# On equivariant formal deformation theory

## TL;DR

This paper generalizes deformation theory results to abstract categories, showing how group actions and coverings extend infinitesimally, with implications for algebraizability and Kodaira vanishing.

## Contribution

It reinterprets and extends Rim's results within a categorical framework, demonstrating the algebraizability of infinitesimal coverings and the stability of pre-Tango structures under certain pullbacks.

## Key findings

- Finite étale coverings can be infinitesimally extended.
- Pre-Tango structures are preserved under specific pullbacks.
- Results have implications for Kodaira vanishing in degree one.

## Abstract

Using the set-up of deformation categories of Talpo and Vistoli, we re-interpret and generalize, in the context of cartesian morphisms in abstract categories, some results of Rim concerning obstructions against extensions of group actions in infinitesimal deformations. Furthermore, we observe that finite \'etale coverings can be infinitesimally extended and the resulting formal scheme is algebraizable. Finally, we show that pre-Tango structures survive under pullbacks with respect to finite, generically \'etale surjections $\pi:X\rightarrow Y$, and record some consequences regarding Kodaira vanishing in degree one.