arXiv subjects

Tianwei Liang

Publications and source records attributed to Tianwei Liang.

4 recordsLinked to original sources

arXiv2023

Perfect algebraic spaces and perfect morphisms

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.

math.AG↗

arXiv2023

On the perfection of algebraic spaces

This paper is a subsequent paper of arXiv:2303.07672. We will continue our research on the subject of perfect algebraic spaces that is developed in arXiv:2303.07672. By means of algebraic Frobenius morphisms, we define the perfection of arbitrary algebraic spaces of prime characteristic. There is a natural perfection functors on algebraic spaces. We prove several desired properties of the perfection functor. This extends nearly all previous results of the perfection functor on schemes, including the recent ones developed by Bertapelle et al. in arXiv:1611.02060. Moreover, our theory extends the previous one developed by Xinwen Zhu in arXiv:1707.05700v1, arXiv:1407.8519. The perfection functor rises the notion of perfection of sites, which enables us to restate some of Zhu's theory in arXiv:1707.05700v1. Then we show that our theory of perfect algebraic spaces in arXiv:2303.07672 is equivalent to Zhu's theory in arXiv:1707.05700v1 when the base scheme is a perfect field of prime characteristic.

math.AG↗

arXiv2023

Perfect algebraic stacks

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.

math.AG↗

arXiv2023

A generalization of anti-homomorphisms

We prove some nice properties of anti-homomorphisms, some of which are analogic to that of homomorphisms. Meanwhile, we develop a new kind of composition called $*$-composition such that the $*$-composition of two anti-homomorphisms is still an anti-homomorphism. Moreover, we develop a certain kind of categories called factorization categories, which generalize anti-homomorphisms and provide a general framework to study "anti" phenomenon. We deduce several results of anti-homomorphisms in great generality.

math.CT↗