What is the purpose of the flat/fppf/fpqc topologies?

Question

There have been other similar questions before (e.g. What is your picture of the flat topology?), but none of them seem to have been answered fully.

As someone who originally started in topology/complex geometry, the étale topology makes some sense to me. It's sort of like the complex topology, in that there are enough "open sets" that things like the inverse function theorem work.

But I don't really understand what these other more "exotic" topologies represent. From the (naive) background of topology, it sounds like we're just "adding more open sets" to our topology (which I know isn't right, since this isn't a topology in the standard sense).

So what do they represent? What do they do for us?

Answer 1

I'm a topologist, and so this answer is going to specifically be about analogies with ordinary topology. I like to think of different Grothendieck topologies as corresponding to different "allowable" ways to build up a space using a quotient topology.

1. The Zariski topology is like recognizing that a space $X$ can be built up from a collection of open subsets $U \subset X$ which cover $X$ and identifying points in common intersections.

2. The etale topology is like recognizing that if a map $Y \to X$ is surjective and a local homeomorphism, then this map makes $X$ homeomorphic to the quotient of $Y$ by an equivalence relation.

3. The fppf topology is similar to the etale topology, but now allowing maps $Y \to X$ which are open.

4. The fpqc topology, for lack of a better analogy, is like allowing arbitrary maps $Y \to X$ which make $X$ into a quotient space of $Y$.

Much of the study of these topologies is the study of sheaves on them. The choice of topology has at several significant consequences.

1. The choice of topology determines how much functoriality your sheaf has. (In particular, choices like "big site" or "small site" determine how many objects $Y$ you can evaluate your sheaf on!)

2. The choice of topology determines what kind of data you need to construct things (which is code for descent theory). We know that if $Y \to X$ is a quotient map, we can build a vector bundle on $X$ by taking a vector bundle on $Y$ and imposing a compatible equivalence relation on it (e.g. from a clutching function). Similarly, to build a sheaf in topology $\tau$ we just need to build it on a $\tau$-cover and glue it together.

3. The choice of topology determines your definition of "local". I really like Simon Pepin Lehalleur's comment here, because it describes exactly the following problem: when is a map ${\cal F} \to {\cal G}$ of sheaves surjective? It is surjective when it is surjective locally. For example, the n'th power map $\Bbb G_m \to \Bbb G_m$ is a surjection of sheaves precisely when, locally, every invertible function is an n'th power of some other invertible function. That's rarely true in the Zariski topology; it's true in the etale topology when $n$ is invertible, because then adjoining a solution of the polynomial $(x^n - \alpha)$ is an etale extension; it's always true in the fppf topology because adjoining a solution of the polynomial $(x^n - \alpha)$ is always fppf.

Answer 2

This is sort of an expansion of Simon's comment above.

I think the purpose of any topology is to give the "correct" cohomology groups, whatever that means. In general you probably want to work with cohomology groups in the coarsest topology you can; for example, computing explicitly in the Zariski topology is much nicer than computing explicitly in the flat topology.

For example, let $X$ be a variety over a field. If $F$ is a sheaf of coherent $\mathcal{O}_X$-modules, then the Zariski topology calculates the "correct" cohomology groups $\mathrm{H}^i(X,F)$, in the sense that going to the finer étale or flat topology gives the same result.

If $F$ is the sheaf represented by a smooth group scheme, then the étale topology gives the "correct" cohomology groups. In particular, as pointed out by Simon in his comment, the Kummer sequence is exact in the étale topology whenever $n$ is coprime to the residue characteristics, so cohomology with values in $\mu_n$ is related in a nice way to Picard groups, Brauer groups and so on. It also agrees with the complex cohomology in characteristic zero. But this doesn't work if the characteristic divides $n$, because then $\mu_n$ is not smooth. To get the "correct" cohomology for $\mu_p$ in characteristic $p$, and in particular for the Kummer sequence to be useful, you really have to work in the flat topology.

Answer 3

The question is what kind of structure you are interested in. For example, the étale topology sees more than you might like as a topologist/complex geometer. If you look at Spec(k) where k is not algebraically closed, the étale topology will see the Galois group. As a topologist/complex geometer you might also like the Nisnevich topology. It has enough covers to make a smooth variety look locally like affine space, but it doesn't see the "arithmetic stuff" that only depends on the ground field and not on the variety itself. But if you are interested in the properties of the ground field, too, you will choose the étale topology.

As for the flat/fppf/fpqc topologies, I think you already got some very convincing answers. I can't add anything to them. Maybe this is rather an overlong comment than an answer.