KU Math 996 (Combinatorial Commutative Algebra): Lecture 1/28/2026
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We began today by watching Numberphile’s video with June Huh on the $g$-conjecture. Then we recalled the theorem we had from last class.
[theorem=%counter%] Let $S = k[x_1, \ldots, x_n]$ and $R = S/I$. Then
\[H_R(t) = \frac{h_R(t)}{(1 - t)^d}\]where $d$ is the Krull dimension of $R$ and $h_R(1) \neq 0$. In this case, it is also true that $h_R \in \ZZ[t]$ rather than $\ZZ[t, t^{-1}]$. [/theorem]
From here, Long added an additional definition. The associated $h$-vector is the coefficients of $h_R(t)$. We then talked about how topological properties inform algebraic properties which inform numerical properties of $h$-vectors (like symmetries, unimodality, etc.).
[definition=%counter%] A simplicial complex on $n$ vertices is a collection of subsets of $[n]$ closed under taking subsets. [/definition]
[example=%counter%] A simplex is defined as $2^{[n]}$. As an example, $2^{[4]}$ is a tetrahedron. The boundary of a simplex is defined as $2^{[n]} \setminus \{[n]\}$. [/example]
[remark=%counter%] Since simplicial complexes are closed under taking subsets, notice that they are fully determined by their largest faces (called facets). [/remark]
[example=%counter%] Graphs are $1$-dimensional simplicial complexes. [/example]
[definition=%counter%] The dimension of a simplicial complex $\Delta$ is the size of the largest face minus $1$. [/definition]
[definition=%counter%] For any simplicial complex $\Delta$, one can define $I_\Delta \subseteq k[x_1, \ldots, x_n]$. It is defined as
\[I_\Delta &= (\text{product of non-faces of } \Delta) \\ &= (\text{minimal non-faces of } \Delta).\]$I_\Delta$ is called the Stanley-Reisner ideal of $\Delta$. [/definition]
[example=%counter%] If $\Delta$ is the simplex on $[n]$, then $I_\Delta = 0$. [/example]
[example=%counter%] If $\Delta$ is the boundary of the simplex on $[n]$, then
\[I_\Delta = (x_1\cdots x_n).\]Consider $R = S/I_\Delta$. Then
\[H_R(t) = \frac{1 - t^n}{(1 - t)^n} = \frac{1 + \cdots + t^{n-1}}{(1 - t)^{n-1}}.\]So the $h$-vector of is $(1, \ldots, 1)$. This tells us $R$ has Krull dimension $n-1$. [/example]