Suppose that is a perfectoid field. In this post, we will introduce algebras over as follows:
A perfectoid -algebra is a Banach algebra such that the subset of consisting of power-bounded elements(as defined in the previous post) is open and bounded, and the Frobenius morphism is surjective(here the element is defined in the first post of this series). Morphisms between perfectoid -algebras are the continuous morphism of -algebras. A perfectoid -algebra is a -adically complete flat -algebra on which the Frobenius morphism induces an isomorphism (here -adically complete means that ); morphisms between perfectoid -algebras are morphisms of -algebras. A perfectoid -algebra is a -adically complete flat -algebra on which the Frobenius morphism induces an isomorphism ; morphisms between perfectoid -algebras are morphisms of -algebras.
We will use Scholze’s notations, that is, we write denotes the category of perfectoid -algebras and denotes the category of perfectoid -algebras, and so on.
The following proposition shows that perfectoid -algebras and perfectoid -algebras are closely related:
If is a perfectoid -algebra, then the Frobenius morphism induces an isomorphism , and is a perfectoid -algebra.
Proof: By definition, is surjective. Suppose that is sent to , that is , say . Since is power bounded, so is , so . Moreover, it is clear that , so induces an isomorphism. Next we need to show that is -adically complete. This is not difficult. We define , and . It is necessary to verify that is well-defined. Indeed, note that, according to the definition, for , thus is a Cauchy sequence in , as is complete, so is , thus exists. Moreover, if , this means that for all , so . So, is indeed well-defined. And it is not difficult to show that and are inverse one to the other, so we showed that is -adically complete. One last thing is to show that is a flat -module. This is not difficult. To finish the proof, we need the following results:
If is a flat -module, then is a flat -module.
Conversely, we have that
If is a -algebra, and let . If we give the Banach -algebra structure making open and bounded in , then is the set of power-bounded elements, and is perfectoid -algebra, and the Frobenius morphism .