Non-Archimedean Analytic Singular Homology Based On Cosimplicial Perfectoid Spaces and Integration along Cycles

TL;DR

This paper develops a new singular homology theory for non-Archimedean analytic spaces using perfectoid spaces and introduces an integration method along cycles, linking homology with differential forms.

Contribution

It introduces a novel singular homology framework based on cosimplicial perfectoid spaces and defines an integration pairing with differential forms.

Findings

Establishes a new homology theory for non-Archimedean spaces.

Defines an integration along cycles that pairs with differential forms.

Provides tools for studying non-Archimedean analytic geometry.

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.