(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’.


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?

4 thoughts on “(Eran Nevo) The g-Conjecture II: The Commutative Algebra Connection

  1. Pingback: » Algebraic variety Esther McCoy

  2. Pingback: Samson en Gert » Rigid analytic space

  3. Pingback: Satoshi Murai and Eran Nevo proved the Generalized Lower Bound Conjecture. | Combinatorics and more

  4. Pingback: Richard Stanley: How the Proof of the Upper Bound Theorem (for spheres) was Found | Combinatorics and more

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s