(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?

(Eran Nevo) The g-Conjecture I

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


Peter McMullen

The g-conjecture

What are the possible face numbers of triangulations of spheres?

There is only one zero-dimensional sphere and it consists of 2 points.

The one-dimensional triangulations of spheres are simple cycles, having n vertices and n edges, where n\geq 3.

The 2-spheres with n vertices have 3n-6 edges and 2n-4 triangles, and n can be any integer \geq 4. This follows from Euler formula.

For higher-dimensional spheres the number of vertices doesn’t determine all the face numbers, and for spheres of dimension \geq 5 a characterization of the possible face numbers is only conjectured! This problem has interesting relations to other mathematical fields, as we shall see. Continue reading