The short answer: yes.
For the longer answer we should first say what a phantom category is. Recall that it is possible to define the Grothendieck group and Hochschild (co)homomology of (an enhancement of) a triangulated category, see for instance the introduction to Hochschild homology and semiorthogonal decompositions. Moreover, these agree with the more geometric definition that one can give of Hochschild (co)homology, so everything is as nice as one hopes. This way we get some computational tools to calculate these abstractly defined concepts.
Grothendieck groups and Hochschild homology are moreover additive with respect to admissible subcategories, hence a decomposition of our triangulated category yields a direct sum decomposition of the Grothendieck group and Hochschild homology. One could then expect or hope for a certain amount of non-pathological behaviour: if the Grothendieck group and Hochschild homology of a triangulated category is zero, the category itself is zero. In other words, these invariants are conservative.
The vanishing conjecture
This is the content of Kuznetsov’s vanishing conjecture, proposed in Hochschild homology and semiorthogonal decompositions (the Grothendieck group aspect was apparently already proposed by Bondal in the 90s). The following is conjecture 9.1 in the article.
Conjecture Let be a smooth projective variety. If is an admissible subcategory, and then .
Recall that admissible means that the embedding functor has both a left and right adjoint. This is a situation first encountered in étale cohomology, in which we have gluing functors for a decomposition of a scheme into an open subscheme and its complement. So admissibility corresponds to knowing how the subcategory is glued to the bigger category.
If the conjecture were true, we would get some nice properties:
- we can check whether a semiorthogonal collection of subcategories is a semiorthogonal decomposition by checking whether it covers the whole Hochschild homology of the triangulated category
- information on the length of a full exceptional collection
- the ascending chain property for admissible subcategories
Unfortunately, as the short answer might already suggest, this conjecture is false. The interesting way in which this conjecture fails is the subject of the next post, let us first introduce some terminology.
Definition An admissible subcategory of , the bounded derived category of coherent sheaves on a smooth and projective variety , is a quasi-phantom category if
If the second condition is strengthened to
is called a phantom category.
So Kuznetsov’s conjecture asserts the non-existence of (non-zero) (quasi-)phantom categories. And as the next post will show, phantoms exist, and they are even particularly interesting!
Noetherianity and Jordan-Hölder for admissible subcategories
If quasiphantoms did not exist, we would get noetherianity for admissible subcategories (of a derived category of sufficiently nice geometric origin). In other words, if we would have an ascending chain of subcategories it would necessarily stabilise (by finite-dimensionality of the Hochschild homology). This is still an open problem, as quasiphantoms do exist…
A stronger property (which is not implied by the vanishing conjecture, and which is false anyway) is the Jordan-Hölder property. This means that, whenever we have two semiorthogonal decompositions whose components are indecomposable, they are the same up to some permutation of the components. And this is false, even for derived categories of a geometric origin!
- The first counterexample was by considering an example that produces a quasiphantom: the classical Godeaux surface (see the next post for more on this one) has both a length 11 exceptional collection (which is the maximal length in this case), whose orthogonal is a quasiphantom category and a length 9 exceptional collection which cannot be extended any further, whose orthogonal is necessarily not a quasiphantom category.
- A less computation-heavy counterexample was found by considering a two-step blowup of projective 3-space in some curves. This realises a known counterexample which has a non-geometric origin (it is the path algebra of some quiver with relations) as an admissible subcategory inside a derived category of geometric origin.
This lack of a Jordan-Hölder property is annoying, as we could use it to prove nonrationality of cubic 4-folds. But I am getting off track here, and I’ll leave a discussion of rationality problems for another time. In the second post of this two-post series I will discuss some (quasi)phantoms.
I accidentally published a draft of a post, so in case you are confused about where it has gone, you will see it soon.
More interesting news is that we have developed a new feature for the Stacks project. It is now possible to add references to the literature, as outlined in References, slogans and history. You will see it live later today. With this functionality in place, implementing slogans and history is a piece of cake. To see it in action, check out tag 001P (also called Nakayama’s lemma).
The ‘we’ in the previous paragraph is by the way no longer just Johan de Jong and I, but for increased confusion during mail conversations Johan Commelin joined us.
And I have also magically made the Stacks project website up to 15 times faster. Whenever you opened a long tag view page (e.g. one that contains a whole section) the loading times were too long (more than a second, up to 4 in the worst cases…) This is now fixed. In case you think I am a guru programmer, think again, I just realised that I needed to put an index on a certain column…
A while ago (last summer, I guess) I came across the phrase Serre’s diagonal trick in the article Hochschild (co)homology of schemes and with tilting object, by Ragnar Buchweitz and Lutz Hille. This terminology really intrigued me, as I couldn’t understand back then what it actually meant just from the notation, nor did I have the background to know what it was.
In the article the trick is described as
[...] the identification of functors .
At the end of this post I hope this will make some more sense.
One of the many master pieces of Jean-Pierre Serre is his Algèbre locale. In this book from 1965, based on lectures he gave in 1957–1958, the algebraic aspects of intersection theory are treated in a really nice way. The first paragraph in section V.B) is titled La réduction à la diagonale, and in it we read
Soient un corps commutatif algébriquement clos, and deux ensembles algébriques de l’espace affine , et la diagonale de l’espace produit . Alors est évidemment isomorphe à et l’isomorphisme identifie à . Les “géomètres” se servent couramment de cette situation pour ramener l’étude de l’intersection de et à l’étude de l’intersection d’un ensemble algébrique avec une variété linéaire.
If anyone can tell me about an earlier reference to the diagonal trick, please do. The remainder of the paragraph explains the actual workings of the trick. Algebraically speaking the intersection is described by the fiber product (or tensor product) . But we can obtain the isomorphism
So it seems like we have replaced our original tensor product by a more difficult tensor product, as the base ring over which we tensor seems more complicated. Remark that on the right we have identified with the diagonal inside (which is the ring of regular functions on the product of our ambient variety with itself).
But to compute the intersection multiplicity Serre has previously shown that one needs to compute the derived functors of the tensor product. Using this isomorphism we have a good way of computing this: if we work in a sufficiently nice ambient variety (i.e. take we have a truly obvious way of finding a free resolution of the factor in this tensor product, we take the Koszul complex! In some sense we take consecutive hyperplane sections, which is a rather easy thing to do.
So what is sometimes called Serre’s diagonal trick is better known as reduction to the diagonal.
The interpretation of the identification of those two functors remains. The left-hand side of (1) corresponds to : we have replaced the tensor product (of which we wish to compute the Tor’s) by the derived tensor product (the object whose cohomologies are exactly those Tor’s), which in the philosophy of derived categories is the correct thing to do. So remains to interpret . The funny product is given by , where and are the projections . So this computes the product of the intersecting subvarieties inside . Then is the base change to the diagonal, which is isomorphic to . Hence we obtain something on , which will represent the intersection we are trying to compute. So the identification of functors is just a fancy way of writing the isomorphism of algebras (1) in terms of derived categories.
This trick also pops up in the context of exceptional collections on , a subject I might write about later.
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!