2023/03/16 by Tianwei Liang, Liang, Tianwei
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2303.09951
openalex publication_date 2023/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop a theory of perfect algebraic stacks that extend our theory of perfect algebraic spaces in arXiv:2303.07672, arXiv:2303.08502 to the setting of algebraic stacks. We prove several desired properties of perfect algebraic stacks. This extends some previous results of perfect schemes and perfect algebraic spaces, 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:1707.05700. Our method to define perfect algebraic stacks differs from all previous approaches, as we utilize representability of algebraic spaces. There is a natural notion of algebraic Frobenius morphisms of algebraic stacks. The algebraic Frobenius morphism provides one with an explicit description of the perfection of an algebraic stack. This gives rise to the perfection functor on algebraic stacks, which enables us to pass between the usual and the perfect world.