion of descent for schemes and descent of properties of morphisms of schemes.
  4. We define algebraic spaces as fppf sheaves which etale locally look like a scheme.
  5. We study the notion of stacks fibred in groupoids over a site. We study properties of algebraic spaces and morphisms of algebraic spaces. We study descent for algebraic spaces.
  6. Finally we use this material to define algebraic stacks as stacks fibred in groupoids on the category of schemes in the fppf topology whose diagonal is representable by algebraic spaces and which have a smooth covering by a scheme.

This seems like a bit of overkill at first. Why can’t we do a little bit of commutative algebra, and then go straight to the definition of algebraic stacks as certain stacks fibred in groupoids over the opposite category of the category of rings endowed with the fppf topology?

The answer is of course that in principle you can do this. One advantage of doing this is that you might not have the kind of repetition that the progression 1,2,3,4,5,6 above shows. Moreover, every geometric object would be an algebraic stack and you would not have to use the customary abuse of notation (such as statements of the form “a scheme is an algebraic stack”) which one finds in papers on algebraic stacks. In fact, feel free to clone the stacks project and to rewrite it in this way.

On the other hand, any full discussion of the theory of algebraic stacks is going to mention affine schemes, schemes, and algebraic spaces. It will still be the case that the most interesting objects of study are algebraic varieties, and their moduli spaces. For example curves, abelian varieties, and Jacobians of curves, moduli spaces of such, Shimura varieties, K3-surfaces, etc, etc. Proofs of foundational theorems such as “coherence of proper pushforward”, or Zariski’s Main Theorem will likely still be proved by proceeding via arguments through the case of schemes.

Currently, the arguments dealing with schemes and morphisms of schemes are of a different nature than those for algebraic spaces and the arguments dealing with algebraic spaces will be I am sure of a different nature again. This means that there is actually less repetition in the sequence 1,2,3,4,5,6 as one expects at first. Also, when working on a new result for say algebraic spaces, by first proving the needed algebra lemmas, then proving the results for schemes, and finally the result for algebraic spaces, we automatically organize the material, and we prove each result in its natural setting.

A related observation is that the reader need only know about schemes when reading any of the results in the theory of schemes. (Of course eventually we will prove results which can be formulated in the language of schemes but whose proof uses algebraic stacks.) Similarly for the algebra results and the results on algebraic spaces.

There is also a psychological component. Sure you can define algebraic stacks without first defining any intermediate geometric objects. However, once this is done, there you are, and there is nothing that you can hold onto and relate the objects to… it seems  a bit similar to introducing quantum physics without first talking about classical mechanics. Sure it is fundamentally more important, but what is it really telling us about the real world?

5 thoughts on “Functorial point of view

  1. ZMT has a very nice proof that relies only on commutative algebra. Mel Hochster has it up in the notes for Math 615, which is a commutative algebra topics course he’s teaching this semester at umich.

    • Actually that depends a little on what form of ZMT you mean. In the stacks project there is an algebraic form in the Chapter on algebra (Algebra, Theorem Tag 00Q9) which of course has a completely algebraic proof. This is basically the same as the one in Mel Hochster’s lecture notes you mention. For algebraic geometers this is reformulated for morphisms of schemes in the chapter on morphisms of schemes. In “More on Morphisms” we elaborate on this theme. Among other things you can find there the statement that a proper morphism with finite fibers is a finite morphism, which no longer even has a formulation purely in terms of algebra. Technically (for the stacks project) a very useful version is that a quasi-finite and separated morphism is quasi-affine, since it guarantees effective descent of such morphisms later. In fact for qf+sep morphism f : X –> S the Stein factorization X –> S’ –> S has the properties

      X –> S’ is open, and
      S’ –> S is integral.

      If S is qc+qs then you may also factor any qf+sep morphism X –> S as X –> Y –> S with X –> Y open and Y –> S finite by applying a simple limit argument to the result above (this is for the moment missing from the stacks project as far as I remember).

  2. Sure, but the amount of scheme theory you need to prove that statement about proper+finite fibers => finite morphisms is very limited. I feel like you could still reduce this statement to the commutative algebra version + a general statement about grothendieck topologies, and a specific statement about the behavior of the global zariski topology. The idea is that taking Sh(CRing_Zar^op) should carry certain properties of the Zariski topology up to all of its sheaves that satisfy the local requirements for being schemes (as functors of points, of course.)

  3. Is the claim that “the amount of scheme theory you need … is very limited” really true? I always thought of this result as being pretty substantial. What sort of statement about the global Zariski topology do you have in mind?

  4. proper and finite fibers are precisely the kind of argument you reduce to affines. It’s not hard to prove finite fibers from Hochster’s proof (this was on the homework this week). I don’t exactly know what properness means (<=== not a geometer), but I do know that it's local, which means that we can reduce arguments about it to the affine case.

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