KU Math 996 (Combinatorial Commutative Algebra): Lecture 1/23/2026
On this page:
Today, Long had us work in groups on the following four problems. For everything below, assume that $S = K[x_1, \ldots, x_n]$.
[exercise] Let $F \in S$ be homogeneous of degree $d$ and $R = S/(F)$. Prove that
\[H_R(t) = \frac{1-t^d}{(1 - t)^n}.\][/exercise]
[remark] Notice that we can force out a hint. Given that the exercise is actually true, then
\[H_R(t) = H_S(t) - t^dH_S(t).\]So then moving everything to the left gives
\[t^dH_S(t) - H_S(T) + H_R(t).\]The intuition that we want to have here is that the $t^d$ corresponds to a “shift” in degree on the homogeneous components. [/remark]
[proof] Consider the sequence
\[0 \to S \xrightarrow{\cdot F} S \to S/(F) \to 0.\]Since $S$ is a domain, $\cdot F$ is injective. So it immediately follows that the sequence above is short exact. Passing to the graded components, we have the SES
\[0 \to S_i \xrightarrow{\cdot F} S_{i+d} \to (S/(F))_{i+d} \to 0.\]Note that exactness at $S_{i+d}$ can be seen by noting that the grading on $S/(F)$ implies
\[(S/(F))_{i+d} = S_{i+d}/(F)_{i+d}.\]So exactness amounts to proving
\[(F)_{i+d} = F\cdot S_i\]which is fairly immediate. So passing to $K$-vector space dimension gives
\[\dim S_i - \dim S_{i+d} + \dim (S/(F))_{i+d} = 0.\]So then
\[\sum_{i\ge 0} \dim S_i t^{i+d} - \sum_{i\ge 0} \dim S_{i+d} t^{i+d} + \sum_{i\ge 0}\dim (S/(F))_{i+d} t^{i+d} = 0.\]The LHS can be simplified to
\[t^dH_S(t) - \left(H_S(t) - \sum_{0\le i < d} \dim S_i\right) + \left(H_R(t) - \sum_{0 \le i < d} \dim S_{i}/(F)_{i}\right).\]Now we claim that
\[\sum_{0\le i < d} \dim S_i = \sum_{0 \le i < d} \dim S_{i}/(F)_{i}.\]The RHS is
\[\sum_{0 \le i < d} \dim S_i/(F)_i = \sum_{0 \le i < d} \dim S_i - \sum_{0 \le i < d} \dim (F)_i.\]So it suffices to show that $\dim (F)_i = 0$ for all $0 \le i < d$. But that’s immediate because every nonzero element of $F$ is degree $d$ or higher. So our overall expression simplifies as
\[t^d H_S(t) - H_S(t) + H_R(t) = 0.\]The claimed identity immediately follows. [/proof]
[exercise] In the exercise above, let $F = x_1^{a_1} \cdots x_n^{a_n}$ with $\sum a_i = d$. Find a combinatorial proof for the formula for the Hilbert series of $R$. [/exercise]
[exercise] Let $F_1, F_2 \in S$ be homogeneous of degree $d_1$ and $d_2$, respectively. Find $H_R(t)$ where $R = S/(F_1, F_2)$.
Hint: Need to use $\deg \gcd(F_1, F_2)$. [/exercise]
[exercise] Find a combinatorial interpretation of the previous exercise. [/exercise]