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 
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 .
Face ring
Fix a field 
. The face ring (Stanley-Reisner ring) of over is ![k[K]=k[x_{1},..,x_{n}]/I_{K}](http://s0.wp.com/latex.php?latex=k%5BK%5D%3Dk%5Bx_%7B1%7D%2C..%2Cx_%7Bn%7D%5D%2FI_%7BK%7D&bg=ffffff&fg=333333&s=0 "k[K]=k[x_{1},..,x_{n}]/I_{K}")
where 
is the homogenous ideal generated by the monomials whose support is not in , 
. For example, if is the boundary of a triangle, then ![k[K]=k[x,y,z]/(xyz)](http://s0.wp.com/latex.php?latex=k%5BK%5D%3Dk%5Bx%2Cy%2Cz%5D%2F%28xyz%29&bg=ffffff&fg=333333&s=0 "k[K]=k[x,y,z]/(xyz)")
. ![k[K]](http://s0.wp.com/latex.php?latex=k%5BK%5D&bg=ffffff&fg=333333&s=0 "k[K]")
is graded by degree (variables have degree one, 
has degree zero), and let’s denote the degree 
part by ![k[K]_i](http://s0.wp.com/latex.php?latex=k%5BK%5D_i&bg=ffffff&fg=333333&s=0 "k[K]_i")
. 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,
![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}](http://s0.wp.com/latex.php?latex=hilb%28k%5BK%5D%29%3A%3D%5Csum_%7Bi%5Cgeq+0%7D%5Cdim_k+k%5BK%5D_i+t%5Ei+%3D+%5Cfrac%7Bh_0%28K%29%2Bh_1%28K%29t%2B...%2Bh_d%28K%29t%5Ed%7D%7B%281-t%29%5Ed%7D&bg=ffffff&fg=333333&s=0 "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 is -dimensional).
Cohen-Macaulay (CM) ring
The ring is called Cohen Macaulay (CM) if there are 
elements 
in ![k[K]_1](http://s0.wp.com/latex.php?latex=k%5BK%5D_1&bg=ffffff&fg=333333&s=0 "k[K]_1")
such that is a free ![k[\Theta]](http://s0.wp.com/latex.php?latex=k%5B%5CTheta%5D&bg=ffffff&fg=333333&s=0 "k[\Theta]")
-module. As ![hilb(k[\Theta])=\frac{1}{(1-t)^d}](http://s0.wp.com/latex.php?latex=hilb%28k%5B%5CTheta%5D%29%3D%5Cfrac%7B1%7D%7B%281-t%29%5Ed%7D&bg=ffffff&fg=333333&s=0 "hilb(k[\Theta])=\frac{1}{(1-t)^d}")
, the numerical consequence is that ![hilb(k[K]/(\Theta))=h(K)](http://s0.wp.com/latex.php?latex=hilb%28k%5BK%5D%2F%28%5CTheta%29%29%3Dh%28K%29&bg=ffffff&fg=333333&s=0 "hilb(k[K]/(\Theta))=h(K)")
(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 
.
Hard Lefschetz
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 ![\omega\in k[K]_1](http://s0.wp.com/latex.php?latex=%5Comega%5Cin+k%5BK%5D_1&bg=ffffff&fg=333333&s=0 "\omega\in k[K]_1")
such that the maps
![w^{d-2i}: (k[K]/(\Theta))_i\rightarrow (k[K]/(\Theta))_{d-i}](http://s0.wp.com/latex.php?latex=w%5E%7Bd-2i%7D%3A+%28k%5BK%5D%2F%28%5CTheta%29%29_i%5Crightarrow+%28k%5BK%5D%2F%28%5CTheta%29%29_%7Bd-i%7D+&bg=ffffff&fg=333333&s=0 "w^{d-2i}: (k[K]/(\Theta))_i\rightarrow (k[K]/(\Theta))_{d-i}")
are isomorphisms between those vector spaces for any integer 
. In particular, ![w: (k[K]/(\Theta))_{i-1}\rightarrow (k[K]/(\Theta))_{i}](http://s0.wp.com/latex.php?latex=w%3A+%28k%5BK%5D%2F%28%5CTheta%29%29_%7Bi-1%7D%5Crightarrow+%28k%5BK%5D%2F%28%5CTheta%29%29_%7Bi%7D+&bg=ffffff&fg=333333&s=0 "w: (k[K]/(\Theta))_{i-1}\rightarrow (k[K]/(\Theta))_{i}")
is injective for 
. Thus, the quotient ring ![k[K]/(\Theta, \omega)](http://s0.wp.com/latex.php?latex=k%5BK%5D%2F%28%5CTheta%2C+%5Comega%29&bg=ffffff&fg=333333&s=0 "k[K]/(\Theta, \omega)")
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’.
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?
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