Remark 85.2.3 (McQuillan's variant). There is a variant of the construction of formal schemes due to McQuillan, see [McQuillan]. He suggests a slight weakening of the condition of admissibility. Namely, recall that an admissible topological ring is a complete (and separated by our conventions) topological ring $A$ which is linearly topologized such that there exists an ideal of definition: an open ideal $I$ such that any neighbourhood of $0$ contains $I^ n$ for some $n \geq 1$. McQuillan works with what we will call weakly admissible topological rings. A weakly admissible topological ring $A$ is a complete (and separated by our conventions) topological ring which is linearly topologized such that there exists an weak ideal of definition: an open ideal $I$ such that for all $f \in I$ we have $f^ n \to 0$ for $n \to \infty $. Similarly to the admissible case, if $I$ is a weak ideal of definition and $J \subset A$ is an open ideal, then $I \cap J$ is a weak ideal of definition. Thus the weak ideals of definition form a fundamental system of open neighbourhoods of $0$ and one can proceed along much the same route as above to define a larger category of formal schemes based on this notion. The analogues of Lemmas 85.2.1 and 85.2.2 still hold in this setting (with the same proof).
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