April 17, 2012

Topological invariance of the etale site via category theory

Posted by Akhil Mathew under algebra, category theory, commutative algebra | Tags: , |
[6] Comments 

Let {R} be a ring. An {R}-algebra {S} is said to be étale if it is finitely presented and for every {R}-algebra {S'} and every nilpotent ideal {I \subset S'} (or ideal consisting of nilpotents), we have an isomorphism

\displaystyle \hom_R(S, S') \simeq \hom_R(S, S'/I).

In other words, given a homomorphism of {R}-algebras {S \rightarrow S'/I}, we can lift it uniquely to the “nilpotent thickening” {S'}.

The algebras étale over {R} form a category, {\mathrm{Et}(R)}; this is the étale site of {R}. For example, for a field, it consists of the category of all finite separable extensions. \’Etaleness is preserved under base-change, so for any morphism {R \rightarrow S}, there is a functor

\displaystyle \mathrm{Et}(R) \rightarrow \mathrm{Et}(S).

A basic property of étale morphisms is the following:

Theorem 1 Let {R} be a ring and {J \subset R} a square zero (or nilpotent) ideal. Then there is an equivalence of categories

\displaystyle \mathrm{Et}(R) \rightarrow \mathrm{Et}(R/J)

given by tensoring with {R/J}.

This result is often proved using the local structure theory for étale morphisms, but this is fairly difficult: as far as I know, the local structure theory requires Zariski’s Main Theorem for its proof. (Correction: as observed below, one only needs a portion of the local structure theory to make the argument, and that portion does not require ZMT.) Here is a more elementary argument. (more…)

 

July 30, 2011

Quasi-coherent sheaves, presentable categories, and a result of Gabber

Posted by Akhil Mathew under algebraic geometry, category theory | Tags: , , , , |
[10] Comments 

So I’ve failed in my duty as a math blogger. I actually have been writing stuff up, notes on my project, but most of them are not suitable (e.g., too detailed) for a blog. What I really should be doing is blogging about more elementary stuff. I have been trying to learn about Kähler manifolds lately; maybe I can start a short series on them.

Today, I would like to point an interesting observation due to Gabber, which I learned about through a discussion with Theo Buehler on math.SE. Given a scheme {X}, one can consider the category {\mathrm{Qco}(X)} of quasi-coherent sheaves on {X}. This is an abelian category, a subcategory of the category of all sheaves on {X}. Moreover, it is closed under colimits. Now the latter category is wonderfully nice: it’s a Grothendieck abelian category. In other words, it has a system of generators and filtered colimits are exact. One consequence of being a Grothendieck abelian category is that there are automatically enough injectives, by a result in the famous “Tohoku” paper.

But the observation of Gabber shows that {\mathrm{Qco}(X)} is a complete, cocomplete Grothendieck abelian category too. There are a few reasons one might care about this. For one, Grothendieck abelian categories are presentable categories. Basically, presentability means that arguments such as Quillen’s small object argument in homotopy theory work out: namely, one has a “small” set of “compact” objects that generates the category under colimits. (This is in fact the idea behind Grothendieck’s proof that such a category has enough injectives.) Moreover, and perhaps more importantly in this case, the adjoint functor theorem becomes nicer for presentable categories. (more…)