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Derived ring theory

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Derived ring theory (edit | talk | history | protect | delete | links | watch | logs | views) – (View log | edits since nomination)
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I'm having some trouble seeing how this meets any definition of notability (WP:N) when there are no hits when searching "Derived ring theory" in Google Scholar. Perhaps some parts of this can be merged with Ring theory and related pages. Afonso Dimas Martins (talk) 17:31, 6 May 2025 (UTC)[reply]

Delete. The term does not exist in the literature, "derived ring" is used, but infrequently and in particular contexts. Викидим (talk) 18:39, 6 May 2025 (UTC)[reply]
  • Delete As others have noted, the term does not seem to be used. Citations on the page that should be pointing to external sources are just cite notes making verification difficult if not impossible. As-written is also much too technical and may be too niche for an encyclopedic article. Anonrfjwhuikdzz (talk) 00:18, 7 May 2025 (UTC)[reply]
    Many of our serious math articles are written without any examples, especially when they touch the category theory. As a famous saying goes, "A monad is just a monoid in the category of endofunctors, what's the problem?" (for the avoidance of doubt, I am no expert in the category theory, but can understand this half-joke). This is just the style the mathematicians prefer nowadays, with our articles reflecting the real-world (IMHO, actually a problem in the real world). This approach of brevity and total lack of introductory context causes our mathematical articles to become totally unverifiable by any person who is not a true expert in a particular field. I hope for the author to come here and explain the text to us. Викидим (talk) 02:51, 7 May 2025 (UTC)[reply]
    There are 30K math articles, of which over 6K are stubs, and another 12K ranked as "start". Almost all of these fail to include appropriate ledes, introductory paragraphs, never mind worked examples and articulated expositions. Personally, over twenty years, I've "fixed" over 600 articles by adding paragraphs, examples and rewriting ledes. I've done major rewrites of several dozen articles. The content here is not something you can flick off in an afternoon; its hard work, and it takes time. This article is a fairly typical of a stub/start-class article. It will take time to bake into something a beginner could hope to grok in short order. 67.198.37.16 (talk) 19:09, 7 May 2025 (UTC)[reply]
  • Keep Yes, the term derived ring theory is not used much in literature. But the concept, the derived version of ring theory clearly does exist; it's called under various names like derived commutative algebra. (Kind like, we have Timeline of category theory and related mathematics even though the term is not used much. Similarly, derived ring theory merely meant a derived version of ring theory.) So, it makes sense to have an article on the topic. We have derived algebraic geometry, which is closest but is the focus is different. Ring theory is already long so a spin-off like this is a good idea. -- Taku (talk) 04:16, 7 May 2025 (UTC)[reply]
    • Comment I have not managed to understand the article in the 30 seconds or so I have devoted to the attempt so I can't really speak to the subject matter. But if the name is not used much in literature, we shouldn't have an article at that name. I don't think the comparison with a "timeline" article is really on-point; in that case the word "timeline" is describing the article, not the subject matter, and is not in danger of creating a neologism. Would it make sense to move to derived commutative algebra, if that term is used more? --Trovatore (talk) 04:52, 7 May 2025 (UTC)[reply]
    Definitely that’s one option. It’s just there is also a derived noncommutative algebra of sort used in noncommutative algebraic geometry. So my thinking was "derived ring theory" can be a broad enough name to cover various derived generalizations of rings. If the neologism is a concern, maybe "derived version of ring theory" or something may be another option. —- Taku (talk) 05:04, 7 May 2025 (UTC)[reply]
    If "derived ring" is a standard term, then my concern about "derived ring theory" being a neologism is significantly lessened. That said, is there any need for the "theory" bit? If a derived ring is a well-specified sort of object, maybe lead with object, and discuss the theory in the body. As an aside, I've never quite made my mind up about object-versus-field-of-study articles. We have, for example, both group (mathematics) and group theory; the division of labor between them is a little hard to work out. The hatnotes claim that the "theory" article is "more advanced", but it's not at all clear why exactly that should be. Arguably the "theory" article should be more general in scope and more focused on applications than the "object" article. But for this case, it seems doubtful that there's enough material to split up that way. --Trovatore (talk) 06:08, 7 May 2025 (UTC)[reply]
    No, "derived ring" isn't some well-established object studied by many. I suppose it's more like a platonic concept; people study something more tangible like spectral ring or ring spectrum, which are concrete incarnations of a hypothetical derived ring. That was actually a reason for the "theory" bit in the title (so not to suggest there is a thing called a derived ring). You're right that in Wikipedia we don't quite have a good naming convention for a topic in math instead of more concrete stuff like objects or theorems. That's probably not surprising given the encyclopedic (thus dictionary-like) nature of Wikipedia. -- Taku (talk) 06:24, 7 May 2025 (UTC)[reply]
    If no one can even come up with a standard name, then I have my doubts that this is a mature topic to write an article about. Even worse than a neologism is an article that abstracts out a concept not abstracted out by the sources. But as I say I don't really grok the topic (once I see the word "cohomology" I know I'm out of my depth; I should really find out what that's about) so I could be off base as to whether that's what's happening. --Trovatore (talk) 06:36, 7 May 2025 (UTC)[reply]
    I have responded to this below. —- Taku (talk) 10:38, 7 May 2025 (UTC)[reply]
  • Comment: Searching for something like "derived ring" algebra instead of "derived ring theory" will return the relevant results. Google Scholar results. — MarkH21talk 04:58, 7 May 2025 (UTC)[reply]
    I have stated that above. A $64,000 question is, do these results match the subject of this article? Do, for example, the following definitions match ours?
    • "A derived ring is a differential graded ring concentrated in positive cohomological degrees"
    • "a derived ring is a commutative ring object in non-positively graded cochain complexes over k" (IMHO, these definitions point to a commutative situation only, while ours clearly tiptoes around the issue of commutativity, but then I am not an expert, and will happily accept whatever judgment the experts render)
    It would be nice to add a cite with a chapter and verse pointing to our variant of the definition in any literature at the very least (and drop the word "theory" altogether). Викидим (talk) 06:36, 7 May 2025 (UTC)[reply]
    As I wrote above, I deliberately named the article so it’s not about some particular object (e.g., a dg-ring, am E_n-ring, spectral ring, etc. etc.) studied in the field, which we might call the derived ring theory or some other names. What we have are various kinds of ways and attempts to generalize ordinary ring theory to the derived context. There is no one universal standard way to do that. I didn’t think and still don’t think having an article under some generic name means that that generic name has to be some specific math term referring to some specific object. While we may not have the universal answer to what a “derived ring” is, if there is sufficient literature on the topic (which seems to be the case), I think we can and should cover the topic (of course, I can see why others might disagree though). -- Taku (talk) 13:00, 7 May 2025 (UTC)[reply]
  • Comment. There is definitely an underlying topic, as others have noted (e.g., the derived stacks paper). This leaves a question of what to name the article. Here there seem to be unfortunate ambiguities. Such as I understand it, a derived algebra (or derived ring) is often just an algebra with a derivation. Perhaps an instance of "derived ring theory", but not the most general "simplicially enriched category of rings". On the other hand, the content of the article is extremely vague and not well-developed. I am left wondering specifically what the point is (how does it relate to the work of Maxim Kontsevich, for example.) The cited sources do not appear to support the overall subject well. I could be persuaded that either keep/delete/userfy are appropriate, but the article needs an expansion and rewrite. Tito Omburo (talk) 11:36, 7 May 2025 (UTC)[reply]
    • Comment There being an underlying topic is different from an underlying topic having been identified in the literature. We should not abstract out an underlying topic ourselves; that's not our role. If an underlying topic has been identified in the literature, I would have expected someone to have given it a name. --Trovatore (talk) 16:59, 7 May 2025 (UTC)[reply]
    It has been identified in the literature, though, so this is unconvincing. E.g., Toën, Bertrand. "Derived algebraic geometry." EMS Surveys in Mathematical Sciences 1.2 (2014): 153-240. My comment is more that the article lacks enough sources and context to make this determination. Tito Omburo (talk) 21:00, 7 May 2025 (UTC)[reply]
  • Keep. I just now added wikilinks and a note that places this stubby article firmly into the middle of a constellation of inter-related articles, including E∞-operad and A∞-operad and ∞-category and differential graded algebras over R and simplicial commutative algebras over R. This is not some topic sitting out in a corner of hyperspace, all by itself. Its smack-dab in the middle of a collection of inter-related topics. ncatlab has a matrix that covers the inter-relationships here: En-algebra. In short this is a stub, but it can be expanded with additional content. 67.198.37.16 (talk) 18:46, 7 May 2025 (UTC)[reply]
  • Delete barring developments I haven't seen yet. Taku has been admirably straightforward in agreeing that no one has really pulled together this collection of considerations into a systematized whole. Doing that is not the job of the encyclopedist; we need to wait for it to be done in the research community. We don't have to lose any material; the subtopics can be treated in their own articles. We just shouldn't be the first to abstract them out into a common whole. --Trovatore (talk) 19:11, 7 May 2025 (UTC)[reply]
    • Comment: I've spent the last week reviewing, updating and fixing up every article I can find on stochastic differential equations. This exercise has taught me that there is a lot of overlap between various articles, often covering the same material, using different notation, and written to audiences of varying mathematical sophistication. Notable is that a "systematized whole" emerges only if you read the dozen or two of the inter-related articles; otherwise, the only picture that emerges is that of piecemeal aggregation by different WP editors adding individual sand-grains into their preferred sand-pile. More or less all of WP seems to be constructed this way: editors who "pulled together this collection of considerations", oddly failing to create a "systematized whole", except in hind-sight.
  • So when I look at this article, and specifically, the section named "Foundations", I see an editing pattern analogous to that seen in SDE's: some crumbs of information, arranged in a fashion that is similar to what can be found in other articles, but speaking at a different level of mathematical sophistication, using a mildly different set of notational conventions and terminology. It's an attempt to explain a collection of relationships. It's incomplete, its stubby, but also, there's no particularly obvious way of merging it into any of the other articles that it's linked to. What I'm saying is "let it grow". Much like other stubby or start-class articles in WP, let it rest as an accumulation point, a nucleation seed for facts to accumulate. I mentioned the ncatlab article, to make it clear that lots of additional info could be added; its not like a systematic description of this topic is absent. The info is there, its just not here in WP.
  • (Also, I think it's notable. Please note that Urs Schreiber is deep into a program that redefines what the notion of a "point" is (and I mean "literally, a point", like what Euclid talked about. Like elementary-school, high-school "point", now redefined as an infinity category), which then, like dominoes, percolates into everything. Sure, many decades ago, the theory may have started with infinity groupoids, but now, its redefining core concepts of what a topological space is, what a group is, what a ring is, what a vector space is, what an algebra is. It's a daunting program, because its highly technical. But its also exciting, because it's laying a brand new foundation. To me, this is captured in the moniker "brave new ring" given in the lede. (Its actually quite revolutionary, and the relation to the string revolutions is not accidental.)) 67.198.37.16 (talk) 20:51, 7 May 2025 (UTC)[reply]
    • Responding to the first para: Just to be clear, I meant that no one in the outside world had "pulled together this collection of considerations into a systematized whole", but the article appeared to be an attempt to do so. If that systematization hasn't been done in the literature, then it shouldn't be done here; this was my point. Now Tito seems to be suggesting that it has been done in the literature, and if that is so, then I would withdraw the objection. --Trovatore (talk) 04:30, 8 May 2025 (UTC)[reply]
Yes, I think the above puts the issue very nice: we are at the moment when the history is being written. We don't know what the end product (derived ring theory) would be like but it seems almost guaranteed we will have something in the future. The question is then when is an appropriate point for encyclopedic treatment? For me, now is ok; kind like, it's perfectly ok to write about a developing event; Wikipedia is different in that regard from traditional encyclopedias. But clearly the others like Trovatore think otherwise. -- Taku (talk) 07:37, 8 May 2025 (UTC)[reply]
My view is it's perfectly find to cover the parts, but we should not attempt to abstract a whole, because that could wind up influencing developments in the outside world, which we are not supposed to do. We are followers, not leaders. Yes, we can write about developing events, but only to the extent that it has been done in reliable sources. --Trovatore (talk) 20:47, 8 May 2025 (UTC)[reply]
I basically agree with this. If sources cannot be found that directly support the text, it should be userfied (or merged to derived algebraic geometry). Tito Omburo (talk) 22:07, 8 May 2025 (UTC)[reply]
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