D-Module Basics II

    In my first post, we showed how important information about a PDE on could be turned into a certain module over the ring of regular differential operators on .  Our focus today will be about extending the idea of differential operators to other kinds of objects.

    The first step will be to define the ring of regular differential operators on an arbitrary complex affine variety.  The construction is a bit mysterious, but for smooth varieties it is the ring generated by multiplying by regular functions and differentiating along regular vector fields.  Then, we will show how there is a natural restriction map from the differential operators on an affine variety to the differential operators on a Zariski open set in that variety.  With this important tool, we can compare differential operators in two different affine subsets of a non-affine variety, and see if we can patch them together into a differential operator over the union.  Effectively, this will define the sheaf of differential operators on any variety.  We will close by computing the global differential operators on , and showing that they are in agreement with the universal enveloping algebra of the Lie algebra of regular vector fields on .

D-module Basics I

    Its a bit embarassing for me, but I am only now really getting around to learning the theory of D-modules.  Given that my research area is Noncommutative Geometry, Homological Algebra or maybe Representation Theory depending on when you ask me, this just shouldn’t be.  I made slow progress through Bernstein’s notes last week, and then I found Schneiders’ notes a few days ago.  They make a good compliment to Bernstein’s notes; where Bernstein is good about mentioning the underlying ways you should think about various tools, Schneiders is good about saying things clearly and writing out proofs.  I tend to prefer the former kind of paper, until the moment I get lost and the lack of nitty-gritty to pour over trips me up.

    Anywho, I am fond of explaining things as a method of solidifying thoughts in my head and turning vague concepts into concrete statements (one motivation for having the blog).  It feels a little silly talking about something that I have already linked to better resources for, but I have never let that stop me in the past.  Besides, Matt requested it, and I think it would be good to foster a bit of discussion, since one key factor in understanding why D-modules are cool is understanding how they connect to far-flung reaches of math.  For more D-module stuff, especially a lengthy list of why mathematicians care about them, see the Secret Blogging Seminar’s pair of posts on the subject, especially the comments on the first post.

    Today, I will define D-modules, and show how certain D-modules can describe solutions to various homogenous and inhomogenous partial differential equations.

Filtrations in Algebraic Geometry

    When learning scheme theory, the first important functor one learns about is probably , and the second is likely .  is important but bland, since scheme theory can’t say anything new; for instance, the cohomology of any coherent module must vanish.  is a whole different story, where graded algebras are turned into spaces with interesting topology and invariants that can be useful in solving some very fundamental problems (see, for example, the Jacobian of an elliptic curve).  I have always found it a little bit mysterious that something as simple as a grading, which I usually regard as a computational tool, can have such important geometric implications.

    A natural question then is, “How far can I weaken things and still construct interesting schemes?”  The particular weakening I have in mind is a filtered algebra.  Instead of the algebra being the direct sum of vector spaces with compatible multiplication, we will look at algebras which are the union of a chain of nested vector spaces with compatible multiplication.  Then, we will look at a couple of associated schemes, and discover that familiar things like projective compactifications arise naturally in this setting.

My Favorite Random Fact About Abelian Categories

  This weekend, I am off to MSRI for a two week mini-course on moduli-spaces and deformation theory.  I am very excited since it comes perfectly timed with a crescendo in my interest in the tangent space to a stack.  I can also assure all of you that this is IN NO WAY a cover for a clandestine sabotage mission aimed at the proprietors of the Secret Blogging Seminar, in a misguided attempt to foster a rivalry between the two blogs.  Absolutely none of that.

  When I get there, I will be expected to give a half hour introductory talk on one of the background concepts that we were all supposed to know.  I signed up to talk about the easiest of the availible topics, abelian categories.  This decision was in part motivated by the obscene degree to which I planned to be busy the week beforehand. However, it was also because I have a random fact about abelian categories that I enjoy sharing with otherwise knowledgable people.

Algebraic Codimension

  When exploring varieties, one of the most important questions is to understand how one variety is sitting inside another.  Equivalently, it is useful to understand all the possible ways of putting subvarieties into a fixed variety; this is how Krull dimension, smoothness, irreducibility, blow-ups, etc are defined.  Stating things in terms of subvarieties is useful because, on the algebraic side of things, these are quotient algebras and hence very tractable.  However, it often happens that the nearest algebraic approximation to a geometric notion isn’t perfect; compare for example the Zariski tangent space to the usual geometric tangent space.

  This is the case for what I want to talk about today: codimension.  The algebraic counterpart is ‘depth’, which is measured by ‘regular sequences’ (which are the algebraic counterpart to flags/bases on the tangent space).  The varieties where depth correctly measures codimension of subvarieties are called ‘Cohen-Macauley’; these varieties have innumerable nice properties such as being about the weakest class that some analog of Serre duality holds for.

Chain complexes as Graded C[epsilon]-modules, part 3: Bicomplexes and Superalgebras

< See Part 2

The last two posts in this topic were mostly exploring what homology looks like in the language of modules. This time, we will return to the task of shoving homological algebra into the theory of module categories, by considering another fundamental tool: bicomplexes. This is particularly fruitful, since it reveals that wants to anticommute with itself (which it technically does, because it squares to zero). To get the right framework for this, we will introduce the notion of a superalgebra.

Chain Complexes as Graded C[epsilon] modules (2)

< See Part 1

In this post, we look at the existance of long exact sequences for a given crude homology functor. I will give a sufficient (but possibly not necessary) condition for when such a functor will have always have them.

Chain Complexes as Graded C[\epsilon] Modules

My first (real) post will be devoted to a pet project of mine that has been on a backburner for several months.

The general idea is that a chain complex can be thought of as nothing more than a graded module, with being a degree 1 or -1 element (depending on whether you want your boundary maps going up or down). This begs the natural question, how much of homological algebra can be stated module-theoretically?