# Inverse Limits in the category of Perfectoid Spaces

First I apologize as the following is likely littered with misunderstanding.

My understanding at least from the discussion in Scholze-Weinstein is that inverse limits in the category of adic spaces rarely exist because there isn't a canonical way to define a topology on the direct limit of affinoid rings. Because of this, Scholze defines a weak notion of inverse limit which he denotes by $\sim$. If such a weak inverse limit $X$ is perfectoid, the situation is better as $X$ satisfies a universal property and thus is unique in the category of perfectoid spaces.

My question is do inverse limits always exist in the category of perfectoid spaces? My guess would be no because of the above. However, I am confused by language immediately before II.3.11 and before III.2.34 in Scholze's torsion paper seeming to suggest otherwise.

• It may come down to what morphisms one allows in 'the' category of perfectoid spaces, since this can greatly affect whether limits exist or not. Nov 2, 2016 at 10:02
• Would working in a corresponding pro-category be useful in your setting as that has often been the way forward when/if inverse limits as such are not there or are ill behaved? It has the advantage that it often has geometric significance, and then classes of pro-objects for which certain conditions are satisfied naturally come to the surface of the theory. Of course this avoids the question completely! That being so, the morphisms that you allow will be important as David points out. Nov 2, 2016 at 12:14
• I believe (this agrees with Peter Scholze's "Perfectoid Spaces," at least, and a quick glance at Torsion didn't find anything suggesting otherwise) that the morphisms in the category of perfectoid spaces are simply the morphisms of adic spaces between perfectoid spaces. Nov 3, 2016 at 0:30

## 1 Answer

As mentioned by Scholze in this lecture (around 40:00; uniform Tate rings are called "spectral rings" in the lecture), given a directed system of maps of uniform Tate-Huber pairs $$(R_i,R_i^+) \to (S,S^+)$$, the map $$\varinjlim_i R_i \to S$$ is continuous for the $$\varpi$$-adic topology on $$\varinjlim_i R_i^+$$, where $$\varpi$$ is the image of a pseudouniformizer of some $$R_i$$. Here we need to use the fact that $$S$$ is uniform, so $$S^+$$ is a ring of definition and it has the $$\varpi$$-adic topology.

So filtered inverse limits should exist in the category of (sous)perfectoid spaces since (sous)perfectoid rings are stably uniform. Unfortunately I do not know of a reference that explicitly proves this, but it does seem to be used implicitly in Etale cohomolgy of diamonds.