Fix a complete nonarchimedean field equipped with a fixed norm, with residue field . Let be a -affinoid algebra in the sense of classical rigid geometry. Here’s a funny definition I learned recently.
Definition. A surjection is distinguished if the associated residue norm equals the supremum seminorm . A -affinoid algebra is distinguished if it admits a distinguished surjection from a Tate algebra.
Being distinguished imposes some obvious conditions on : since the supremum seminorm is a norm iff is reduced, it certainly it implies
1) is reduced.
Since any residue norm takes values in , it also implies
If is stable (which holds if is discretely valued or algebraically closed), then the converse is true: 1) and 2) imply that is distinguished. Since 2) is automatic for algebraically closed, we see that any reduced -affinoid is distinguished if is algebraically closed. It is also true that if is a distinguished surjection, then is surjective. Moreover, if satisfies 2), or is not discretely valued, then a surjection is distinguished iff is surjective. Either way, if is distinguished then is a tft -algebra.
All of this can be found in section 6.4.3 of BGR.
Question 1. If is reduced, is there a finite extension such that is distinguished as an -affinoid algebra?
This should be easy if it’s true. I didn’t think much about it.
Now suppose is distinguished, and let be its reduction to a finite type -algebra. As usual we have the specialization map . It is not hard to see that if is a principal open, then is a Laurent domain in . Much less obvious is that for any open affine , the preimage is an affinoid subdomain such that is distinguished and . This is buried in a paper of Bosch.
Loosely following Bosch, let us say an affinoid subdomain is formal if it can be realized as for some open affine . Now let be a reduced quasicompact separated rigid space over . Let us say a finite covering by open affinoids is a formal cover if
1) all are distinguished, and
2) for each , the intersection , which is automatically affinoid, is a formal affinoid subdomain in and in .
This is a very clean kind of affinoid cover: we can immediately build a formal model for by gluing the tft formal affines along their common formal affine opens . Moreover, the special fiber of this formal model is just the gluing of the schemes along the affine opens .
Question 2. For a reduced qc separated rigid space over , is there a finite extension such that admits a formal affinoid cover?