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 , and define to be sets of elements of which are -simplices, that is, every subset of elements of an element of is an element of . 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 , we have a ring , which is the free -algebra on the vertices. We can take each element of to be a monomial of degree in the vertices, in fact, every possible -simplex is such a monomial, and squarefree! So we define an ideal to be generated by the non-faces of , and we define the Stanley-Reisner ring of to be .
So now, a couple of examples. The first is the trivial example: the -simplex. For the -simplex, every subset is a face, and so , so . 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 , and these will be the generators of .
So now, we define the -vector of , which has for the number of -faces, and we consider to be a face. So the octohedron has .
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 , we have and , and so determines the -vector.
Now, it’s clear what date the -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 -vector is defined to be and can be recovered as .
So…where does this come from? This is just a strange thing to do. Lets look at 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 .
For the octohedron, the -vector is , 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 , and this is additive in exact sequences. So the resolution above lets us compute , so the -vector is the set of coefficients of the reduced Hilbert series!
Now, it’s a nontrivial theorem that if is a -polytope. So then, we take to be such a polytope. Then it defines a simplicial projective toric variety , and it can be proved that . So this gives a weaker version of Poincare duality for “nice” but possibly singular toric varieties!