When fine just ain’t enough

If you use sheaves to study differential geometry, one of the basic lemmas you’ll want is the following: Let X be a smooth manifold and let \mathcal{E} be a sheaf of modules over C^{\infty}(X). (For example, \mathcal{E} might be the sheaf of sections of a vector bundle.) Then all higher sheaf cohomology of \mathcal{E} vanishes.

The proof of this theorem is basically homological algebra plus the existence of partitions of unity. This gives rise to a slogan “when you have partitions of unity, sheaf cohomology vanishes.” One way to make this definition precise is through the technology of fine sheaves.

As Wikipedia says today, “[f]ine sheaves are usually only used over paracompact Hausdorff spaces”. That means they are not used when working with the Zariski topology on schemes, for example. When I started digging into this, I realized there were good reasons: The technology of fine sheaves (and the closely related technology of soft sheaves) does not include the scheme theory cases which we would want it to.

However, there are theorems of the form “when you have partitions of unity, sheaf cohomology vanishes” on schemes and on complex manifolds. I put up a question at MathOverflow asking whether there were better formulations that included these examples, but I probably didn’t formulate it well. I think spelling out all my issues would be too discursive for MathOverflow, so I’m bringing it over here.

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A hunka hunka burnin’ knot homology

One of the conundra of mathematics in the age of the internet is when to start talking about your results. Do you wait until a convenient chance to talk at a conference? Wait until the paper is ready to be submitted to the arXiv (not to mention the question of when things are ready for the arXiv)? Until your paper is accepted? Or just until you’re confident you’ve disposed of any major errors in your proofs?

This line is particularly hard to walk when you think the result in question is very exciting. On one hand, obviously you are excited yourself, and want to tell people your exciting results (not to mention any worries you might have about being scooped); on the other, the embarrassment of making a mistake is roughly proportional to the attention that a result will grab.

At the moment, as you may have guessed, this is not just theoretical musing on my part. Rather, I’ve been working on-and-off for the last year, but most intensely over the last couple of months, on a paper which I think will be rather exciting (of course, I could be wrong). Continue reading

The Witt group, or the cohomology of the periodic table of n-categories

A very popular topic at the Modular Categories conference was the a generalization of the Witt group which is being developed by Davydov, Mueger, Nikshych, and Ostrik. What is this Witt group? Well it’s the simplest case of the cohomology of the periodic table of n-categories!

In this post I want to explain the definition of this cohomology theory and explain why it generalizes the classical Witt group.

First recall the Baez-Dolan periodic table.

Periodic Table

Periodic Table

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Hall algebras and Donaldson-Thomas invariants I

I would like to tell you about recent work of Dominic Joyce and others (Bridgeland, Kontsevich-Soibelman, Behrend, Pandaripande-Thomas, etc) on Hall algebras and Donaldson-Thomas invariants.  I don’t completely understand this work, but it seems very exciting to me. This post will largely be based on talks by Bridgeland and Joyce that I heard last month at MSRI.

In this post, I will concentrate on different versions of Hall algebras. Let us start with the most elementary one. Suppose I have an abelian category \mathcal{A} which has the following strong finiteness properties: namely Hom(A,B) and Ext^1(A,B) are finite for any objects A, B . Then one can define an algebra, called the Hall algebra of \mathcal{A}, which has a basis given by isomorphism classes of objects of \mathcal{A} and whose structure constants c_{[M], [N]}^{[P]} are the number of subobjects of P which are isomorphic to N and whose quotient is isomorphic to M .

The main source of interest of Hall algebras for me is the Ringel-Green theorem which states that if you start with a quiver Q, then the Hall algebra of the category of representation of Q over a finite field \mathbb{F}_q is isomorphic to the upper half of the quantum group corresponding to Q at the parameter q^{1/2}.

The obvious question concerning Hall algebras is to come up with a framework for understanding them when the Hom and Ext sets are not finite. This is what Joyce has done and he has applied it where A is the category of coherent sheaves on a Calabi-Yau 3-fold.

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Reference Hunt I

Does anyone know where the following useful facts were first proved? A lot of papers just say, "It is known that…" and I’d like to give proper attribution in some future work.

Let A be an abelian group, and let Vect^A denote the monoidal category of A-graded complex vector spaces. Then:

  1. Equivalence classes of braided structures on Vect^A are classified by elements of H^4(K(A,2),\mathbb{C}^\times).
  2. H^4(K(A,2),\mathbb{C}^\times) also classifies \mathbb{C}^\times-valued quadratic forms on A.
  3. H^4(K(A,2),\mathbb{C}^\times) = H^3_{ab}(A, \mathbb{C}^\times), where the right side is "Eilenberg-MacLane abelian group cohomology" (defined in MacLane’s 1950 ICM address).

There is an additional neat interpretation involving double loop maps and multiplicative torsors on A, but I don’t need that level of sophistication for the near future.

Derived categories bleg

Is the following theorem true, and if so, where is the reference?

Theorem? Let D be a triangulated category with finite homological dimension, and let \{L_1,\ldots, L_n\} be a finite set of objects which generate D and whose classes are a basis of the Grothendieck group, such that \mathrm{Ext}^{-k}(L_i,L_j)=0 for all k>0. Then there exists a t-structure on D such that the L_i are the set of simple objects in the heart.

These conditions are obviously necessary, but I’m a bit less confident that they are sufficient.

Hmmm, I guess this means we’re going to have to explain what a t-structure is now, doesn’t it?

EDIT: I found the theorem I wanted in the paper David mentioned, by Bezrukavnikov.  Roughly, the right theorem is that the L_i‘s are the simple objects in the heart of a t-structure if there is a semi-simple abelian subcategory of  the triangulated category in which they are the simple objects, and they satisfy the conditions above.

(Anton Geraschenko) The Salamander lemma

[I’m happy to introduce Anton, our very first guest blogger.]

A couple of years ago, George Bergman gave me a copy of a fun preprint that he never got around to preparing for publication. A scan of it is posted here. It starts

The “magic” of diagram-chasing consists in establishing relationships between distant points of a diagram—exactness implications, connecting morphisms, etc.. These “long” connections are in general composits of “short” (unmagical) connections, but the latter, and even the objects they join, are frequently not visible in the diagram-chasing proof. We attempt to remedy this situation here.

If you don’t like diagram chases, it’s likely that you still won’t like them once you know the Salamander lemma. The salamanders chase the diagrams for you, but you still have to chase the salamanders. I think the salamander proofs are easier to explain (once you know the Salamander lemma), and it’s easier to see where you use the hypotheses. For example, it is totally clear that the argument for the 3\times 3 lemma can prove the “20\times 20 lemma” as well.
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Koszul algebras and Koszul duality

One of the famous theorems that tend to crop up in undergraduate algebra classes is the Artin-Wedderburn theorem, which says

Theorem. Any semi-simple ring is a product of matrix algebras over division algebras. In particular, if k is an algebraically closed field, any semi-simple k-algebra is a product of matrix algebras over k.

(We say that an algebra A is semi-simple if any submodule of any A-module has a complement, that is, if every short exact sequence of A-modules splits).

Now, looking at this theorem, one might imagine that we now know a lot about finite-dimensional algebra. After all, there are only two kinds of finite-dimensional algebras, semi-simple and non-semi-simple, and we understand one of those halves quite well. Better yet semi-simplicity is an “open” condition. If we think about the set of associative products a finite dimensional vector space could have, the set of such products which are semi-simple is an open set in the Zariski topology, which those of us who like algebraic geometry know means it is pretty darn big, provided it is non-empty (which is it is, since every vector space has a semi-simple product as the sum of a bunch of copies of the field).

But, of course, this is ridiculous. To borrow a metaphor, dividing algebras into semi-simple and not-semi-simple is like dividing the world into bananas and non-bananas. Continue reading

Real Curves, Open Strings, and A-infinity Algebras

Kevin Costello gave a talk last week in one of Peter Teichner’s many seminars, explaining A_\infty-algebras with a view towards his papers on topological string theory. It was the sort of talk that might have interested a lot of people, so (with Kevin’s permission), I’m posting my .pdf scanned notes here. I’ve added some physics interpretation that Kevin didn’t make explicit.

Defining A_\infty-algebras

Kevin also gave a shorter talk (in a different Teichner seminar) about his characterization of the homotopy type of a moduli space of genus zero open string worldsheets. My notes here are less detailed, but maybe someone will enjoy them.

The Homotopy Type of a Certain Moduli Space of Open String Worldsheets