Informal Seminar on Stanley-Reisner Theory, UMN, Fall 2002 17 October 2002

Introduction and motivation for Stanley-Reisner rings, I Speaker: Vic Reiner

Scribe notes by Jeremy Martin

1. Definitions

Definition 1. Let V be a finite set of vertices. An abstract simplicial complex Δ on V is a subset of the power set 2^{V} which is closed under inclusion, that is,

F ∈ Δ, G ⊂ F ⟹ G ∈ Δ.

The elements of Δ are called faces. The dimension of a face, dim F, is defined as |F | − 1.

We can often represent Δ pictorially. For instance, if V = {a, b, c, d} and Δ is the abstract simplicial complex

(1)  Δ = {∅, a, b, c, d, ab, bc, cd}

(abbreviating the face {a} by a, {a, b} by ab, etc.), then the corresponding figure is

• a • b  • c • d

One can think of a, b, c, d as orthonormal basis vectors in |V |-space, so that the face ab (which has dimension 1) represents the affine span of the vectors a, b (which is a line segment), etc.

Fix a field k, and let

S = k[x_{v} : v ∈ V],

the (commutative) polynomial ring in variables corresponding to the vertices. Definition 2. The Stanley-Reisner ideal of Δ is

I_{Δ} := (∏_{j=1}^{r} x_{v_{i_{j}}} : {v_{i_{1}}, …, v_{i_{r}}} ∉ Δ) .

Note that I_{Δ} is a monomial ideal (that is, it is generated by monomials) and that the generators are squarefree (they are not divisible by the square of any variable). A minimal set of generators is given by the minimal nonfaces of Δ. Definition 3. The Stanley-Reisner ring (or face ring) of Δ is

k[Δ] := S/I_{Δ}.

Note that the set of monomials

{x_{v_{1}}^{e_{1}} … x_{v_{r}}^{e_{r}} : {v_{1}, …, v_{r}} ∈ Δ, e_{1}, …, e_{r} > 0}

is a basis for k[Δ] as a k-vector space. In particular, k[Δ] is a graded ring.

For example, if Δ is the simplicial complex given in (1), then

I_{Δ} = (ac, ad, bd)

(the minimal nonfaces of Δ) and k[Δ] is the k-linear span of

(2)  \{ \begin{matrix}1, a, a^{2}, a^{3}, …, & ab, a^{2}b, ab^{2}, …, \\ 1, b, b^{2}, b^{3}, …, & bc, b^{2}c, bc^{2}, …, \\ 1, c, c^{2}, c^{3}, …, & cd, c^{2}d, cd^{2}, …, \\ 1, d, d^{2}, d^{3}, … \end{matrix} \} .

This construction actually gives a bijection between simplicial complexes on V and ideals of S generated by squarefree monomials. The simplicial complex corresponding to such an ideal is its Stanley-Reisner complex.

2. Motivations

1. In algebraic geometry, one wants to study rings of the form R = S/I, where S is a polynomial ring over a field k and I is an ideal of S. That is, R is the coordinate ring of the affine algebraic variety defined by I. To study R using Stanley-Reisner rings, we may proceed as follows:

First, “deform” I as follows. Fix some monomial order < on S and compute the initial ideal in_{<}(I) ⊂ S (this is equivalent to computing a Gröbner basis of I). By definition, in_{<}(I) is generated by monomials, However, the generators need not be squarefree, so a second step, polarization, may be required. The idea of this step is to get rid of high powers of variables by the following trick: if one of the generators of in_{<}(I) is, say, x^{2}, then we adjoin a new variable x′, replace x^{2} with xx′, and mod out by x − x′. The result is a squarefree monomial ideal of some polynomial ring S′ ⊃ S, which we may regard as the Stanley-Reisner ideal I_{Δ} of a simplicial complex Δ on the variables of S′. The ideal in_{<}(I) and its polarization are very closely related, so we don’t have to worry too much about this second step.

The passage from I to in_{<}(I) does not preserve all structure, but it is pretty good (in the language of algebraic geometry, it is a flat deformation). Lots of geometric/ring invariants of R are closely related— often equal—to those of k[Δ] = S′/I_{Δ}. For instance, the dimension, degree and Hilbert series of R are the same as for k[Δ], and these can be computed combinatorially from Δ. In addition, some homological-type properties, such as Cohen-Macaulayness, can only get worse–e.g., if k[Δ] is Cohen-Macaulay then so is R. (For an example, see the scribe’s Ph.D. thesis.)

Here’s an elementary example. Let S = k[a, b, c, d], I = (ac − b^{2}, bd − c^{2}, ad − bc), and R = S/I. (In fact, Proj(R) is the twisted cubic, the image of the degree-3 Veronese embedding ℙ^{1} → ℙ^{3} mapping [s : t] to [s^{3} : s^{2}t : st^{2} : t^{3}] in homogeneous coordinates.) The given generators of I form a Gröbner basis with respect to lexicographic order on the monomials of S, with a < b < c < d, so the initial ideal is

in_{<}(I) = (ac, bd, ad),

the Stanley-Reisner ideal of the simplicial complex Δ given in (1). Geometrically, the twisted cubic is being “flattened out”:

[figure: ⇝; •; •]

We compute the dimension, degree and Hilbert series of the twisted cubic using k[Δ].

Fact: dim k[Δ] = 1 + dim Δ = 1 + max{dim F : F ∈ Δ}.

In this case, the largest faces have dimension 1, so dim R = 2. It seems as though R should have dimension 1, but it is really the affine coordinate ring of the cone over the twisted cubic, so dim R = 2 makes sense.

Fact: deg k[Δ] = number of facets (maximal faces) of Δ.

Here, that number is 3.

Now for the Hilbert series. By definition, this is

Hilb(k[Δ], t) := ∑_{m≥0} (dim_{k}(k[Δ])_{m}) t^{m}

where (k[Δ])_{m} denotes the mth graded piece of k[Δ] (that is, the k-linear span of the monomials of degree m. We have already written down a monomial basis (2) for k[Δ]. The minimal elements of the basis correspond to faces of Δ, and we have

Hilb(k[Δ], t) = 1 + 4 (\frac{t}{1 − t}) + 3 (\frac{t^{2}}{(1 − t)^{2}} )
= \frac{1 + 2t}{(1 − t)^{2}} .

The dimension and degree can be extracted from the Hilbert series: the dimension is the order of the pole at t = 1 (here 2), and the degree is the sum of the coefficients in the numerator (here 3). The Hilbert series has the following combinatorial interpretation in terms of Δ. Definition 4. The f-vector of Δ is

f(Δ) = (f_{−1}, f_{0}, f_{1}, …, f_{dim Δ}) ,

where f_{i} is the number of i-dimensional faces. (So f_{−1} = 1, since ∅ is the unique face of dimension −1, and f_{0} is the number of vertices.) Definition 5. The h-vector of Δ is

h(Δ) = (h_{0}, h_{1}, …, h_{dim Δ+1}) ,

where h_{i} is defined as follows. You can use the formula

∑_{i} h_{i}t^{i} = ∑_{i=0}^{dim Δ+1} f_{i−1}t^{i}(1 − t)^{dim Δ−i+1}

(Bruns and Herzog, Lemma 5.1.8).

A more fun way to compute the h-vector is as follows. First, draw a triangle with the coefficients of the f-vector down the rightmost diagonal, and 1’s down the leftmost diagonal:

f_{−1}
1  f_{0}
1  ■  f_{1}
1  ■  ■  f_{2}
⋮

Then, fill in the ■’s down to the row below f_{dim Δ} as though you were constructing Pascal’s triangle—but subtracting instead of adding. That is, replace each ■ with the number to the northeast minus the number to the northwest. The bottom row will be the h-vector.

For example, for the complex Δ we have been working with, we start with

1

1  4
1  ■  3
1  ■  ■

Filling in the boxes, we get

1

1  4
1  3  3
1  2  0

The numbers below the line form the h-vector, in this case (1, 2, 0). The trailing 0’s are frequently dropped, so we would write h(Δ) = (1, 2).

Now back to the Hilbert series. The connection is the following:

Hilb(k[Δ], t) := ∑_{m≥0} (dim_{k}(k[Δ])_{m}) t^{m}

= ∑_{i≥0} f_{i}(Δ) (\frac{t}{1 − t})^{i+1}

= \frac{∑_{i=0}^{dim Δ+1} h_{i}t^{i}}{(1 − t)^{dim Δ+1}} .