Geometric motivation for the Stanley-Reisner correspondence

Question

The Stanley-Reisner ring of an abstract simplicial complex $\Delta$ on the vertex set $\{1,...,n\}$ is the $k$-algebra $$ k[X_1,...,X_n]/I_\Delta $$ where $I_\Delta$ is the ideal generated by the $X_{i_1}...X_{i_r}$ with ${i_1,...,i_r}\notin \Delta$.

Somebody told me that this construction helps to study varieties $k[X_1,...,X_n]/I$ for an arbitrary ideal $I$ as follows (if I am not missing something): Let $X_1<...< X_n$ be a monomial order and consider the initial ideal $I':=in_{<}(I)$ of $I$. The passage from $k[X_1,...,X_n]/I$ to $k[X_1,...,X_n]/I'$ is called 'flat deformation' and this term makes sense if I draw pictures of the varieties. Many properties of $I$ (like dimension) are directly related to properties of $I'$.

The aim is to find a to $I'$ related ideal $I_\Delta$ for an abstract simplicial complex $\Delta$.

I was told that the problem that $I'$ has a generator like $X_1^2$ could be resolved by introducing a new variable $X_1'$, replacing $X_1X_1$ by $X_1X_1'$ and mod out $X_1-X_1'$ of $k[X_1,...,X_n,X_1']$. First, I don't understand why this should be closer to the form $I_\Delta$ because $I_\Delta$ is generated by monomials.

My main question is:

Can one see in a concrete affine example how the geometry of $\Delta$ relates to the initial variety $I$?

If you take for example the simplicial complex $\Delta$ (I apologize that I can not typeset the brackets) $\emptyset,X, Y, Z, XY, XZ, YZ$ which looks like a one sphere, the associated variety is the union of the $XY$, the $XZ$ and the $YZ$ hyperplane in $\mathbb{R}[X,Y,Z]$. Why is this reasonable? On the other hand, I would like to know how the simplicial complex (after the transformation indicated above) corresponding to the circle $I=(X^2+Y^2-1)$ in $\mathbb{R}[X,Y,Z]$ looks like. It would be nice if this is associated to the $\Delta$.

Please tell me, if this question is completely unreasonable or doesn't make any sense. I am an absolute beginner in algebraic geometry.

Answer 1

First when it comes to comparison with the simplicial complex it should be realised that the Stanley-Reisner ring corresponds to the cone over the complex. There is a non-homogeneous version of it where one replaces the linear subspace through the origin by the affine space parallel to the linear space but passing through $1$ (the way the standard simplices are defined). If the simplicial complex is a cone with a fixed apex, the Stanley-Reisner ring of the base of the cone is isomorphic to the non-homogeneous ring of the cone. (Incidentally, or maybe not so incidentally, for a field of characteristic zero this non-homogeneous ring is the degree zero part of the Sullivan differential graded algebra associated to the complex.)

The Stanley-Reisner rings also pop back up for a general complex as being related to the stars of vertices, the stars being cones with apex the point. More precisely the localisation of the Stanley-Reisner ring at the irrelevant maximal ideal is equal to the localisation of the non-homogeneous ring at the point. As the Stanley-Reisner ring is graded this localisation is "essentially" the same as the graded Stanley-Reisner ring. This is why the properties of the Stanley-Reisner ring reflect the local properties of the complex, i.e., of the links of vertices as well as the global properties as the link of the apex of the cone of a complex is the complex itself.

If the base field is the real numbers, then the space of real points of the non-homogeneous ring of a complex is homotopy equivalent to the complex itself. Indeed, the real points is the union of the affine span of the simplices of the complex and one may retract each such affine space to the simplex in a way which is compatible with passing to the subsimplices.

Finally, as for the flat deformation an arbitrary ideal gives rise to a monomial ideal, an ideal generated by monomials. The monomial ideals associated to the a simplicial complex are special monomial ideals; they are exactly those monomial ideals generated by square free monomials. Depending on the ordering of the variables the monomial ideal associated to $X^2+Y^2-1$ is $(X^2)$ or $(Y^2)$ so no you do not get the Stanley-Reisner ring of the circle.

Addendum: Note that cones mean somewhat different things in the simplicial and affine world. For a simplicial complex its cone is obtained by picking a new point, the apex, and connecting with all points on the complex by using $1$-simplices, i.e., closed intervals. For the affine picture corresponding to the Stanley-Reisner ring one picks an apex and takes a line for every point on the affine hull of the simplicial complex passing through the point and the apex but being a line continues on the other side of the apex.

Hence starting with a $1$-simplex one draws the interval between each point on the simplex and a point outside of the affine span of the $1$-simplex. In the affine picture, the $1$-simplex is replaced by a line and one then draws all the lines through a point on that line and the apex getting the plane.

The non-homogeneous ring is obtained from the Stanley-Reisner ring by dividing out by the relation that all the variables sum up to $1$.

Answer 2

I think you're asking if there's a direct geometric relationship between an algebraic variety $X=V(I)$ (i.e., the zero set of an ideal $I$) and the Stanley-Reisner complex $\Delta_I$ or $\Delta_{in(I)}$ --- in other words, does the variety look like the simplicial complex? In general, I think the answer can be quite subtle, and is probably best approached using the algebra as an intermediary.. For example, if $\Delta$ is a simplicial sphere, then the number $d$ of its facets will correspond to the degree of $X$, but you will not necessarily be able to ``see'' directly from $X$ how those facets fit together combinatorially in $\Delta$ (e.g., if $d=20$, is $\Delta$ a decagonal bipyramid or an icosahedron?). On the other hand, you can calculate $d$ and similar invariants fairly easily from the ideal $I$ (in this case, compute the Hilbert series as a rational function in $q$, and plug $q=1$ into the numerator).

For "flat deformation", here's the example I always keep in mind (very similar to Torsten's): take the hyperbola $xy=c$ and let $c\to0$. In the limit, the hyperbola degenerates into a pair of lines $x=0$ and $y=0$. This corresponds to replacing the ideal $(xy-c)$ (or $(xy-cz^2)$, if you want to think projectively) with the monomial ideal $(xy)$, which is its initial ideal under the right term ordering on $\Bbbk[x,y,z]$. The general principle is that some invariants (e.g., Hilbert series) don't change when you degenerate, and some (e.g., singularities) can only get \emph{worse}, so if you can prove that the degeneration is, say, Cohen-Macaulay then so was the original thing.

As the operation of replacing, e.g., $X_i^2\in I$ by $X_iX_i'$, it is called ``polarization'' and has similarly mild effects. See, e.g., \S3.2 of Miller and Sturmfels \emph{Combinatorial Commutative Algebra}.