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.
First, rigorous definitions. An filtered algebra is an algebra , together with a sequence of vector spaces such that:
An element of which is not in is called an element of degree n. A filtered -algebra will be called positive if for , and connected if it is positive and . All filtered algebras from here on are connected.
Filtered algebras are very easy to construct; in particular, they are easier to construct than graded algebras. A graded algebra is usually defined by picking a collection of generators and assigning them each degrees , such that all the relations are homogeneous. To make a filtered algebra, define the degree of an element to be the smallest degree of any polynomial in the which represents . As long as the degrees of all the generators were positive, this will be a connected filtered ring.
The goal now is to find schemes that contain information about the ring and its filtration. To do this, we will construct two graded algebras whose tries to accomplish this feat.
The first algebra is called the associated graded algebra, usually denoted :
The multiplication is effectively the same as in , and the filtered condition implies it is well-defined. There is a set map, called the symbol map, from to which sends an element of degree n to its image in , but this map is not even an additive homomorphism. One useful ability of the associated graded algebra is for checking when a map of filtered rings is an isomorphism, since it will necessarily induce an isomorphism between the associated graded algebras. Can you see why this is true? (Answer)
The second algebra we want to look at, called the Rees algebra, is often denoted :
The first important observation is that the Rees algebra contains all the information of both the original ring and of the associated graded algebra. To see this, notice that
Actually, this tells us more. is naturally a -algebra, and so has a map to the affine line. This map is usually flat and if so, this turns into a flat family. The fibers of this map are precisely described by the above two equations. Above , the fiber is , while the fiber over every other point is . Thus, the associated graded algebra is a flat limit of the filtered algebra, so any property that is continuous with respect to flat families is the same for both.
What about ? Let’s split it up into two pieces, the closed subscheme given by and its compliment. The closed subscheme is , which is . Its compliment is given by , but since is invertible, this open subscheme is isomorphic to , which is . Thus, is the union of two pieces, the of the original ring, and the of the associated graded algebra.
Geometrically, we can see what happened by picking generators for . This embeds in affine -space. The assignment of degrees for each generator determines a projective closure of affine -space. of the Rees algebra is then the closure of inside this projective closure. This also shows that of the associated graded algebra is the pieces you must add to complete the scheme .
The moral of the story then is that giving a ring a connected filtration is the same as choosing a compactification of the of that ring. Thus, two important schemes to look at are the compactification, and the compactification minus the original points, which correspond to the Rees algebra and the associated graded algebra.
Noncommutative Remark: One type of filtered algebra that people care quite a bit about are rings of differential operators. Their construction and importance is a post in of itself, but I can’t not mention the following nifty fact for people who have seen them already. Rings of differential operators are non-commutative, but their associated graded algebras with respect to the natural filtration are commutative. Therefore, if you believe that most things that work in commutative algebraic geometry should work in some sort of non-commutative algebraic geometry (at least on the philosophical level), then rings of differential operators have the property that their ‘spectrums’ are naturally compactified by commutative spaces. Specifically, the ‘non-commutative scheme’ of differential operators on a smooth scheme is compactified by the projectivized cotangent bundle. See, for instance, http://arxiv.org/abs/math/0304320.
Geometric Remark: One way of characterizing a graded -algebra that I think is too often overlooked is that they are algebras with an action of . These are the same, because the action splits the algebra into character spaces (eigenspaces), which are parameterized by elements of . Conversely, given a -graded algebra, a can be defined by having act on an element of degree by multiplication by .
This is productive to think about because it is closer to the geometric nature of . Given a graded (connected) -algebra , , where is the point determined by the irrelevant ideal.
But then the problem is that there is no analogous characterization of a filtration (that I know of). I used to believe that it was the action was really the reason graded algebras were so geometric, but if filtrations also carry interesting geometric information, there has to be something more.