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Comparison of topologies

Last year I made a graph comparing a bunch of Grothendieck topologies, and I had been toying with the idea of making an interactive web version of it. Two weeks ago Shane Kelly published a preprint Points in algebraic geometry, while Stefan Schroër tackled in a preprint that appeared on the same day the intriguingly hard case of Points in the fppf topology.

The article by Kelly contains a comparison graph, just like the one I made last year (mine emphasised the “usual” topologies more, and had less topologies), but also describes what type of rings act as stalks for this topology, and gave short definitions of each of the topologies in use. Hence I had all the data to make a complete comparison!

There is a very preliminary draft available. I’ll be away from the internet for 2 weeks, hence if you are looking for a summer programming project and feel like contributing, there is the GitHub repository. I want to:

  1. Improve the layout (I’m a horrible designer, as you can tell, and my SVG skills are rather sketchy).
  2. Add explanations on how to use the comparison graph.
  3. Fix all the TODOs, the code is riddled with them…
  4. Add more data (you don’t have to be a programmer for this): I am especially looking for improved comparison results, more topologies, … I know there is a fpuo and a l' topology out there. Are there more topologies in use?
  5. Add the canonical and extensive topology to the graph, and indicate these properties in the “passport” as well.
  6. I want a tighter integration with the Stacks project, preferably all the result that are used should turn up there, this requires both writing for the Stacks project and providing the functionality in the comparison graph.
  7. Right now everything is for \mathrm{Sch}/S, the category of separated schemes of finite type over S a separated noetherian scheme: I don’t know what goes wrong if we take a different category, hence
    • I would like to be able to switch between categories, and suitably update the comparison graph.
    • I have no clue regarding the actual results that should be there…

New feature for the Stacks project: slogans

Update from the Stacks project: there is a new feature, namely slogans. Johan de Jong has discussed this idea before, and right now they are implemented on the website *and* there is a nice web page (compatible with your smartphone, I hope) to suggest slogans. Moreover, there are numerous small bug fixes (and hopefully not too many new bugs).

Hence: first suggest slogans and then see them appear on the Stacks project website.

I would like to thank Johan and Grietje Commelin for providing an excellent place to stay during the programmingathon Johan (Commelin, having multiple Johans makes for confusing mail conversations) and I did. And special thanks to their 2.5 year old daughter Hannah for the great conversations one can have with toddlers during breaks :).

Notes on stratification of triangulated categories

The last week of June I attended a summer school on derived categories in Nantes. Really interesting summer school, and a truly brilliant city. During the long and boring train ride home I decided to TeX up my notes for Henning Krause‘s lecture series on stratification of triangulated categories. The result of that is now proofread and corrected (thanks Henning!), and can be found on my personal webpage (or his, or hopefully the summer school’s page soon).

New version of Hodge diamonds and Serre duality tables for complete intersections

Recall that a few months ago I wrote a tool to visualise / compute Serre duality and Hodge diamonds for complete intersections.

Using it yesterday I realised that I should beef it up a little, so I have:

  • added more numerical invariants in the case of surfaces, they are defined in terms of the Hodge diamond anyway but it saves me from miscomputing things (I should do it for more general varieties too, but at the moment I care only about surfaces)
  • improved the layout, no more incessant scrolling if you care about Hodge diamonds

The result still won’t win a medal for its visuals. If you have suggestions regarding layout or functionality, just contact me.

Table of contents for Thomason–Trobaugh

Yesterday I got fed up with the lack of a table of contents for Thomason–Trobaugh’s Higher algebraic K-theory of schemes that I decided to make one myself. The result is available as a pdf, or as a table in this blog post for your (and my) convenience:

1 Waldhausen K-theory and K-theory of derived categories 250
2 Perfect complexes on schemes 283
3 K-theory of schemes: definition, models, functorialities, excision, limits 312
4 Projective space bundle theorem 329
5 Extension of perfect complexes, and the proto-localization theorem 337
6 Basic fundamental theorem and negative K-groups, KB 351
7 Basic theorems for KB, including the localization theorem 363
8 Mayer–Vietoris theorems 367
9 Reduction to the affine case, and the homotopy, closed Mayer–Vietoris, and invarience-under-infinitesimal-thickenings properties of K-theory with coefficients 375
10 Brown-Gersten spectral sequences and descent 382
11 Éale cohomological descent and comparison with topological K-theory 391
A Exact categories and the Gabriel–Quillen embedding 398
B Modules vs. quasicoherent modules 409
C Absolute noetherian approximation 418
D Hypercohomology with supports 424
E The Nisnevich topology 427
F Invariance under change of universe 431

Phantom categories do exist

In a previous post I introduced the notion of (quasi-)phantom categories, and discussed Kuznetsov’s conjecture stating that they should not exist. This post has lain dormant for too long, but I decided to finally finish it this evening. The goal is clear: indicate how we can construct phantom categories, tell how results in classical algebraic geometry help us in doing so and explain what the abstract properties of some of the (quasi)phantom categories indicate about the surface.

The counterexamples

They are in roughly chronological order, based on the date of the arXiv preprints. The easiest case to understand is probably the Beauville surface, which is not the first example that was obtained (at least in the form of a publicly available preprint).

Classical Godeaux surface

This surface is a \mathrm{Cyc}_5-quotient of the Fermat quintic surface in \mathbb{P}^3, i.e. we take the fifth roots of unity acting on the four coordinates. The explicit construction as a quotient allows for strong structural results, a phenomenon we will see again for the Beauville surface.

In this case, the length of the exceptional collection is 11, which is the longest such a sequence can get. The complement in the Grothendieck group is (by the quotient construction) \mathrm{Cyc}_5. The complement in the Hochschild homology on the other hand is zero, which is a Hochschild-Kostant-Rosenberg computation.

Burniat surfaces

A first class of quasi-phantoms is obtained by considering the Burniat surfaces of degree 6 (i.e. \mathrm{K}_X^2=6). This is done in Derived categories of Burniat surfaces and exceptional collections. This type of Burniat surfaces can be described as Galois covers (with group \mathrm{Cyc}_2^2, i.e. 4-fold covers) of a del Pezzo surface of degree 6, which are the blow-ups of \mathbb{P}^2 in 3 points not on a line. We have strong structural results on the derived category of del Pezzo’s, and these can be lifted to the Burniat surfaces of degree 6.

The idea is to use the 3-block exceptional sequence of length 6 on such a del Pezzo. One can then prove that the lift of this sequence to the cover is still an exceptional collection, but now with some nonzero \mathrm{Ext}^2‘s. This tells us that the endomorphism algebra, which describes the triangulated category generated by the exceptional collection, is formal, and independent of the Burniat surface under consideration (there is a 4-dimensional family of these). This has the funny consequence that the quasi-phantom category (which was conjectured to not exist) actually contains all the non-trivial information on the derived level!

The fact that the Grothendieck group of the orthogonal complement is \mathrm{K}_0(\mathcal{A}) is \mathrm{Cyc}_2^6 is a classical result, which is related to the degree of the surface as explained in section 3 of On certain examples of surfaces with \mathrm{p}_g=0 due to Burniat.

Determinantal Barlow surfaces

These surfaces (constructed using a \mathrm{Dih}_{10}-action on \mathbb{P}^3) also admit an exceptional sequence of length 11. And what is more interesting about these surfaces is that their Grothendieck group is free of rank 11. Hence the orthogonal complement of the sequence of length 11 is a true phantom category!

Beauville surfaces

This is a class of surfaces obtained by taking a suitable quotient of the product of two curves. The example considered in Exceptional collections of line bundles on the Beauville surface is of a \mathrm{Cyc}_5\times\mathrm{Cyc}_5-action on the Fermat quintic curve X^5+Y^5+Z^5=0 (which is of genus 6). Because of the construction as a quotient it is again possible to obtain strong structural results for these surfaces.

The interesting property of Beauville surfaces is that they can be considered as fake quadrics: they look and behave like \mathbb{P}^1\times\mathbb{P}^1 in many respects. But now we take non-trivial curves, and incorporate a group action. One of the similarities is that they have an exceptional collection of line bundles of length 4. But unlike the quadric surface case this collection is not full: its orthogonal complement will be a quasi-phantom category, whose Grothendieck group is the torsion part of the Picard group, which is \mathrm{Cyc}_5\times\mathrm{Cyc}_5.

Fake projective planes

We have had fake quadrics, but there are also fake projective planes, which are surfaces that share numerical invariants with the projective plane, yet are of general type.

Fake del Pezzo surfaces

When the first examples had been obtained, the question became whether it is possible to unify these examples and see why they are (quasi)phantoms. This has been done in Enumerating exceptional collections on some surfaces of general type with \mathrm{p}_{g} = 0. What is so interesting about this preprint is that the code is publicly available.


To summarise the results we give the following table. We obtain the (quasi-)phantom category \mathcal{A} as the orthogonal complement of an exceptional collection, whose length gives us the rank of the freely generated part of the Grothendieck group.

type \mathrm{K}_0(\mathcal{A}) length of the exceptional collection
Godeaux surfaces quasi-phantom \mathrm{Cyc}_5 11
Burniat surfaces of degree 6 quasi-phantom \mathrm{Cyc}_2^6 6
Barlow surfaces phantom 0 11
Beauville surface quasi-phantom \mathrm{Cyc}_5^2 4

This table is incomplete, but the number of known quasiphantom categories increases rapidly, hence only the (as far as I can tell) first four examples are listed. Maybe I will elaborate further on the subject of (quasi)phantoms later, right now I just want to get this post finished.

TikZ workshop: the slides

Yesterday I gave the TikZ workshop announced in the previous post. I liked it, I hope the same applies to the people in the audience. It certainly wasn’t intended as a thorough introduction to the intricacies of the TikZ syntax, more a “learn to use these resources”, a show-off of what is possible, and a freewheeling through some examples allowing me to explain some of the important things in creating TikZ / pgfplots figures.

As promised: the slides (and the non-handout version). For those who are interested in the actual source code required to produce these slides: I have put the code on GitHub. LuaLaTeX is required.


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