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Alternative visualisation of the structure of the Stacks project

It turns out that someone independently made visualisations of the structure of the Stacks project.

He or she also performed some statistical analysis, but I’m not sure the metrics of how important a result is are chosen appropriately. For instance tag 07FJ appears in the top 10 list, just because it refers to so many other tags in its proof. That does not necessarily mean it is important because it heavily depends on how a statement and its proof are written: if a statement is a long list of substatements, or its proof does not contain any interesting sublemmas (at first sight) then by this metric it will become important just because it refers to many other tags. The metric confounds importance and difficulty I think. Although in this case it actually is an important result because it forms the main technical part of the proof of Popescu’s theorem.

I really do like the big chord diagram showing the relationships and sizes of all the parts and chapters in the Stacks project. Go check it out!

On a conjecture by Novakovic

In his recent preprint No full exceptional collections on non-split Brauer–Severi varieties of dimension \leq 3 Sasa Novakovic conjectured the following:

Conjecture Let X\neq\mathbb{P}_k^n be a Brauer–Severi variety. Then X does not admit a full (strong) exceptional collection consisting of arbitrary objects.

In his paper he proves this (as the preprint’s title already suggests) in dimension less than 3, by using the transitivity of the braid group action on the set of exceptional collections. Bondal and Polishchuk conjectured this for arbitrary triangulated categories, but it is only known in very special cases, mainly for the projective line, plane and three-space (over an algebraically closed field). From this he can prove the conjecture in the special cases where the degree of the central simple algebra is at most 4.

Recently Theo Raedschelders showed how one can use the results of the preprint Noncommutative motives of separable algebras by Tabuada–Van den Bergh to prove the conjecture in complete generality. He has written a note Non-split Severi–Brauer varieties do not admit full exceptional collections about it, available on his website. It is a very cool application of some heavy and abstract machinery (namely noncommutative motives)!

Semi-orthogonal decompositions

Recall that there is a well-known semi-orthogonal decomposition for the Brauer–Severi variety of a central simple algebra. In A semiorthogonal decomposition for Brauer–Severi schemes Bernardara gave a semi-orthogonal decomposition generalising Orlov’s projective bundle formula (or Beilinson’s decomposition for projective space if you like), categoryfing Quillen’s result for algebraic K-theory. For a central simple algebra A of degree n it says that \mathbf{D}^{\mathrm{b}}(A^{\otimes i}) is an admissible subcategory, and that there exists a semi-orthogonal decomposition

\mathbf{D}^{\mathrm{b}}(\mathrm{BS}(A)=\langle\mathbf{D}^{\mathrm{b}}(k),\mathbf{D}^{\mathrm{b}}(A),\ldots,\mathbf{D}^{\mathrm{b}}(A^{\otimes n})\rangle

Observe that \mathbf{D}^{\mathrm{b}}(A^{\otimes i}) in general is not equivalent to the subcategory generated by an exceptional object, sometimes one says that it is given by a semi-exceptional object. So Novakovic’s conjecture tells you that (non-trivial) Brauer–Severi varieties cannot have a better choice of objects giving rise to a full exceptional collection. For the matrix algebra the Brauer–Severi variety is just \mathbb{P}_k^{n-1}, for which Beilinson’s collection is a full collection.

On the proof

Observe that it is even possible to immediately appeal to a result which is already in the Tabuada–Van den Bergh preprint. For this it suffices to realise that having a full exceptional collection in the derived category of the Brauer–Severi variety indeed means that its noncommutative motive is isomorphic to the noncommutative motive of the Brauer–Severi variety of the matrix algebra (of the same degree). But by theorem 3.12 (whose proof goes along the lines of the argument in Theo’s note) this means that the central simple algebra must generate the same subgroup of the Brauer group as the matrix algebra, hence is necessarily trivial itself.

To get a feel for how one proves such a result regarding noncommutative motives, I advise you to take a look at Tabuada’s Additive invariants of toric and twisted projective homogeneous varieties via noncommutative motives, where all the details are collected in section 9 and the proof of Novakovic’s conjecture follows immediately from proposition 9.2 (bypassing some of the steps in the more general Tabuada–Van den Bergh paper). Given the framework of noncommutative motives, it boils down to the fact that everything in the derived category of a central simple algebra is formal, that the Grothendieck group of a central simple algebra is always just the integers, and some very explicit linear algebra over the integers.

Fortnightly links (10)

New paper: Noncommutative quadrics and Hilbert schemes of points

Theo Raedschelders and I just uploaded our newest preprint: Noncommutative quadrics and Hilbert schemes of points. The main goal of the paper is to extend the result of Orlov’s paper Geometric realizations of quiver algebras: we prove that (the derived category of) a noncommutative quadric embeds in (the derived category of) a commutative deformation of the Hilbert scheme of two points on a quadric. Orlov showed this for noncommutative planes (i.e. quadratic Artin–Schelter regular algebras), we now do the cubic Artin–Schelter regular case.

We also raise a question regarding the infinitesimal version of the picture above, whose statement works in smooth families. Namely, their common starting point is a fully faithful embedding \mathbf{D}^{\mathrm{b}}(S)\hookrightarrow\mathbf{D}^{\mathrm{b}}(\mathrm{Hilb}^nS) for a sufficiently nice smooth projective surface. In this special case (I blogged before about lack of functoriality in general) we can induce a morphism \mathrm{HH}^n(\mathrm{Hilb}^nS)\to\mathrm{HH}^n(S) , and we have an Hochschild–Kostant–Rosenberg for both sides, hence a block decomposition for these linear maps.

The question is now easy: in which cases do we have a surjection (or even isomorphism)

\mathrm{H}^1(\mathrm{Hilb}^nS,\mathrm{T}_{\mathrm{Hilb}^nS})\twoheadrightarrow\mathrm{H}^0(S,\bigwedge^2\mathrm{T}_{S})?

This would mean that we can lift every infinitesimal noncommutative deformation of S to a commutative deformation (as in Kodaira–Spencer) of the Hilbert scheme (ignoring obstructions). If this were indeed the case, it would be interesting to see whether it is possible to generalize Toda’s result, showing that fully faithful functors (and not just equivalences) lift to deformations.

I have some work in progress addressing these questions, also for other moduli spaces (i.e. not just Hilbert schemes of points on surfaces). Stay tuned for more, and do not hesitate to ask questions.

Fortnightly links (9)

Time for another list of things I found interesting on the web the past 2 weeks.

  • Peter Krautzberger, Math on the web: time to step up! in which he announces the creation of a group of people interested in getting mathematics on the web. I’ll keep you posted on its activities.

  • arXiv2BibTeX.org is a tool to quickly create Bib(La)TeX snippets for an arXiv entry, or a list thereof. The cool is thing that it also works for BibLaTeX, which happens to implement special support for the arXiv, automatically hyperlinking things. One not very crucial thing that is missing is the category (for the example there should be eprintclass = {math.CT}), but anyone can implement this and do a pull request.

  • Gonçalo Tabuada, Jacques Tits’ motivic measure links the author’s theory of noncommutative motives to the Grothendieck ring of varieties (also known as baby motives, or naive motives in the next preprint I’ll link to), proving properties about the classes of Brauer–Severi varieties of central simple algebras in the latter from their properties as noncommutative motives. The codomain of the motivic measure is a ring that measures properties of all the possible subgroups the Brauer group of the base field, it being the quotient of the ring freely generated by all elements in the Brauer group modulo the relation that the sum of the unit and a product of two elements of coprime index is equal to their sum.

    There are more cool things you can do with noncommutative motives and Brauer–Severi varieties, about which I hope to blog soon.

  • Lieven Le Bruyn, Brauer-Severi motives and Donaldson-Thomas invariants of quantized 3-folds is also about the Grothendieck ring of varieties and Brauer–Severi schemes, but with the twist that it concerns Brauer–Severi schemes of orders. In February Brent Pym gave a very cool lecture in our departmental seminar about this preprint and it got Lieven interested in studying some of the conjectures in that preprint. He tackles the problem by using his awesome machinery of Cayley smooth orders, which happens to be the current topic of our student seminar, so maybe I’ll blog a little about the details later on (if he doesn’t already on his own blogs).

  • Daniel Chan, 2-hereditary algebras and almost Fano weighted surfaces discusses the generalisation of weighted projective lines in dimension 2. I’ve been thinking a little about this type of things lately, and I hope to blog about weighted projective lines and their deformation theory somewhere in the near future. For now, you’ll have to do with this interesting preprint.

Do you want to be my colleague?

The mathematics department has two openings for PhD students in pure mathematics. They are teaching positions, i.e. you have a bigger teaching load than your average PhD student, but to compensate this you get 6 years (3 times 2 years). This also means that you will be required to teach in Dutch the moment you start. Probably that is why there is no English version of the job openings available.

So if you are a 2nd year Master student in Flandres or the Netherlands (or happen to speak fluent Dutch for some other reason) looking for a PhD position, you can apply for these. If you know someone who fits this description, please spread the word.

More cheat sheets

A while ago I created a cheat sheet on recollements. I haven’t improved at remembering things, hence I created two more cheat sheets:

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