Hilbert–Serre theorem

Remarks

The proof goes by induction on $r$. Multiplication by $x_r$ gives an exact sequence of graded modules $0 \to K(-1) \to M(-1) \to M \to C \to 0$, where $K$ and $C$ are killed by $x_r$ and so are finitely generated graded modules over $k[x_1, \dots, x_{r-1}]$. Taking dimensions degree by degree gives $(1 - t)$ times the Hilbert series of $M$ in terms of those of $C$ and $K$, and the induction hypothesis finishes the proof. Part 2 follows from part 1 by expanding $1/(1-t)^r = \sum_{n \ge 0} \binom{n + r - 1}{r - 1} t^n$. Example: for $M = S$ one has $Q = 1$ and $H_S(n) = \binom{n + r - 1}{r - 1}$ for $n \ge 0$. When the Krull dimension of $M$ is at least $1$, the degree of $P_M$ is one less than this dimension.

Going deeper

For a student

The polynomial $P_M$ is the Hilbert polynomial of $M$. In projective geometry, applied to the coordinate ring of a projective variety, it gives the dimension and the degree of the variety. The local analogue is the existence of the Hilbert–Samuel polynomial, which is proved by applying this theorem to the associated graded ring; it underlies the dimension theorem for local rings.