Non-archimedean integration on totally disconnected spaces

Abstract

We work in the category ${\mathcal{CLM}}_k^u$ 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 ${\mathcal{C}}_{?}(X,k)$ 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 ${\mathcal{D}}_{?}(X,k)$ 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 $${\mathbf{A}}_{\inf }=\mathrm{W}\left(\widehat{{\mathbb{F}}_p[[t^{1/p^{\infty }}]]}\right)={\mathcal{D}}_{\mathrm{unif}}({\mathbb{Q}}_p,{\mathbb{Z}}_p).$$ We discuss Fourier duality between ${\mathbf{A}}_{\inf }$ and ${\mathcal{C}}_{\mathrm{unif}}({\mathbb{Q}}_p,{\mathbb{Z}}_p)$ and exhibit a remarkable Fréchet basis of ${\mathcal{C}}_{\mathrm{unif}}({\mathbb{Q}}_p,{\mathbb{Z}}_p)$ related to the classical binomial coefficients.

74 pages