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ag.algebraic geometry – Unique route to topologise finite algebra over Huber ring

I cerebrate I can show assertion 2. (I too should not have entry to this bespeak “Bewertungsspektrum und rigide Geometrie” and was attempting to look it up…)

Let $B$ breathe generated as an $A$-module by $b_1,dots,b_n$.

Write $b_ib_j=sum a_{ijk}b_k$ for $a_{ijk}in A$.

Pick a diminutive sufficient model of definition $I$ of $A_0$ such that $Ia_{ijk}subset A_0$ for all $i,j,ok$. Then we behold that

$$

B_0=A_0+Ib_1+dots+Ib_n

$$

is a subring of $B$.

Now point to that if $Usubset A^n$ is launch additive subgroup containing $0$, then $q^{-1}(q(U))$ is an additive subgroup of $A^n$ containing $U$, so it’s launch, and thus $q(U)$ is launch. We now behold that $B_0$ is an additive subgroup of $B$ containing the launch additive subgroup $q(Itimesdots occasions I)$, so $B_0$ is launch.

Now if $Vsubset B$ had been an launch neighborhood of $0$, then $q^{-1}(V)$ accommodates $I^Ntimes dots occasions I^N$ for some sizable $N$, so $U$ accommodates $q(q^{-1}(V))=I^Nb_1+dots+I^Nb_n$. Write $1=a_1b_1+dots+a_nb_n$ for $a_iin A$. Pick a sizable sufficient $M$ such that $M>N-1$ and $I^Ma_isubset I^N$. Then

$$

I^MB_0 = I^MA_0+I^{M+1}b_1+dots+I^{M+1}b_n subset I^Nb_1+dots+I^Nb_nsubset V.

$$

We resolve that the beliefs $(IB_0)^m$ figure an launch neighborhood foundation of $0$ in $B$. Thus $B$ is a Huber ring with ring of definition $B_0$ and model of definition $IB_0$.

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