Formal schemes and infinitesimal neighbourhoods

Abstract

The aim of this article is to give a rigorous geometric interpretation of the completion of a ring with respect to an ideal. To this end, we define the infinitesimal neighbourhood of an immersion of formal schemes as the largest possible thickening. Further, we show that immersions of formal schemes locally of formal finite presentation admit the existence of infinitesimal neighbourhoods, which are locally given by completing with respect to an ideal.