Non-archimedean integration on totally disconnected spaces

Abstract

We work in the category $\mathcal{CLM}^u_k$ of [5] of separated complete bounded $k$-linearly topologized modules over a complete linearly topologized ring $k$ and discuss duality on certain exact subcategories. We study topological and uniform structures on locally compact paracompact $0$-dimensional topological spaces $X$, named $td$-spaces in [11] and [17], and the corresponding algebras $\mathscr{C}_?(X,k)$ of continuous $k$-valued functions, with a choice of support and uniformity conditions. We apply the previous duality theory to define and study the dual coalgebras $\mathscr{D}_?(X,k)$ of $k$-valued measures on $X$. We then complete the picture by providing a direct definition of the various types of measures. In the case of $X$ a commutative $td$-group $G$ the integration pairing provides perfect dualities of Hopf $k$-algebras between $$\mathscr{C}_{\rm unif}(G,k) \longrightarrow \mathscr{C}(G,k) \;\;\;\mbox{and}\;\;\; \mathscr{D}_{\rm acs}(G,k) \longrightarrow \mathscr{D}_{\rm unif}(G,k) \;.$$ We conclude the paper with the remarkable example of $G= \mathbb{G}_a(\mathbb{Q}_p)$ and $k = \mathbb{Z}_p$, leading to the basic Fontaine ring $${\bf A}_{\rm inf} = {\rm W} \left(\widehat{\mathbb{F}_p[[t^{1/p^\infty}]]}\right) = \mathscr{D}_{\rm unif}(\mathbb{Q}_p,\mathbb{Z}_p) \;.$$ We discuss Fourier duality between ${\bf A}_{\rm inf}$ and $\mathscr{C}_{\rm unif}(\mathbb{Q}_p,\mathbb{Z}_p)$ and exhibit a remarkable Fr\'echet basis of $\mathscr{C}_{\rm unif}(\mathbb{Q}_p,\mathbb{Z}_p)$ related to the classical binomial coefficients.