Let $X=\text{Sp}(A)$ be an affinoid $K$ space, where $K$ is a $p$-adic field. If $f\_0, f\_1,..., f\_s \in A$ generate the unit ideal then we can define the rational subdomain $U= X(f\_0, f\_1..., f\_s) = \{ x \in X: \vert f\_i(x) \vert \leq \vert f\_0(x) \vert \text{ for } i=1...s \}$ of $X$ with coordinate ring $\mathcal{O}(U)= A \langle x\_1,...,x\_s \rangle/(f\_1-f\_0x\_1, ..., f\_s-f\_0x\_s)$.


If $\rho \in \sqrt{\vert K^\times \vert}$ and $\rho >1$ then, by definition, there exists a natural number $n$ such that $\rho^n= \vert t \vert$ for some $t \in K^\times$. We can define another rational subdomain $U(\rho)= X(\rho f\_0, f\_1..., f\_s) = \{ x \in X: \vert f\_i(x) \vert \leq \rho \vert f\_0(x) \vert \text{ for } i=1...s \}$ of $X$. I think its coordinate ring is $\mathcal{O}(U(\rho))= A \langle t^{-1}x\_1,..., t^{-1}x\_s \rangle/(f^n\_1-f^n\_0 x\_1, ..., f^n\_s-f^n\_0 x\_s)$.


As $U \subset \subset\_{X} U(\rho)$ we have, in particular, a sheaf restriction map
$ r\_{U(\rho)U}: \mathcal{O}(U(\rho)) \rightarrow \mathcal{O}(U)$. I think this is just given by $t^{-1} x\_i \mapsto x\_i$ for all $i=1...s$. *I have been trying to work out whether this restriction map is necessarily injective*. So far, my attempts have been direct, i.e. trying to prove that the kernel is zero, but I haven't got anywhere. Irritatingly, I also cannot find any counterexample to the statement I am trying to prove (I might be being stupid).


Does anybody know whether what I am trying to prove is true? If so, could you please point me towards a proof? I would also be very interested to know whether the more general statement $U \subset \subset\_{X} V$ (wide open neighbourhood) with $U$ an affinoid subdomains of $X$ implies that the structure sheaf restriction map $\mathcal{O}(V) \rightarrow \mathcal{O}(U)$ is injective. My original question is a special case of this one. I had a go at the special case first because things seemed more explicit!


Thank you so much in advance for any help!



Injectivity of sheaf restriction maps for wide open neighbourhoods of rational subdomains
If your space has several connected components, then a rational domain may well isolate one of them and you could get a non-injective map.


For an explicit example, you can choose $A = \mathbb{Q}\_p \langle pT \rangle/(T(pT-1))$. Its spectrum has exactly two points: $0$ and $1/p$. To find a example of a non-injective map, you can consider the Weierstrass domains $\{|T| \le 1\}$ (which contains only $0$) and $\{|T| \le 1/p\}$ (which is the whole space).


Also, the map you write is wrong. When $n=1$, you want to send $x\_i$ to $x\_i$ in order to preserve the relations in the quotient.


