Recently I have been reading through the PhD thesis of Dr. Louis Brewis, "Ramification theory of the p-adic open disc and the lifting problem", which is available free here:
http://webdoc.sub.gwdg.de/ebook/dissts/Ulm/Brewis2009.pdf?origin=publication_detail
The set-up for my question is based on the third chapter at the bottom of the 58th page. Namely, we have a complete DVR R with maximal ideal $(\pi)$, field of fractions K, and algebraically closed residue field k of prime characteristic p. We let $Y = (Spf R[[z]]) \otimes K$ be an open rigid disk over K, with $G$ a finite p-group of automorphisms of $Y$ that has a nonempty number of fixed points. We denote the set of fixed points by $\Delta$. We let $D \subset Y$ be the smallest closed disk that contains $\Delta$.
The part I don't understand is that Brewis writes we can find a finite family $(E_j)_{j \in J}$ of residue classes $E_j \subset D$ such that $E_j \cap \Delta \neq \emptyset$ and that $\cup_j E_j = D$. How do we know that this can be done? In the context of this paper, we are inducting on the cardinality of $\Delta$, and so this finite cover by residue classes cannot just be one set equal to $D$.
I assumed residue class refers to the residue class of an element, i.e. the set of points that have the same image in the special fiber as a given point. Via translation, we should be able to assume WLoG that $D = \{z : v(z) \geq n\}$ where $v$ is the order of $z$ with respect to the uniformizer $\pi$, and where $n \in \mathbb{N}$ is fixed. In such a case, I don't see how we can get a finite cover by residue classes.
As n is positive ($Y$ is open) and $E_j$ is assumed to be in $D$, it would seem to me that we may only consider the residue class of an element in the ideal $(\pi)$, at which point we will have all of $D$ (and possibly more) in the residue class. Might the residue class refer instead to open sets about a point of the form $\{z : |z-a| < p^{-n}\}$? In that case, because the residue field is infinite, I am still unsure how to get a finite cover by residue classes.
Thank you for reading,
Garnet