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.