Talk:Derived noncommutative algebraic geometry

Todo

• https://en.wikipedia.org/wiki/Help:Displaying_a_formula#Commutative_diagrams for uploading tikz diagrams — Preceding unsigned comment added by Wundzer (talk • contribs) 20:18, 28 February 2020 (UTC)
• https://arxiv.org/abs/math/0302304 , https://arxiv.org/pdf/math/0302304.pdf
• https://arxiv.org/pdf/hep-th/0210296.pdf
• https://arxiv.org/pdf/math/0510670.pdf -> include results from Kuznetsov and key theorems
• monodromy "Some Applications of Commutative Algebra to String Theory" https://arxiv.org/pdf/hep-th/0110071.pdf and https://arxiv.org/pdf/math/0305042.pdf
• Add more on triangulated categories of singularities https://arxiv.org/pdf/math/0503632.pdf
• Derived affine - https://arxiv.org/abs/math/0204218 — Preceding unsigned comment added by Wundzer (talk • contribs) 23:31, 10 July 2020 (UTC)
• Add matrix factorizations and equivalence of categories checkout https://mathoverflow.net/questions/9733/matrix-factorizations-and-physics
• tri cats and LG model equivalence https://arxiv.org/abs/hep-th/0210296v2
• tri cats of singularities and d-branes in LG models https://arxiv.org/abs/math/0302304
• Mention tilting objects and equivalences of derived category with derived category of modules over a non-commutative algebra, the quiver algebra
• Mention noncommutative deformations of quiver algebra, and as an application, giving deformations of the derived category of projective space, https://arxiv.org/pdf/1001.4400 and https://arxiv.org/pdf/math/0404281.pdf
• https://arxiv.org/pdf/1809.10919.pdf -> K-theory of singularity category
• https://arxiv.org/abs/0801.2014 -> Meet HMS
• https://arxiv.org/pdf/1202.6297.pdf -> noncommutative motives and SO decompositions
• https://link.springer.com/article/10.1007/s00029-019-0482-y
• https://pbelmans.ncag.info/assets/thesis.pdf — Preceding unsigned comment added by Wundzer (talk • contribs) 23:13, 26 February 2020 (UTC)
• https://mathoverflow.net/questions/203979/could-we-extend-the-exact-sequence-k0x-to-k-0x-to-k-0d-sgx-to-0-to?rq=1
• https://mathoverflow.net/questions/181581/is-there-an-infty-2-category-with-morphisms-given-by-db-textcoh?rq=1

Orbifold example

http://www.claymath.org/library/monographs/cmim04.pdf - pg 386 - Orbifold $\mathbb {C} ^{3}/\mathbb {Z} _{5}$

• Periods
• Monodromy - of the derived category
• Stability — Preceding unsigned comment added by Wundzer (talk • contribs) 18:21, 10 July 2020 (UTC)

Matrix Factorizations

Maybe this should be its own page...

• https://web.math.princeton.edu/~takumim/takumim_Spr13_JP.pdf
• http://www.math.ucr.edu/~jbergner/burke
• https://arxiv.org/abs/1110.2918

Main reference needed

I think this draft article still needs a reference that says there is a field called "Derived noncommutative algebraic geometry". I cannot find one myself. But I found this [1] which has this sentence: "The above point of view agrees with the philosophy of derived noncommutative algebraic geometry." with the footnote "We are not sure whether this name is commonly accepted or not." So, it might be a stretch to claim there is a field called "Derived noncommutative algebraic geometry". Hence, one option is to rename the article to something like Draft:Geometry of derived categories; another is to merge it with existing articles like noncommutative algebraic geometry. (The materials of the article themselves look notable enough to me.) Pinging Wundzer. -- Taku (talk) 00:21, 21 February 2020 (UTC)

Here's a couple links: https://arxiv.org/abs/1808.02287, http://imperium.lenin.ru/~kaledin/math/tokyo/final.pdf Wundzer (talk) 05:56, 21 February 2020 (UTC)

No, no, what I mean is that those references don’t use the term “Derived noncommutative algebraic geometry”. In Wikipedia, we are not allowed to introduce a new terminology. We need some reliable ref that introduces the name “Derived noncommutative algebraic geometry”. I cannot find one myself; can you? or anyone else? Pinging Wundzer. —- Taku (talk) 09:19, 21 February 2020 (UTC)

Ah, okay. I think the options are derived noncommutative schemes, or derived noncommutative geometry, although I cannot find the exact words of the latter in the literature. TakuyaMurata Wundzer (talk) 22:11, 21 February 2020 (UTC)

A remark

Not sure if you are aware: recently user BTotaro has created a very satisfactory wikipage for semiorthogonal decompositions, which includes the discussion of exceptional collections, Beilinson's theorem, etc. There is a significant intersection between the two articles. Perhaps it makes sense to link to semiorthogonal decomposition, e.g., using "Main article: ..." template inside of the sections.

I think that's a great idea! Can you add this to the draft article? Wundzer (talk) 21:58, 23 February 2020 (UTC)

Also, re the title of the article: I agree with Taku that, unfortunately, this field does not seem to have a standard name. "Derived noncommutative algebraic geometry" is the best variant I know, and I support this choice 100%, but I'm not sure if that's the standard yet. Some people call it "derived algebraic geometry", which is certainly not a good idea, and if I remember correctly, Orlov sometimes calls this approach just "noncommutative geometry", which maybe is not descriptive enough. When I need to specify the field, I usually say "I study derived categories of coherent sheaves". However, that's not a good title for the article. Dpirozhkov (talk) 03:49, 23 February 2020 (UTC)

Do you know of a reference which contains the name "Derived Noncommutative Geometry"? I think that's a sufficient change which keeps most of the original title. Dpirozhkov -- Wundzer (talk) 21:57, 23 February 2020 (UTC)

Some papers ([2], [3]) by Orlov prominently use the words "derived noncommutative schemes", but I think the closest they come to introducing the name for the field is "studying the world of derived noncommutative schemes". Dpirozhkov (talk) 22:24, 23 February 2020 (UTC)

I'm not sure where to go from here... Ping Dpirozhkov and TakuyaMurata Wundzer (talk) 23:40, 23 February 2020 (UTC)

It is not the strict requirement that an article title in Wikipedia must be the name of the field; if there is the standard one, then, to avoid confusion, the standard one must be used. But if there is none, I think editors are allowed to choose an article title that best approximates the names used in literature (since the have to pick some name). In this particular case, Derived noncommutative algebraic geometry may be ok. The intro should however make it clear that the field does not yet have the standard name. —- Taku (talk) 19:09, 24 February 2020 (UTC)

Alternatively, how about Geometric study of derived categories?, which does not have the naming issue. Or derived categories as noncommutative spaces? Wundzer —— Taku (talk) 19:11, 24 February 2020 (UTC)

I added a sentence about disambiguation in the intro. I'm happy with either keeping the name or using Geometric study of derived categories Wundzer (talk) 05:49, 25 February 2020 (UTC)

Sorry; with reading the literate further, I have come to think "noncommutative" should not be omitted from the article title; which leaves us only one option "DNAG". I have added the note that the name may not be standard and hopefully that is good enough to convey our position that we are not asserting the DNAG is a standard name. If there is a better article title, we can always use that later. Since the draft looks complete, I took a liberty of moving it to mainspace as well. Ping Wundzer -- Taku (talk) 06:40, 26 February 2020 (UTC)

Awesome! Thanks! That's a good idea about saving the potential naming convention problem for later. Wundzer (talk) 19:56, 26 February 2020 (UTC)