Non-Archimedean Analytic Singular Homology Based On Cosimplicial Perfectoid Spaces and Integration along Cycles
Tomoki Mihara
TL;DR
This paper develops a non-Archimedean analytic singular homology theory for adic spaces over a field $k$ by leveraging a cosimplicial perfectoid object $Δ_{K/k}^{ullet}$ built from the completed group algebra of $D^n_k$, encoding $p$-power exponential maps on affine polytopes. It then defines an integration along cycles valued in the de Rham period ring $B_{dR}$, pairing cycles with differential forms through a $p$-adic logarithm of $p$-power roots and a $S$-compatible integration mechanism, and establishes a non-Archimedean analogue of the Stokes theorem. The framework rests on constructing sheafy Banach algebras, $p$-divisible normed groups, and perfectoid algebras, together with analytic standard simplexes and dual objects, to realize a Galois-represented homology theory and a concrete integration theory along rigid analytic cycles. A Tate-curve example demonstrates nontrivial cycles detected by the integration, illustrating the potential to compute $p$-adic period pairings in non-Archimedean geometry. Overall, the work provides a rigorous toolkit for non-Archimedean analytic homology and period-integral theory with clear paths to applications in cycle nontriviality and $p$-adic integration.
Abstract
We introduce singular homology for non-Archimedean analytic spaces using a cosimplicial perfectoid space as a Galois representation. We define an integration along a cycle, which gives a pairing with the singular homology and the space of differential forms.