Conical calculus on schemes and perfectoid spaces via stratification
Abstract
In this paper we show that, besides the usual calculus involving K\"ahler differentials, it is also possible to define conical calculus on schemes and perfectoid spaces; this can be done via a stratification process. Following some ideas from [1-2], we consider some natural stratifications of these spaces and then we build upon the work of Ayala, Francis, and Tanaka [3] (see also [4-5] and [18]); using their definitions of derivatives, smoothness and vector fields for stratified spaces, and thanks to some particular methods, we are able to transport these concepts to schemes and perfectoid spaces. This also allows us to define conical differential forms and the conical de Rham complex. At the end, we compare this approach with the usual one, noting that it is a useful addition to K\"ahler method.