This is a scheduled post, I’ve been on a vacation for the past week, and I will be for the next. So I might’ve missed some really cool preprints and other links.

• Anthony Blanc, Marco Robalo, Bertrand Töen and Gabriele Vezzosi: Motivic realizations of matrix factorizations and vanishing cycles is a very exciting paper. First it generalises the correspondence between singularity categories and Landau–Ginzburg models, then it constructs a functor from noncommutative motives to (Morel–Voevodsky) commutative motives, and finally it gives a deep comparison result between the inertia invariant part of the vanishing cohomology of a morphism and the realization of the category of a Landau–Ginzburg pair. All very complicated, and I would love to understand more about this. But for that I will have to immerse myself a little more in the recent work of the authors.

• Bertrand Toën and Gabriele Vezzosi, The l-adic trace formula for dg-categories and Bloch’s conductor conjecture is a research announcement of results that build on the previous fortnightly link. I had missed it when it came out a few weeks ago, but it deserves to be mentioned. Bertrand Toën lectured about this in Stuttgart, and it looks really exciting to see how noncommutative geometry can be used to prove arithmetic results.

• Goncalo Tabuada, A note on secondary K-theory II is a nice continuation of an earlier paper by Tabuada. The secondary K-theory of the title is also known as the Grothendieck ring of pretriangulated dg categories à la Bondal–Larsen–Lunts. The goal of the paper is to study the map from the derived Brauer group of a ring (or scheme) to the secondary K-theory, and show that in many cases it is injective.

• Piotr Achinger and Maciej Zdanowicz, Some elementary examples of non-liftable schemes constructs examples in characteristic $p$ of schemes that do not admit a lifting to characteristic 0 or the ring of second Witt vectors. I always find it funny how you can abuse the Frobenius morphism in ever more twisted ways, or decide to blow up all $\mathbb{F}_p$-rational points, and get something interesting.

• Jack Hall and David Rydh: The telescope conjecture for algebraic stacks is a preprint by two nice gentlemen that I met in Banff two weeks ago. The telescope conjecture is a relationship between small and big versions of triangulated categories, and should nowadays rather be called the telescope conjecture as there are counterexamples (the first one due to Keller if I’m not mistaken). The paper shows that in the cases where one expects the hypothesis to hold (one needs noetherianity) it does indeed hold.

• Jack Hall and David Rydh: Mayer–Vietoris squares in algebraic geometry is a second preprint they produced recently. It also does what the title promises: discuss Mayer–Vietoris squares. Usually such a square is given by two open subschemes and their intersection, so that you are gluing objects in a topologically intuitive sense. This paper studies many other settings, unifying some of the already existing descent situations.

Today I decided to spend some time improving the website for the Stacks project, fixing bugs and implementing some small improvements. I grouped the issues in the GitHub issue tracker that I wanted to address. Here are two highlights, not because the bugfix was interesting, but because you might learn something about how to use the website.

• the search now works better with quotes: If you want to learn about quasicoherent sheaves in the Stacks project, you should first of all know that the spelling quasi-coherent is used. Then it is important to know that if you want to use the search function to find results about quasi-coherent sheaves, you should enclose it with quotes. I.e. the following would be a way of looking for tags mentioning quasi-coherent sheaves on a stack:

"quasi-coherent" stack

Because of a silly implementation mistake by yours truly, the search would be successful, but the keywords would disappear from the search field. This is now fixed.

• the captcha is now easier to read: I love it when you can make something better by such a little change. We are now using a font that distinguishes between ‘0’ and ‘O to display the tag identifier. This was a nice idea by Brian Conrad, thanks!

The other bugs are not very interesting, and most of them had a nearly trivial fix anyway.

Because I just keep on forgetting the conditions under which it is known when $\mathbf{D}(\mathop{\mathrm{Qcoh}}X)$ is generated by a single perfect complex (where $X$ is allowed to be an algebraic stack) I decided to write a quick summary post about the state of the art.

I phrase the question in a specific way: for the sake of this blogpost I am only interested in $\mathbf{D}(\mathop{\mathrm{Qcoh}}X)$. One can also consider $\mathbf{D}_{\mathrm{qc}}(\mathcal{O}_X)$, the derived category of $\mathcal{O}_X$-modules whose cohomologies are all quasicoherent, which under suitable hypotheses is equivalent to the derived category of quasicoherent sheaves via the natural functor $\mathbf{D}(\mathop{\mathrm{Qcoh}}X)\to\mathbf{D}_{\mathrm{qc}}(\mathcal{O}_X)$. But for the sake of this post, $\mathbf{D}(\mathop{\mathrm{Qcoh}}X)$ is the main player.

## When $X$ is a scheme

In this case the result holds if $X$ is quasicompact and has affine diagonal.

It is already classical, and due to Neeman (1996), although for the statement that there is a single perfect complex that acts as a compact generator one has to refer to Bondal–Van den Bergh (2003).

In the Stacks project it is given in tag 09IS. Observe that the statement is for $\mathbf{D}_{\mathrm{qc}}(\mathcal{O}_X)$, but assuming quasicompact and having affine diagonal these categories turn out to be equivalent, which was proven in Bökstedt–Neeman (1993) for the quasicompact separated case, and in tag 08DB in the Stacks project assuming affine diagonal.

## When $X$ is an algebraic space

In this case the result holds if $X$ is quasicompact and has affine diagonal.

This result is not published, but is due to Van den Bergh, from 2005.

In the Stacks project it is given in tag 09IY. Observe that the statement is for $\mathbf{D}_{\mathrm{qc}}(\mathcal{O}_X)$, but assuming quasicompact and affine diagonal these turn out to be equivalent, which is given in tag 08H1.

## When $X$ is an algebraic stack

In this case the result holds if $X$ is quasicompact and has affine1 and quasi-finite diagonal.

In particular, if I’m not mistaken the result applies to quasicompact Deligne–Mumford stacks having affine diagonal: by definition these have an unramified diagonal, and unramified implies quasi-finite (everything being quasicompact).

The result in this case is again a combination of knowing when the natural functor is an equivalence, and when the codomain is compactly generated by a single perfect complex. The results in the literature are:

1. $\Psi_X$ is an equivalence if $X$ is quasicompact and has affine diagonal, provided (!) $\mathbf{D}_{\mathrm{qc}}(\mathcal{O}_X)$ is compactly generated. So compact generation of this category becomes now important for two reasons. This is theorem 1.2 in Hall–Neeman–Rydh.

2. $\mathbf{D}_{\mathrm{qc}}(\mathcal{O}_X)$ is compactly generated by a single perfect complex if $X$ is quasicompact and has quasifinite separated diagonal. This is theorem B in Hall–Rydh.

There are more general situations in which $\Psi_X$ is an equivalence, the harder question is to have a single compact generator, and one cannot expect such a thing in general for an algebraic stack not having suitable finiteness properties, as one can conclude by considering $\mathrm{B}\mathbb{G}_{\mathrm{m}}$ which needs a countable set of compact generators.

The main conclusion of this post is probably that I should dedicate a blogpost (or even several) to the wonderful properties of diagonals.

1 I assume here that affine implies separated for algebraic stacks, a statement I couldn’t find in the Stacks project.

I was wondering how often the Stacks project gets cited nowadays. I wanted to check this both on arXiv and MathSciNet, but the fulltext search of arXiv only returns its results in a way not suitable for easy processing (and also not all of the results it seems, because the oldest preprint is from 2014), so I had to stick to MathSciNet.

So what I did is look at how often “stacks project” appears in the references on MathSciNet, and after some parsing (and with the sample being taken today) this gives the following results, broken down by year:

year number of citations
2010 1
2011 2
2012 6
2013 10
2014 14
2015 30
2016 14

In total, the Stacks project has been cited 77 times.