Today, we’re going to do something completely different, but which most of my peers seem not to have seen, but is a very cool application of algebraic geometry.

A simplicial complex is a nice type of topological space.  Take a collection of points, called the vertices X_0, and define X_n to be sets of (n+1) elements of X_0 which are n-simplices, that is, every subset of (k+1) elements of an element of X_n is an element of X_k.  We can construct these geometrically by looking in a really large Euclidean space, picking points, and for each simplex, taking their simplicial span, that is, all linear combinations of the points with coefficients nonnegative and summing to one.

Now, we start with the algebra.  Given a simplicial complex \Delta, we have a ring k[X_0], which is the free k-algebra on the vertices.  We can take each element of X_n to be a monomial of degree n+1 in the vertices, in fact, every possible n-simplex is such a monomial, and squarefree!  So we define an ideal I_\Delta to be generated by the non-faces of \Delta, and we define the Stanley-Reisner ring of \Delta to be k[\Delta]=k[X_0]/I_\Delta.

So now, a couple of examples.  The first is the trivial example: the n-simplex.  For the n-simplex, every subset is a face, and so I_\Delta=0, so k[\Delta]=k[v_0,\ldots,v_n].  Now, a less trivial example is in order: the octohedron.  It has six vertices, and we can label them so that the non-edges are v_2v_6, v_1v_3, v_4v_5, and these will be the generators of I_\Delta.

So now, we define the f-vector of \Delta, which has for f_i the number of i-faces, and we consider \emptyset to be a -1 face.  So the octohedron has (1,6,12,8).

Next, we look at polytopes.  These are the convex hulls of a finite collection of points in some Euclidean space.  We’ll actually care only about the case where the boundary is a simplicial complex, we’ll call these simplicial polytopes.  We even get a couple of nice formulas: there’s Euler’s formula that for a simplicial polyhedron in \mathbb{R}^3, we have f_0-f_1+f_2=2 and 3f_2=2f_1, and so f_0 determines the f-vector.

Now, it’s clear what date the f-vector describes, but we’re going to transform it into a much less obvious form, and from that, pull a nice theorem out of thin air.  The h-vector is defined to be h_j=\sum_{i=0}^j (-1)^{j-i}\binom{d-i}{j-i}f_{i-1} and f_j can be recovered as \sum_{i=0}^{j+1}\binom{d-i}{j+1-i}h_i.

So…where does this come from? This is just a strange thing to do.  Lets look at k[\Delta] itself.  We want to try to make a graded free resolution (which we’ll treat largely as a black box).  This amounts to there being an exact sequence with maps of degree 0 which is free modules, except the last term.  For the octohedron, we have 0\to R(-6)\to R(-4)^3\to R(-2)^3\to R\to R/I_\Delta\to 0.

For the octohedron, the h-vector is (1,3,3,1), which you can see are the exponents here, and that’s not a coincidence! But more fundamentally, we define the Hilbert series of a graded module to be h_M(t)=\sum (\dim M_i)t^i, and this is additive in exact sequences.  So the resolution above lets us compute h_{k[\Delta]}(t)=\frac{1-3t^2+3t^4-t^6}{(1-t)^6}=\frac{1+3t+3t^2+t^3}{(1-t)^3}, so the h-vector is the set of coefficients of the reduced Hilbert series!

Now, it’s a nontrivial theorem that h_i=h_{d-i} if \Delta is a d-polytope.  So then, we take \Delta to be such a polytope.  Then it defines a simplicial projective toric variety X_\Delta, and it can be proved that h_i=\dim H^{2i}(X_\Delta,\mathbb{C}).  So this gives a weaker version of Poincare duality for “nice” but possibly singular toric varieties!

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