In this post, we will say something on -algebras. At first sight, this notion is a bit mysterious, because we do not know whether the category has any ring or field serving as base ring or base field, just like where is a ring or a field. In fact, the essential point in defining an algebra is a morphism, , to serve as the multiplication operation on latex $A$(of course, if the algebra has a unit, more conditions should be posed). This is bi-additive(just like the bilinear forms), so this inspires us to redefine the multiplication operation as . So, here the tensor product appears naturally, and as we have said, is a tensor category, that is, the notion of tensor product makes sense in this category. As a result, we can safely talk about -algebras.

Suppose that is a -algebra, then as a -module, . Surprisingly, we can make it into an -algebra. The point is, suppose two -morphisms, then we can define the product of as . Note that this is indeed a -morphism, and the product thus defined indeed gives a structure of -algebra. Using the last proposition of the precedent post, we see that , which means that any -algebra comes from a -algebra. Similarly, for any -module , we have that is an -module, and shows that -modules come from -modules. We can also show that * the category is again an abelian tensor category*, so again we can define -algebras, and things the like.

Now we give several definitions stemmed from commutative algebra:

**Suppose is a -algebra, then(1) an -module is flat if the functor on is exact;(2) an -module is projective if the functor on is exact;(3)if is a -algebra and is an -module, then is an almost finitely generated(resp. almost finitely presented) -module if and only if for all , there is a finitely generated(resp. finitely presented) -module with a morphism such that are annihilated by . We say that is uniformly almost finitely generated if there is some integer such that can be chosen to be generated by elements for all .**

Here are some easy consequences of these definitions:

If is a -algebra and is an -module, then the -module is flat if and only if is almost zero for all -module and for all ; is almost projective if and only if is almost zero for all -module and for all . For their proofs, see Gabber and Ramero’s book, ‘almost ring theory’ section 2.3 and 2.4.

We consider the following example: take to be the completion of and . Then is the completion of and its maximal ideal consists of those elements of whose valuations are , that is, is generated by . As for , we know that . For any , we can find an injection , then we see easily that , thus , so this means that , as a is uniformly almost finitely generated.

There are other concepts to be introduced later.

In the next post, we will introduce the perfectoid -algebras and show some equivalences of categories.