Now that we have algebra on our side, we can start doing some real geometry. Today we’ll look at Hilbert Functions, Hilbert Polynomials, and all the very nice invariants we can define with them. These invariants are key to the study of varieties (and, in fact, schemes, which we’ve yet to mention) and are easily calculated with the Hilbert Polynomial, though we will not touch on computation now.
Now let be an algebraically closed field, and let and let be a graded -module. We define the Hilbert Function of to be . That is, the dimension, as a -vector space, of the degree part of .
Theorem: Let be a finitely generated -module. Then there exists a unique polynomial with rational coefficients such that, for all sufficiently large, . Furthermore, if is the ideal , then .
Proof: To begin, we define a short exact sequence to be , a pair of module homomorphisms such that the is injective, is surjective, and the image of consists precisely of the elements of which are sent to 0 by . Later, when we talk about exact sequences in general, it will be more obvious why the zeros are there, but for now we’ll just take it as a convention.
We first note the fact that Hilbert Functions and the varieties mentioned in the theorem are both additive in short exact sequences. That is, knowing them for the first and last module gives complete knowledge for the second one. Specifically, if the theorem holds for , then it holds for . Taking into account the existence of filtrations discussed last time, we in fact only have to look at modules of the form for some prime ideal .
Now, if , then it is true, provided that we define the degree of the zero polynomial to be -1 as well as the dimension of the empty set, as the Hilbert Function is zero for positive input. Now we assume that is not , and that the theorem holds for ideals with varieties of lower dimension. Then there is a short exact sequence for given by having the first map be multiplication by and the second being the natural quotient map. The last module we will denote by .
Using the additivity in short exact sequences, this tells us that and that , where . This tells us that the dimension of . So the result holds for .
Finally, a function such that is a polynomial must be a polynomial itself, and so is a polynomial of the appropriate degree. QED
This introduces us to the importance of short exact sequences, things being additive in short exact sequences, and the power of induction. So now, we can define the Hilbert Polynomial to be the polynomial of the theorem.
So now let be a projective variety. Then we define to be the Hilbert Polynomial of the projective coordinate ring of . We call this the Hilbert Polynomial of the variety. It encodes a LOT of information. We’ll now define a bunch of invariants that it gives us.
First, we can use the Hilbert polynomial to DEFINE the dimension. Just say that the dimension of a variety is equal to the degree of its Hilbert Polynomial.
Second, we can define the arithmetic genus by . This invariant is especially useful for curves, for reasons that we’ll encounter in the future.
Third, we define the degree of a variety of dimension to be multiplied by the lead coefficient of the Hilbery Polynomial. The degree is roughly the number of times that the variety will intersect a hyperplane of complementary dimension in projective space. We’ll not make this rigorous right now, because rather we’ll do that next time, and use it to prove Bezout’s Theorem.