sheaves of modules on an $\ell$-space

Question

Let $X$ be a Hausdorff, locally compact, and totally disconnected topological space, which I call an $\ell$-space, and write $A = C^{\infty}_C(X)$ for the algebra of locally constant complex-valued functions on $X$ with compact support (under pointwise multiplication). The algebra $A$ is nonunital when $X$ is noncompact, so let's look at $A$-modules $M$ which are nondegenerate in the sense that $A \cdot M = M$. Now there is an equivalence between the category of nondegenerate $A$-modules and the category of sheaves of complex vector spaces on $M$, which brings me to my question: can anyone improve on the following construction of this equivalence? I know this is vague, but I would very much like to have a slightly more concrete or "geometrical" equivalence.

As I understand it, given a nondegenerate $A$-module $M$, we start by forming the constant sheaf $[M]$ of locally constant $M$-valued functions on $X$. Denote by $\mathcal{A}$ the sheaf of locally constant complex-valued functions on $X$, so that $\mathcal{A}$ has the structure of a sheaf of $[A]$-modules (if this is as confusing to you as it was to me, notice that a section of $[A]$ can be "evaluated twice" at a point in its domain to give a complex number). Finally we define a sheaf of vector spaces $\widetilde{M} = \mathcal{A} \otimes_{[A]} [M]$, at which point I claim $M \mapsto \widetilde{M}$ is the desired equivalence. Does this seem unnecessarily complicated? Can anyone think of a way to streamline this construction, or for that matter any alternative construction?

By way of motivation, maybe I should mention that this stuff is useful in the study of smooth (sometimes called algebraic) representations of $\ell$-groups (topological groups whose underlying space is an $\ell$-space), such as $p$-adic linear algebraic groups. If you have access to it, Rodier has a nice paper called "Decomposition Spectrale des Representations Lisses" which is unfortunately behind a paywall on Springerlink.

Answer 1

You probably know about this already, but this is discussed in the great paper of Bernstein and Zelevinsky, "Representations of the group GL(n, F), where F is a non-Archimedean local field" to some extent. They define an "l-sheaf", which is a sort of intermediate concept between sheaves (which make sense on any space) and nondegenerate $A$-modules. It consists of the following data:

• Stalks (vector spaces) $\mathcal{F}_x$ at every point $x$ and a vector space $\Gamma(X, \mathcal{F})$ of global sections, consisting of functions $s$ with $s(x) \in \mathcal{F}_x$, such that:

• If you have a function which is not necessarily one of these distinguished cross-sections, but which is locally equal to cross sections everywhere on $X$, then it is in fact in $\Gamma(X, \mathcal{F})$;

• If some $s \in \Gamma(X, \mathcal{F})$ vanishes at $x$, then it vanishes in a neighborhood of $x$;

• The evaluation maps $\Gamma(X, \mathcal{F}) \to \mathcal{F}_x$ are surjective.

Since $X$ is an l-space, in fact this data is enough to reconstruct an entire sheaf as it is usually defined, so this is nothing really new. Proposition 1.14 of that paper asserts that an l-sheaf is "the same" as a nondegenerate $A$-module, in that every such module is obtained as the space of compactly supported global sections of an l-sheaf.