On the gluing of formal schemes

TL;DR

This paper proves that the process of gluing formal schemes results in a formal scheme, providing algebraic conditions and highlighting that the glued scheme is always singular, independent of the original schemes' structures.

Contribution

It establishes that gluing formal schemes yields a formal scheme and characterizes when this applies to $k$-formal schemes, also showing the inherent singularity of the glued scheme.

Findings

Gluing of formal schemes results in a formal scheme.

Gluing of $k$-formal schemes is a $k$-formal scheme under certain conditions.

The glued formal scheme is always singular, regardless of initial structures.

Abstract

The main focus of this paper is to show that the gluing of formal schemes is also a formal scheme. The algebraic approach established here also leads us to conclude when the gluing of $k$-formal schemes is a $k$-formal scheme. In addition, we derive that the gluing of formal schemes is always singular, regardless of whether we know the structure of the schemes involved.