Local systems: the path groupoid approach April 21, 2009Posted by David Speyer in Algebraic Geometry, D-modules.
This is the first of the series of posts I promised, on different ways of getting local systems.
In this section, we’ll explain the approach which leads to étale sheaves. I’ll start out by describing the analogous ideas in the topological setting; and then sketch how to make them fully algebraic.
I’ve realized that I need a word for the data which I use to obtain a local system. Because I’m feeling uncreative, I’ll call it the input. Again, is a space of some sort on which we want to build a local system.
For any two points and in , let denote the set of paths from to , modulo homotopy. By concatenating paths, we get a multiplication . In particular, is the fundamental group .
Definition A.1: An A.1 input consists of a vector bundle on and, for every and in and every path from to , an isomorphism , such that .
Let be an A.1 local system and let be an open subset of . Define to be the vector space of sections such that , for and are any two points of and is any path from to that stays within . Notice that, if is a ball, then the dimenion of is the rank of . Also, is a containment of two balls, then the restriction map is an isomorphism.
There is a general philosophy in mathematics that a bundle can be recovered from knowing its sections over open sets. The key technical definition here is that of a sheaf.
So here is a definition which uses the ‘s.
Definition A.2 An A.2 input of rank is the data of (1) for any open subset of , a vector space and (2) for any inclusion of open sets, a map . It is required that (1) satisfy the axioms of a sheaf (2) whenever is a ball, is an -dimensional vector space and (3) whenever is a containment of balls, is an isomorphism.
This is the definition wikipedia gives for a local system.
Making this definition algebraic: To make this definition algebraic, one modifies definition A.2. Open sets are replaced by étale maps.
When working over , we can describe an étale map as an algebraic map such that, for any , we have some open neighborhood of such that is a homeomorphism. (Notice that this definition is local on the source; if we made the definition local on the target, we’d be defining a covering space.)
The reason to introduce étale maps is that there aren’t enough Zariski open sets to “see” the topology of but there are enough étale maps. For example, let be and let’s try to see the nontrivial cycle. We cannot find Zariski open sets , and with nonempty pairwise intersections, but triplewise empty intersection. However, we can find the -fold cover of by itself. This latter étale map let’s us see that has a cycle.
One has to rework the definition of a sheaf to work with maps rather than open sets. This is abstract but not difficult; the precise definition you need is that of a sheaf on a Grothendieck topology.
You’ll notice that there was no reason to work with real vector spaces here; vector spaces over a finite field would have done just as well in the topological discussion, and turn out to do much better once we shift to the fully algebraic setting. It is common to take to be an algebraic variety over a field of characteristic and to be a bundle of vector spaces over a field of a different characteristic . When you hear people talking about -adic methods, that’s what they are talking about.
Finally, I’ll remark that there is a definition of the étale fundmental groupoid, and one could use this to mimic definition A.1. If you unfold what that definition means, however, you’ll see that you are really just working with definition A.2.