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it seems we were lacking an entry completion of a ring. I started a bare minimum and cross-linked with p-adic integers and localization of a ring. Wanted to do more, but am being interrupted now.
Urs: is there a need for a wider interpretation of ‘completion’ in this context? This deals with completion in the $I$-adic topology, but ring theorists also look at other topologies given by other topologies / filtrations.
Yes, no time, please do
I have created linear topological ring. Just a stub for the moment. It should link with the ideas around completion of a ring eventually, but also with the mention under Grothendieck category…. that is, sort-of, the plan.
A completion of a ring is a completion of a ring to a complete topological ring.
It makes sense only if the original ring is a topological ring. So I am adding word topological and two more sentences to Idea.
A nice subcategory of topological rings suitable for a more general variant of formal schemes and alike objects are pseudocompact rings.
I have changed somewhat linear topological ring to have it compatible with uniform filter. Unfortunately, I made some confusion creating topologizing filter entry with nonstandard definition; I think now it is accepted that uniform and topologizing should be the same. In fact, all three entries should be eventually merged with redirects.
I added a few words to explain the $I$-adic topology on $R$, so as to make sense of the first example (or zeroth example as the examples come after it!)
Thanks for the additions!
I made adic topology a hyperlink… to discover that it already redirects to adic noetherian ring
Do we want to have two different entries for these two keywords?
I suggest just putting an entry for linear topological ring at adic noetherian ring and getting the group of entries interrelating well, possibly with rewording in places. There is no real conflict, except that if one takes the $I$-adic topology on $R$, it is not, to start off with, necessarily complete, hence the need for a completion. (This may suggest that the redirect needs to be more via the ‘Related Ideas’ than in the form of an actual redirect.
There are at least two separate directions to develop from this area of the Lab, so there will be readjustments needed later on so not to worry too much at the moment.
One question we DO have to address is when the rings are commutative, as in general they should not be, but for some applications the main examples are.
Going in this spirit, I have added a remark to p-adic integer on their interpretation as the formal neighbourhood of a prime.
9, Tim, linear topologizing rings have NOTHING to do with noetherianess!
And not that much to do with adic topologies on commutative rings (the latter may provide an example, though).
I note the mention in formal spectrum of ind-schemes. Does anyone know if anyone has explored the square of ideas linking ind-object in $C$ with pro-objects in $C^{op}$ via a duality, then pro-artinian rings are pseudocompact, and pseudo-compact (coherent?) modules over such form the dual of a Grothendieck category. I feel that there should be some missing bits to this and the each part needs categorifying slightly to get a useful NCG situation. (I am being vague because I do not understand NCG.)
I have edited at completion of a ring a bit.
First I expanded what is still titled the Idea-section, trying make clear that to get the underlying ring of a formally completed ring, that limit over quotient rings is taken indeed in the category of (just) commutative rings.
In the course of this I ended up making the whole Idea-section a bit more verbose. And then I expanded the Examples-section a fair bit.
I am just to going away to travel so no time. The general completion is relative: ring completes in more general ring, topological formal whatever context. So let us have that in mind (the current picture is rather special case).
added pointers at_completion of a ring_ to page and verse of the excellent (and apparently more or less original) text (Sullivan 1970/2005).
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