Remark 87.2.3 (McQuillan's variant). There is a variant of the construction of formal schemes due to McQuillan, see . 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 87.2.1 and 87.2.2 still hold in this setting (with the same proof).

Comment #2165 by David Hansen on

A very minor possible tweak in the end of the third sentence: "... such that any neighborhood of 0 contains I^n for some n \geq 1."

There are also:

• 12 comment(s) on Section 87.2: Formal schemes à la EGA

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AI3. Beware of the difference between the letter 'O' and the digit '0'.