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 be a triangulation of a -dimensional sphere. Stanley’s idea was to associate with a ring , and study the relations between algebraic properties of and combinatorial properties of .
Fix a field . The face ring (Stanley-Reisner ring) of over is where is the homogenous ideal generated by the monomials whose support is not in , . For example, if is the boundary of a triangle, then . is graded by degree (variables have degree one, has degree zero), and let’s denote the degree part by . This part is a finite dimensional -vector space and we can collect all these dimensions in a sequence, or a series, called the Hilbert series of , which carries the same information as . More precisely,
(recall that is -dimensional).
Cohen-Macaulay (CM) ring
The ring is called Cohen Macaulay (CM) if there are elements in such that is a free -module. As , the numerical consequence is that (we use 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 -vectors (sequences).
A theorem of Riesner characterizes the simplicial complexes with a CM face ring over a fixed field in terms of the homology of and its face links (with $k$-coefficients). It follows that if is a simplicial sphere then is CM, hence is an $M$ vector! This gives more inequalities on . This is also how Stanley proved the Upper Bound Conjecture, for face number of spheres: It follows that if is a -sphere with vertices, and is the boundary of the cyclic -polytope with vertices, then for every , . This is as .
Let be the boundary of a simplicial -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 as above and such that the maps
are isomorphisms between those vector spaces for any integer . In particular, is injective for . Thus, the quotient ring has Hilbert series starting with . This means, again by Macaulay theorem, that is an -vector!
Later, in 1993, McMullen gave a different proof of this part of his conjectured -theorem. His proof actually proves hard Lefschetz for this case. See McMullen’s survey paper (Polyhedra and polytopes: algebra and combinatorics. Algebraic and geometric combinatorics, 235–267, Contemp. Math., 423, Amer. Math. Soc., Providence, RI, 2006. (short summary)). (Update Feb. 2019: See also Balin Fleming and Kalle Karu’s paper with a greatly simplified version of McMullen’s proof: https://www.math.ubc.ca/~karu/papers/simple.pdf.)
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?