June 12, 2011

Cohomological dimension

Posted by Akhil Mathew under topology | Tags: , , |
[3] Comments 

(This is the fourth in a series of posts intended to cover the basics of Verdier duality, starting here.)

One of the features of derived categories that seems to require the most bookkeeping is the size. Many results apply specifically to the bounded-below or bounded-above derived categories, for instance; the problem is, in general, with statements like the following. If F is a left-exact functor on some abelian category with enough injectives and T^\bullet is an acyclic complex consisting of F-acyclic objects, then F(T^\bullet) is not necessarily acyclic (though it is if the complex is bounded below). Dimensionality bounds will, for apparently similar reasons, play a crucial role in the proof of Verdier duality, and it will be necessary to show that the spaces in question are fairly nice. I will try to explain the necessary tools in this post, after which we can actually start the proof.

1. Cohomological dimension

The Verdier duality theorem will apply not only to manifolds, but more generally to locally compact spaces of finite cohomological dimension, and it will thus be useful to show that simple spaces (e.g. finite-dimensional CW complexes) satisfy this condition. The resulting theory will also show that much of basic algebraic topology can be done entirely using sheaf cohomology.

Definition 1 A locally compact space {X} has cohomological dimension {n} if {H^k_c(X, \mathcal{F}) =0} for any sheaf {\mathcal{F} \in \mathbf{Sh}(X)} and {k > n}, and {n} is the smallest integer with these properties. We shall write {\dim X} for the cohomological dimension of {X}.

A point, for instance, has cohomological dimension zero. For here the global section functor is an equivalence of categories between {\mathbf{Sh}(\left\{\ast\right\})} and the category of abelian groups. Our first major goal will be to show that any interval in {\mathbb{R}} has cohomological dimension one. (more…)

 

June 10, 2011

Soft sheaves

Posted by Akhil Mathew under topology | Tags: , |
1 Comment 

The statement of Verdier duality was fairly fancy–it used the jazzed up language of derived categories, and was phrased in terms of the existence of an adjoint to a suitable derived functor. Though by the end of it hopefully it’ll become clear that, at least for manifolds, the mysterious f^! adjoint functor has a fairly concrete interpretation that will turn out to be plain old Poincare duality.

But before we get there, we need to discuss some sheaf theory. I have usually  thought of sheaves in the context of algebraic geometry, and I wanted to review (at least for my own benefit) some of the classical theory of sheaves on ordinary non-pathological topological spaces, like locally compact Hausdorff ones. In addition, it will at any rate be necessary to introduce the compactly supported cohomology functors to get Verdier duality. So I’ll do a couple of preparatory posts.

1. Soft sheaves

Let {X} be a locally compact space. If {\mathcal{F} \in \mathbf{Sh}(X)} and {Z \subset X} is closed, we define {\Gamma(Z, \mathcal{F}) = \Gamma(Z, i^*\mathcal{F})} for {i: Z \rightarrow X} the inclusion. This allows us to make sense of a section “over a closed subset.” The interpretation via the espace étale is helpful here: if {\mathfrak{X} \rightarrow X} is the espace étale of {\mathcal{F}}, then a section over {Z} is the same thing as a section of the projection {\mathfrak{X} \rightarrow X} over {Z}. It follows from the latter interpretation that if we have sections {s, t} on closed subsets {F, G} that agree on {F \cap G}, one obtains uniquely a closed subset on {F \cup G}. Now we want to show that we can recover this notion from the familiar idea of sections over an open set.

Lemma 1 If {Z \subset X} is compact, we have

\displaystyle \Gamma(Z, \mathcal{F}) \simeq \varinjlim \Gamma(U, \mathcal{F}),

where {U} ranges over open sets containing {Z}. (more…)