I continue by introducing standard notions from commutative algebra to the context of almost mathematics. For the references to the proof of the following definition/proposition, see Proposition 4.7 in Scholze’s “`Perfectoid spaces”‘. I follow closely the exposition in the Scholze’s paper in this blog post (consult it for all references).

Definition 42Let be a -algebra

- An -module is
flatif the functor on is exact. If is a -algebra and is an -module, then -module is flat if and only if for all -modules and all , the module is almost zero.- An -module is
almost projectiveif the functor on -modules is exat. If is a -algebra and is an -module, then is almost projective over if and only if for all -modules and all , the module is almost zero.- If is a -algebra and is an -module then we say is
almost finitely generated(resp.almost finitely presented) -module if and only if for all , there is some finitely generated (resp. finitely presented) -module with a map such that the kernel and cokernel of are annigilated by . We say is uniformly almost finitely generated if there is some integer such that can be chosen to be generated by elements for all .

Proposition 43Let be a -algebra. Then an -module is flat and almost finitely presented if and only if it is almost projective and almost finitely generated.

We shall call such -modules **finite projective**. If moreover is uniformly almost finitely generated, we say is **uniformly finite projective**. For such modules we have a good notion of rank.

Theorem 44Let be a -algebra and let be a uniformly finite projective -module. Then there is a unique decomposition such that for each the -module has the property that is invertible, and . Here, we call -module invertible if .

We introduce the notion of étale morphisms

Definition 45Let be a -algebra and let be an -algebra. Let denote the multiplication morphism.(1) The morphism is

unramifiedif there is an element such that , and for all .(2) The morphism is

étaleif it is unramified and is a flat -module.

Definition 46A morphism of -algebras isfinite étaleif it is étale and is an almost finitely presented -module. We write for the category of finite étale -algebras.

Observe that any finite étale -algebra is also finite projective -module. There is an equivalent characterization of finite étale morphisms in terms of trace morphisms. For a -algebra and a finite projective -module we define which is also a finite projective -module. Moreover canonically and there is an isomorphism

In particular, we get a trace morphism

Definition 47Let be a -algebra and let be an -algebra such that is a finite projective -module. We define the trace form as the bilinear form

given by the composition of and the map sending any to the trace of the endomorphism of .

We remark that the last map is defined on as we cannot talk about elements of an almost object . That is, we construct a map in the same way and then pass to the almost setting by taking .

Theorem 48If are as in the definition above, then is finite étale if and only if the trace map is a perfect pairing, that is it induces an isomorphism .

We finish by saying that finite étale covers lift uniquely over nilpotents (recall was a uniformizer of our base field):

Theorem 49Let be a -algebra. Assume that is flat over and -adically complete, that is

Then the functor induces an equivalence of categories . Any is again flat over and -adically complete. Moreover is a uniformly finite projective -module if and only if is a uniformly finite projective -module.