I’ve been away from this blog for longer than I should have. I got stuck in my series on the cotangent complex, partially because I’ve been busy doing other things–namely, trying to learn about the foundations of etale cohomology. As I learn more I might write a few posts. And someday the cotangent complex thing will get finished as a short expository note on my website.

One thing I’ve discovered as of late is that many concepts that I learned earlier in life were in fact shadows or special cases of more powerful and general ones. I’ve consequently had to un-learn many such concepts, to replace them with the newer ones.

Sheaves

An example is basic sheaf theory: like many people, I learned this from Hartshorne chapter II, working out the exercises there. But as I have more recently discovered, many of the methods there are not the appropriate ones for the general theory of sheaves on a site. As an example, Hartshorne defines sheafification (and many other things) on a topological space using stalks. However, on a site this is meaningless because there is no analogous notion in general.

The stalk of a sheaf (or presheaf) on a space $X$ at a point corresponds to the inverse image functor via the inclusion $\{\ast\} \to X$. The analogy in the theory of sites would be the inverse image via a morphism from the site with one point (or something equivalent to this). It turns out, fortunately, in etale cohomology this more general notion does make sense, if $\{\ast\}$ is taken to be the spectrum of a separably closed field. So, if $X$ is a scheme, it is not topological points $\{\ast \} \to X$ that lead to the stalk functors in etale cohomology, but the morphisms $\mathrm{Spec} K \to X$ for $K$ a separably closed field (e.g. the separable closure of the residue fields of the topological points).

It is a curious story that there is an even more general theory of points of a (Grothendieck) topos. A point is a geometric morphism (that is, an adjunction where the left adjoint is exact) between the category of sets and the given topos. The direct and inverse image functors obtained from maps $\mathrm{Spec} K \to X$ show that there are lots of “points” in the etale topos. In fact, on general so-called “coherent” topoi there is a general theorem of Deligne that there are always enough points to detect isomorphisms of sheaves. Apparently this is a topos-theoretic reformulation of the completeness theorem in first-order logic! I’m far from understanding the story here though. (more…)