Recall that I have written about the standard conjectures and noncommutative motives before. Now there is a follow-up article by Bernardara–Marcolli–Tabuada: Some remarks concerning Voevodsky’s nilpotence conjecture. The conjecture is proved for some previously unknown cases (as far as I can tell, I’m not an expert on the motives literature, or any other literature) using the equivalent formulation in the world of noncommutative motives. In the situations they cover (which are based on quadrics) we have some known semi-orthogonal decompositions, and these allow them to prove the nilpotence conjecture.
Another interesting article is Orlov’s Smooth and proper noncommutative schemes and gluing of DG categories. It’s a really nice write-up of noncommutative geometry using dg categories (of which the previous paragraph was of course already a manifestation). I’m not sure how important, new or spectacular the result is, but I particularly like the last section, in which an “ad hoc” (or local) approach to noncommutative projective planes is embedded into these general noncommutative schemes.
The last article in this commercial break is Rizzardo–Van den Bergh’s http://arxiv.org/abs/1402.4506, which covers one of my favourite subjects: Fourier-Mukai transforms. The question they tackle is whether Orlov’s representability result of fully faithful functors as Fourier-Mukai transforms still holds when you drop the fully faithfulness. It is shown that in a slightly different context (not beteen the bounded derived categories of coherent sheaves, but perfect complexes to the derived category of quasicoherent sheaves) the representability fails if one drops fully faithfulness. The most awesome thing about the proof is that it entails two totally different subresults:
- comparing scalar extensions of derived categories of Grothendieck categories depends on the Hochschild dimension (section 10)
- proving that this result on comparing scalar extensions cannot be improved by exhibiting a geometric counterexample (and now comes the amazing part) based on the moduli of vector bundles on curves (sections 4 and 8)
In the ANAGRAMS seminar I’m planning to give a talk on moduli of vector bundles on curves. I had decided this before I saw this article, but this really gives me the motivation to fully understand this particular instance of geometric invariant theory.
See Announcing a seminar for a little context.
The past four weeks I’ve given lectures on Grothendieck duality for the ANAGRAMS seminar (yes, this is a redundancy). The goal was to tell people something about Riemann-Roch and Serre duality first, then go on to Grothendieck duality and discuss some of the proofs. I changed the schedule a little while I was giving the lectures (originally I planned more on Grothendieck duality, but the down-to-earth geometric applications were more appropriate for the audience, I think). I have prepared notes for each of these lectures, which are now available at the seminar webpage.
Maybe I will rewrite some things for my weblog, if people want me to elaborate on certain aspects.
And in the philosophy of open research, there is the Git repository for the notes. Feel free to contribute!
To make sense of the next post, and potentially attract more attendees, I want to announce a student seminar on algebra and algebraic geometry at the University of Antwerp. It’s called ANAGRAMS, which is an acronym for “Antwerp algebraic geometry, rings and more seminar”.
All the interesting information can be found on the website, to summarise: PhD students give lectures for PhD students on subjects related to algebraic geometry and algebra (based on the research areas we are active in). We pick a subject, work on it for a few weeks, then take another one. It’s about learning things, it’s not a research seminar.
If you want to know anything, just contact me. I guess the number of potential attendees who are also readers of my blog is rather small (I don’t want to know the limiting factor lest it be the number of readers…), but please prove me wrong!
Summary: look at the page
cohomology-tables/index.html, it gives you numbers if you move the sliders. It’s fun. If you wish to understand what the numbers mean, read on. If you want to take a look at the code: see pbelmans/cohomology-tables.
A few days ago I wanted to visualise Serre duality for twists of the structure sheaf , for a complete intersection. By visualise I mean write down the dimensions of the cohomology spaces. That way it becomes clear which spaces are dual to eachother, and it also shows the vanishing results that are applicable in that situation.
The main theorem that I needed for this to work is proposition 5 (in paragraph 78) of Serre’s Faisceaux algébriques cohérents:
Proposition 5. Soit une intersection complète, définie par des polynômes homogènes de degrés .
- L’application est bijective pour tout .
- pour et tout .
- est isomorphe à l’espace vectoriel dual de avec .
The notation in the statement refers to the degree part of the homogeneous coordinate ring of (i.e. the quotient of by , where is a complete intersection in . This means that the codimension of is as high as it can be: with each equation it can drop with at most 1, and hence if it drops each time we get a -dimensional variety.
This statement allows us easily to give the dimensions of the cohomology spaces, as in degree 0 we get a bijection with an explicit vectorspace, in intermediate degrees we have a vanishing result, and for the top degree we get an explicit duality with degree 0 information. The only part missing is computing the dimension of the degree 0 spaces. For this we use the fact that the Hilbert series of such a complete intersection (in the notation of Serre) is given by
Hence if we can compute the coefficients in this series we have the dimensions we are looking for. This is just manipulating series, and is part of any computer algebra system.
Another class of interesting numbers related to cohomology of complete intersections can be found in the Hodge diamond. Now we don’t look at the structure sheaf and its twists, but at , the differential -forms and its cohomology. This algorithm is based on Nicolas Addington’s
complete_intersection.cpp. It uses:
- Hilbert polynomials are additive with respect to exact sequences
- the Euler exact sequence (and its exterior powers) relating the structure sheaf to differential forms, to get Hilbert polynomials for these objects
- the adjunction formula (and its exterior powers) relating the differential forms of the ambient space to the hypersurface (i.e. this is done inductively for each hypersurface)
- the Lefschetz hyperplane theorem to compute the dimensions of the cohomology spaces (with an error term for the top degree, as this is where the Lefschetz hyperplane theorem doesn’t apply immediately).
Disclaimer It might not work in browsers different from Google Chrome and Safari. And be careful, if you push the limits of the Hodge diamond (high dimensions, high degrees of equations) it might run into (silent) overflows. I might fix this later on.
One last post this year. I came across the following statement in the Stacks project:
This fact is already discussed on the Stacks project blog. Recall that it should be interpreted as “the categories of sheaves for this topology are the same”. A similar phenomenon happens for étale and smooth.
For posterity I would like to collect some facts about this statement:
- in the locally noetherian case finite locally free is the same as finite and flat;
- the fppf topology can also be taken as the “fppfqf topology”, i.e. fppf coverings can be refined by fppf coverings where each morphism is quasi-finite (again an important reduction for the proof);
- the étale topology has a similar description, but instead of the four steps one takes for the fppf topology one only takes 3 steps (and this is crucial in the proof);
- it seems mysterious at first sight that Cohen-Macaulay morphisms pop up, but one first proves that for fppf morphisms Cohen-Macaulayness is an open condition which then allows us to slice morphisms locally of finite presentation into locally quasi-finite morphisms (these two steps seem to be the hard part);
With these interesting facts I will end the blogging year 2013. I hope to write more about Grothendieck topologies soon.
We are having a reading seminar on the book Representation theory of Artin algebras, by Auslander, Reiten and Smalø, and this afternoon it’s my turn to discuss chapter VI, on finite representation type. An important interest of most of the participants (excluding myself) is modular representation theory, and I should be able to tell something interesting about it in the context of finite representation type. Because of my appalling lack of knowledge on the structure of finite groups I had half an hour of fun with GAP to come up with some (trivial) facts.
Recall that by the recognition theorem for modular group algebras of finite type (see Theorem VI.3.3 in the book) it suffices to check whether the -Sylow subgroups are cyclic or not, where is the characteristic of the ground field, which necessarily is a divisor of the order of the group (otherwise things are semisimple). Hence the following oneliner suffices to check being of finite representation type:
IsFiniteRepresentationType := function(G, p) return IsCyclic(SylowSubgroup(G, p)); end;
The question now becomes: which groups are of finite representation type? Is it an easy condition to satisfy, or not? Of course, it would be best to sit down and think first. In case you are not like this, you write the following code
CountFiniteRepresentationType := function(n, p) local finite; finite := 0; for G in AllSmallGroups(n) do if IsFiniteRepresentationType(G, p) then finite := finite + 1; Print(" ", StructureDescription(G)); fi; od; Print("\n"); Print(" Of finite representation type: ", finite, "\n"); Print(" Of infinite representation type: ", NumberSmallGroups(n) - finite, "\n"); end;
You feed it with an order and a prime (which should be a divisor of the order) and it checks how many groups are of finite representation type, for a field of this characteristic. An easy observation one can make, to reduce the number of cases to check, is:
The prime has to be factor of the order with multiplicity at least two.
Otherwise it is of course impossible for the Sylow subgroup to be non-cyclic. Hence using the following code one reduces the output a little
for n in [2 .. 128] do Print("Counting the groups of finite representation type of order ", n, " = ", FactorsInt(n), "\n"); for p in Set(FactorsInt(n)) do if Number(FactorsInt(n), m -> m = p) = 1 then # Print("Not worth checking at ", p, "\n"); else Print("Checking it in characteristic ", p, "\n"); CountFiniteRepresentationType(n, p); fi; od; Print("\n"); od;
One obtains this output, and we conclude that being of finite representation type is indeed a strict condition: as the multiplicity of a factor grows, the groups of finite representation type become scarce. The smallest case is just , a rather small algebra if you’d ask me.
Another question to ask is: what about certain families? Symmetric and alternating groups have non-cyclic Sylow subgroups (if a prime factor has multiplicity greater than 1). Dihedral groups on the other hand have modular group algebras of finite representation type over each field of odd characteristic.
Every conclusion could’ve been made without resorting to a computer algebra system. Unfortunately my computer science background sometimes induces a “code first, think later” way of life.