If you are anything like me you might sometimes struggle with remembering the fine details of a definition. One example of this for me is the notion of recollement in the world of triangulated categories: this is a decomposition of a triangulated category in two smaller triangulated categories with properties that are reminiscent of the decomposition of a (deliberately left vague) triangulated category on a topological space into a piece associated to an open set and a piece associated to its complement.
After reading its definition for the umpteenth time I realised it is easy to reconstruct all parts of the definition, assuming you can remember the shape of the diagram:
- each pair of (vertically) adjacent arrows is an adjoint pair
- the (unique) composition from left to right is zero
- functors towards the big category are fully faithful
- the four non-equivalence (co)unit transformations are used to decompose objects of the big category
From this set of four ingredients you can deduce the explicit axioms for a recollement. Otherwise you can just use the little cheat sheet that I decided to write.
The following is another music theory gem that I want to share with you: an interactive analysis of all fugues in Bach’s Das Wohltemperierte Klavier (both books). When I was taking a class in music analysis I was dreaming of such a thing, and it is awesome to see how it turned out in this case. So after reading why you need something like well (or equal) temperament as linked in the previous blogpost you can now learn about the structure of Bach’s hommage to this tuning system!
The title of my blog suggests that I should be talking about music, but I don’t often do this. For those who wish to read about music and mathematics, consider adding alpof to your aggregators. He writes both about the stuff I would like to write about (tuning theory) and about the stuff I would like to understand (neo-Riemannian music theory).
I just came back from a really good summer school on algebraic stacks, and the good thing is that there are some excellent notes to be shared. Jarod Alper wrote his own notes for his lectures, and these are a brilliant introduction to Artin approximation and Artin algebraization leading to a description of the local structure of algebraic stacks. For the other lectures participants are TeX’ing up their notes, so far Vistoli’s lectures on gerbes (done by David Holmes) are available, stay tuned for the others as all of the lecture series were really good and worth a read if you are interested in algebraic stacks.
What makes the Kronecker quiver special?
In the context of our conversation this meant the following: why is the derived category of the Kronecker quiver (with 2 arrows) equivalent to the derived category of the projective line, yet for the generalised Kronecker quiver (with arrows, for ) there is no equivalence with the derived category of any smooth projective variety?
The exact interpretation of this question is that the path algebra of the Kronecker quiver is the endomorphism algebra of some full and strong exceptional collection of objects in the derived category of a smooth projective variety, whilst the generalised Kronecker can never occur in such a fashion. So why is this the case?
There are many ways in which it is not hard to see that this is indeed the only possibility (arguing on algebraic K-theory, using Okawa’s indecomposability result for curves, studying exceptional objects, representation-theoretic arguments…) and certainly proving that the Kronecker quiver is indeed special wasn’t the issue. But somehow we felt that each of these arguments was like appealing to the classification of finite simple groups to prove some easy lemma in group theory. Yes, it works, but it isn’t the best method of proof.
Luckily, we are not just given a triangulated category: Serre duality imposes extra rigidity on a category, and it turns out that by studying the properties of the Serre functor we can distinguish between the Kronecker quiver and the generalised Kronecker quivers.
So for the actual answer: Bondal and Polishchuk (attributing the result to Suslin) prove in their paper Homological properties of associative algebras: the method of helices that the Serre functor on the level of the Grothendieck group (when equipped with the correct sign) is a unipotent operator. As the Gram matrix of the -Kronecker quiver is given by
the Serre functor is given by
and the characteristic polynomial of is given by . For the matrix to be unipotent we need the characteristic polynomial to be a power of , hence is the only solution.
What I like about this answer is that it opens up questions in higher dimensions, to which I might return at some point.
* I don’t recall his exact words, but I guess we were using a more dg categorical lingo at the time, hence more likely they were something along:
What makes the gluing of two objects along a 2-dimensional vectorspace so special?
which gives a less entertaining title.
- complete intersections
- twists of the structure sheaf
Lately I have been toying a little with Macaulay2, and computations of this sort are very easy in this language, and can be done in far greater generality for arbitrary projective varieties and arbitrary coherent sheaves. As long as you can define the objects you are good to go, because of cohomologyTable. As you can use Macaulay2 online you might as well do it this way, instead of using the limited functionality in my implementation.
As an example you can run the following code (the online interface requires you to put your cursor on the first line and then repeatedly click evaluate):
loadPackage "BGG"; -- P^1 X = Proj(QQ[a,b]) cohomologyTable(OO_X(0) ++ OO_X(1), -4, 4) cohomologyTable(cotangentSheaf X, -4, 4) -- twisted cubic X = Proj(QQ[a,b,c,d] / (a*c-b^2, b*d-c^2, a*d-b*c)); cohomologyTable(OO_X^1, -4, 4) -- Fermat quartic (a K3 surface) X = Proj(QQ[a,b,c,d] / (a^4+b^4+c^4+d^4)); cohomologyTable(tangentSheaf X, -4, 4)
The output for the K3 surface is
ii16 : -- Fermat quartic (a K3 surface) X = Proj(QQ[a,b,c,d] / (a^4+b^4+c^4+d^4)); ii17 : cohomologyTable(tangentSheaf X, -4, 4) -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 oo17 = 2: 124 80 45 20 6 . . . . . . . . . . 1: . 1 4 10 16 20 16 10 4 1 . . . . . 0: . . . . . . 6 20 45 80 124 176 236 304 380
Observe the (weird) indexing: for the K3 surface we have , which sits in position (1,1) and not (1,0) as I would expect. The reason for this can be found in the manual:
This function takes as input a coherent sheaf F, two integers l and h, and prints the dimension dim HH^j F(i-j) for h>=i>=l.
This morning I discovered a super awesome visualisation of Calabi–Yau 3-folds. The very same plot of Hodge numbers of Calabi–Yau 3-folds indicated 25 years ago that something like mirror symmetry could be true, and this tool is a spectacular visualisation of what is already known about mirror pairs for Calabi–Yau 3-folds. Make sure you read the documentation properly, and check out the “Open as static max zoom view” button, which gives you a plot of the whole view!