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 42 Let be a -algebra
- An -module is flat if 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 projective if 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 43 Let 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 44 Let 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 45 Let be a -algebra and let be an -algebra. Let denote the multiplication morphism.
(1) The morphism is unramified if there is an element such that , and for all .
(2) The morphism is étale if it is unramified and is a flat -module.
Definition 46 A morphism of -algebras is finite étale if 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 47 Let 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 48 If 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 49 Let 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.