## (Eran Nevo) The g-Conjecture II: The Commutative Algebra Connection

Richard Stanley

This post is authored by Eran Nevo. (It is the second in a series of five posts.)

## The g-conjecture: the commutative algebra connection

Let $K$ be a triangulation of a $(d-1)$-dimensional sphere. Stanley’s idea was to associate with $K$ a ring $R$, and study the relations between algebraic properties of $R$ and combinatorial properties of $K$.

### Face ring

Fix a field $k$. The face ring (Stanley-Reisner ring) of $K$ over $k$ is $k[K]=k[x_{1},..,x_{n}]/I_{K}$ where $I_{K}$ is the homogenous ideal generated by the monomials whose support is not in $K$, $\{\prod_{i\in S}x_i:\ S\notin K\}$. For example, if $K$ is the boundary of a triangle, then $k[K]=k[x,y,z]/(xyz)$. $k[K]$ is graded by degree (variables have degree one, $1$ has degree zero), and let’s denote the degree $i$ part by $k[K]_i$. This part is a finite dimensional $k$-vector space and we can collect all these dimensions in a sequence, or a series, called the Hilbert series of $k[K]$, which carries the same information as $f(K)$. More precisely,

$hilb(k[K]):=\sum_{i\geq 0}\dim_k k[K]_i t^i = \frac{h_0(K)+h_1(K)t+...+h_d(K)t^d}{(1-t)^d}$

(recall that $K$ is $(d-1)$-dimensional).

### Cohen-Macaulay (CM) ring

The ring $k[K]$ is called Cohen Macaulay (CM) if there are $d$ elements $\Theta=\{\theta_1,..,\theta_d\}$ in $k[K]_1$ such that $k[K]$ is a free $k[\Theta]$-module. As $hilb(k[\Theta])=\frac{1}{(1-t)^d}$, the numerical consequence is that $hilb(k[K]/(\Theta))=h(K)$ (we use $h$ both as a vector and as a polynomial, with the obvious identification).

Macaulay (revisited) showed that the Hilbert series of standard rings (=quatient of the polynomial ring by a homogenous ideal) are exactly the $M$-vectors (sequences).

A theorem of Riesner characterizes the simplicial complexes $K$ with a CM face ring over a fixed field $k$ in terms of the homology of $K$ and its face links (with $k$-coefficients). It follows that if $K$ is a simplicial sphere then $k[K]$ is CM, hence $h(K)$ is an $M$ vector! This gives more inequalities on $f(K)$. This is also how Stanley proved the Upper Bound Conjecture, for face number of spheres: It follows that if $K$ is a $(d-1)$-sphere with $n$ vertices, and $C(d,n)$ is the boundary of the cyclic $d$-polytope with $n$ vertices, then for every $i$, $f_i(K)\leq f_i(C(d,n))$. This is as $h_i(K)\leq \binom{n-d+i-1}{i}=h_i(C(d,n))$.

### Hard Lefschetz

Let $K$ be the boundary of a simplicial $d$-polytope. Stanley observed that the hard Lefschetz theorem for toric varieties, an important theorem in algebraic geometry, translates in the language of face rings as follows: there exists $\Theta$ as above and $\omega\in k[K]_1$ such that the maps

$w^{d-2i}: (k[K]/(\Theta))_i\rightarrow (k[K]/(\Theta))_{d-i}$

are isomorphisms between those vector spaces for any integer $0\leq i\leq \frac{d}{2}$. In particular, $w: (k[K]/(\Theta))_{i-1}\rightarrow (k[K]/(\Theta))_{i}$ is injective for $1\leq i\leq \frac{d}{2}$. Thus, the quotient ring $k[K]/(\Theta, \omega)$ has Hilbert series starting with $g(K)$. This means, again by Macaulay theorem, that $g(K)$ is an $M$-vector!

Later, in 1993, McMullen gave a different proof of this part of his conjectured $g$-theorem. His proof actually proves hard Lefschetz for this case. See McMullen’s survey paper `Polyhedra and polytopes: algebra and combinatorics’.

### Problems

Does hard Lefschetz theorem hold for non polytopal spheres?

Can you think of examples of simplicial spheres which cannot be realized as the boundary of convex polytopes?