When fine just ain’t enough February 2, 2010Posted by David Speyer in Algebraic Geometry, complex analysis, homological algebra, things I don't understand.
If you use sheaves to study differential geometry, one of the basic lemmas you’ll want is the following: Let be a smooth manifold and let be a sheaf of modules over . (For example, might be the sheaf of sections of a vector bundle.) Then all higher sheaf cohomology of 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.
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.
Hall algebras and Donaldson-Thomas invariants I March 25, 2009Posted by Joel Kamnitzer in Algebraic Geometry, conferences, homological algebra, quantum groups, things I don't understand.
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 which has the following strong finiteness properties: namely and are finite for any objects . Then one can define an algebra, called the Hall algebra of , which has a basis given by isomorphism classes of objects of and whose structure constants are the number of subobjects of which are isomorphic to and whose quotient is isomorphic to .
The main source of interest of Hall algebras for me is the Ringel-Green theorem which states that if you start with a quiver , then the Hall algebra of the category of representation of over a finite field is isomorphic to the upper half of the quantum group corresponding to at the parameter .
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 is the category of coherent sheaves on a Calabi-Yau 3-fold.
Reference Hunt I April 10, 2008Posted by Scott Carnahan in Algebraic Topology, Category Theory, homological algebra, quantum algebra.
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 denote the monoidal category of A-graded complex vector spaces. Then:
- Equivalence classes of braided structures on are classified by elements of .
- also classifies -valued quadratic forms on A.
- , 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.
Hypertoric varieties and Koszul duality April 4, 2008Posted by Ben Webster in Algebraic Geometry, category O, combinatorics, D-modules, homological algebra, IAS, representation theory, talks.
So, on Wednesday, I gave a talk with the above title at IAS, about work in progress with Tom Braden, Tony Licata, and Nick Proudfoot. I was hoping to get David Nadler to blog it for me, but he was *ahem* indisposed. Failing that, I’ll direct you all to David Ben-Zvi’s notes (warning: freaking huge PDF). Hopefully, that will whet your appetite for the forthcoming paper.
Derived categories bleg March 24, 2008Posted by Ben Webster in Category Theory, homological algebra.
Is the following theorem true, and if so, where is the reference?
Theorem? Let be a triangulated category with finite homological dimension, and let be a finite set of objects which generate and whose classes are a basis of the Grothendieck group, such that for all . Then there exists a -structure on such that the 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 -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 ‘s are the simple objects in the heart of a -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 November 13, 2007Posted by Noah Snyder in Anton Geraschenko, blog triumphalism, guest post, homological algebra.
[I'm happy to introduce Anton, our very first guest blogger.]
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 lemma can prove the “ lemma” as well.
Koszul algebras and Koszul duality November 1, 2007Posted by Ben Webster in category O, homological algebra.
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 is an algebraically closed field, any semi-simple -algebra is a product of matrix algebras over .
(We say that an algebra is semi-simple if any submodule of any -module has a complement, that is, if every short exact sequence of -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. (more…)
Real Curves, Open Strings, and A-infinity Algebras September 20, 2007Posted by A.J. Tolland in Category Theory, homological algebra, mathematical physics.
Kevin Costello gave a talk last week in one of Peter Teichner’s many seminars, explaining -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.
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.