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The title essentially explains it, but I'll give some background:

I'm giving a talk to some fellow grad students about the relative Picard functor which requires introducing Grothendieck (pre-)topologies, and I'm curious what the historical reason to introduce them is. Particularly I was trying to think of a natural situation in which one would think of this.

What I came up with was essentially the following: if I have an abelian variety $A$ over a field $k$ and I look at multiplication by $m$ coprime to the characteristic of $k$, I get a natural "exact sequence" of varieties

$ 0 \rightarrow A_m \rightarrow A \overset{m_A}{\rightarrow} A \rightarrow 0$

But the real question here is what the exactness means. If we look at the underlying variety(i.e. closed points) we get an exact sequence of abelian groups, but when we think of the corresponding sequence of presheaves

$ 0 \rightarrow h_{A_m} \rightarrow h_A \rightarrow h_A \rightarrow 0 $

This is no longer exact, instead taking sections we get an exact sequence

$ 0 \rightarrow h_{A_m}(V) \rightarrow h_A(V) \rightarrow h_A(V) $

for any $V/k$. This is starting to look sheafy, and so we hope for some general cohomology theory in which I can write $H^1(V,A_m)$ and get some sort of obstruction to exactness on the right. Grothendieck topologies fill this role, and all is right in the jungle.

Is this at all close to the original reasoning?

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up vote 8 down vote accepted

That is not the original motivation, but is similar in spirit. The original motivation is a bit more non-abelian/interesting. For smooth affine groups $G$ and $H$ over a field $k$, one wants to regard the map $G \rightarrow G/H$ as a fiber bundle, which it often is not for the Zariski topology. Serre gave a seminar talk explaining (in modern terms not available at that time) that this is an $H$-torsor for the etale topology on $G/H$. Grothendieck was in the audience. He came up with the idea by the end of the talk, and worked out the basic formalism soon thereafter, and could handle degree-1 cohomologies with constant coefficients using his results on $\pi_1$, but got stuck on getting beyond that until Artin came up with the Brauer group and Kummer sequence ideas.

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Thank you! That is both more non-abelian and more interesting – John Stalfos Sep 28 '13 at 21:19

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