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

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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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