(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…)