Perfect algebraic spaces and perfect morphisms

Abstract

We develop a theory of perfect algebraic spaces that extend the so-called perfect schemes to the setting of algebraic spaces. We prove several desired properties of perfect algebraic spaces. This extends some previous results of perfect schemes, including the recent one developed by Bertapelle et al. in arXiv:1611.02060. Moreover, our theory extends the previous one developed by Xinwen Zhu in arXiv:1407.8519. The perfect morphisms will provide equivalent descriptions to perfect algebraic spaces. Our method to define perfect algebraic spaces differs from all previous approaches, as we utilize representability of functors. There is a natural notion of algebraic Frobenius morphisms of algebraic spaces, which is analogous to the absolute Frobenius morphisms of schemes. In terms of algebraic Frobenius morphisms, one can define perfection of an algebraic space.