Derived Hom spaces in rigid analytic geometry

TL;DR

This paper develops a derived framework for Hom spaces in rigid analytic geometry, revealing deformation-theoretic information and establishing foundational results for derived non-archimedean analytic geometry.

Contribution

It constructs derived Hom spaces in rigid analytic geometry using a representability theorem, and proves key results like derived Tate acyclicity and proper base change.

Findings

Constructed derived Hom spaces encoding deformation data.

Proved derived Tate acyclicity and proper base change.

Established conditions for derived analytic stack representability.

Abstract

We construct a derived enhancement of Hom spaces between rigid analytic spaces. It encodes the hidden deformation-theoretic informations of the underlying classical moduli space. The main tool in our construction is the representability theorem in derived analytic geometry, which has been established in our previous work. The representability theorem provides us sufficient and necessary conditions for an analytic moduli functor to possess the structure of a derived analytic stack. In order to verify the conditions of the representability theorem, we prove several general results in the context of derived non-archimedean analytic geometry: derived Tate acyclicity, projection formula, and proper base change. These results also deserve independent interest themselves. Our main motivation comes from non-archimedean enumerative geometry. In our subsequent works, we will apply the derived mapping stacks to obtain non-archimedean analytic Gromov-Witten invariants.