Informal Seminar on Stanley-Reisner Theory, UMN, Fall 2002 5 December 2002

Local cohomology Speaker: Anton Leykin

Scribe notes by Jeremy Martin

We work with the following objects throughout. Let k be a field and R a finitely generated, ℕ^{n}-graded kalgebra, i.e., R = ⨁_{α∈ℕ^{n}} R_{α} with R_{α}R_{β} ⊂ R_{α+β}. The motivating example is R = k[Δ], the Stanley-Reisner ring of a simplicial complex Δ. Define

R_{+} := ⨁_{α≠0} R_{α}

the so-called irrelevant ideal. Finally, M will be a ℤ^{n}-graded R-module, i.e., M = ⨁_{α∈ℤ^{n}} M_{α} with R_{α}M_{β} ⊂ M_{α+β}.

1. Depth and Cohen-Macaulayness

Definition 1. An element a ∈ R is called a nonzerodivisor (or NZD) on M if m ∈ M, am = 0 implies m = 0. Equivalently, the map

M ⟶^{·a} M

given by multiplication by r is one-to-one. Definition 2. A sequence of homogeneous elements θ_{1}, …, θ_{s} is a regular M-sequence, or M-sequence for short, if θ_{i+1} is a NZD on M/(θ_{1}, …, θ_{i})M for i = 0, …, s − 1. Definition 3. The dimension of M, denoted dim_{R} M or dim M, is the Krull dimension of R/ Ann_{R} M. The depth of M, denoted depth_{R} M or depth M, is the length of a maximal M-sequence. It can be shown that every maximal M-sequence has the same length.

In general depth_{R} M ≤ dim_{R} M (since any M-sequence of length s generates a height-s ideal of R/ Ann M). Equality is an important “niceness” condition which gets its own name: Definition 4. M is Cohen-Macaulay if depth_{R} M = dim_{R} M.

2. Local Cohomology

Define the torsion functor Γ by

Γ(M) := {u ∈ M | R_{+}^{n}u = 0 for n ≫ 0} .

It is routine to check that Γ is a covariant, left-exact functor. That is, a map f : M → N of graded R-modules induces a map Γ(f) : Γ(M) → Γ(N), and if f is injective then Γ(f) is injective.

The ith local cohomology functor H^{i} (more precisely, H_{R_{+}}^{i}) can now be defined as the ith right derived functor of Γ:

H^{i}(M) = R^{i}Γ(M).

That is, one may calculate H^{i}(M) by taking an injective resolution

I^{•} : 0 → M → I^{0} → I^{1} → …,

applying Γ, and defining H^{i}(M) := H^{i}(ΓI^{•}). (See a textbook on homological algebra for more details.) Lemma 5. For all i, the modules H^{i}(M) are R_{+}-torsion, i.e., they are killed by some power of R_{+}.

Proof. By definition of Γ, every module of the form Γ(N) is R_{+}-torsion. In particular, if I^{•} is an injective resolution, then every Γ(I^{i}) is R_{+}-torsion, so the same is true of the cohomology modules of Γ(I^{•}). □

The local cohomology functors are useful because they detect depth and dimension. Specifically, we have the following fact. Theorem 6. Let e = depth_{R} M and d = dim_{R} M. Then:

(1) H^{i}(M) = 0 unless e ≤ i ≤ d.

(2) H^{e}(M) ≠ 0 and H^{d}(M) ≠ 0.

Proof. We’ll prove only (1), which is the case we really need. We proceed by induction on e.

If H^{0}(M) ≠ 0, then every element of R_{+} is a zerodivisor on M, which is exactly the statement that e = 0.

If e = 0, then

R_{+} ⊂ ⋃_{P ∈Ass M} P,

so R_{+} is itself an associated prime. It is immediate that Γ(M) = H^{0}(M) ≠ 0 as desired.

For the inductive step, assume that (1) is true for all R-modules N with depth N < e. Let a ∈ R_{+} be a homogeneous NZD on M and let N = M/aM. Then depth_{R} N = e − 1, so by induction H^{i}(N) = 0 for i < e − 1 and H^{e−1}(N) ≠ 0.

By general homological nonsense, the short exact sequence of R-modules

(1)  0 → M ⟶^{·a} M → N → 0

induces a long exact sequence on cohomology

(2)  … → H^{i−1}(M) → H^{i−1}(N) → H^{i}(M) ⟶^{·a} H^{i}(M) → …

If i < e, then H^{i−1}(N) = 0, so a is a NZD on H^{i}(M). But H^{i}(M) is R_{+}-torsion and so has depth 0. It follows that H^{i}(M) = 0.

On the other hand, if i = e then the first three terms displayed in (2) are

0 → H^{e−1}(N) → H^{e}(M),

and H^{e}(M) ≠ 0 since H^{e−1}(N) ≠ 0. □

The local cohomology functors can be computed using the Čech complex Č^{•}(x_{1}, …, x_{n}; M), which is defined as

(3)  Č^{•}(x_{1}, …, x_{n}; M) := ⨂_{i=1}^{n} ((0 → R → R_{x_{i}} → 0) ⊗ M).

where R_{+} = \sqrt{(x_{1}, …, x_{n})} and R_{x_{i}} = R[x_{i}^{−1}]. The ith Čech module Č^{i}(x_{1}, …, x_{n}; M) may be described explicitly as follows. For F ⊂ [n], define

x_{F} = ∏_{i∈F} x_{i}

and let

R_{F} = R[x_{F}^{−1}].

Then

(4)  Č_{i}^{•}(x_{1}, …, x_{n}; M) = ⨁_{\begin{matrix}F ⊂[n] \\ |F |=i \end{matrix}} M_{F}

where M_{F} = M ⊗ R_{F} and the maps between adjacent terms in the Čech complex are given by the usual Koszul maps (just like simplicial cohomology.)

3. Hochster’s Theorem

Let Δ be a simplicial complex on vertices x_{1}, …, x_{n}, and R = k[Δ] = k[x_{1}, …, x_{n}]/I_{Δ} its Stanley-Reisner ring. With respect to the obvious ℕ^{n}-grading, we have R_{+} = (x_{1}, …, x_{n}). Definition 7. Let F ∈ Δ. The star of F with respect to Δ is

st_{Δ} F := {G ∈ Δ | G ∪ F ∈ Δ}

and the link of F with respect to Δ is

lk_{Δ} F := {G ∈ Δ | G ∪ F ∈ Δ, G ∩ F = ∅}.

We suppress the subscript when possible.

Note that both st F and lk F are simplicial complexes, and that st F = ⟨F ⟩ ∗ lk F. For instance, if Δ = ⟨123, 14, 24⟩ and F = 12, then lk F = ⟨3⟩ and st F = ⟨123⟩.

Let q_{1}, …, q_{n} be indeterminates. Denote by Hilb(M; q) the finely graded Hilbert series of M, i.e.,

Hilb(M; q) := ∑_{α∈ℤ^{n}} q^{α} dim_{k} M_{α},

where q^{α} = q_{1}^{α_{1}} … q_{n}^{α_{n}}. Also, let H̃_{i}(Δ; k) denote the ith reduced simplicial homology of a simplicial complex Δ with coefficients in k.

For α ∈ ℤ^{n}, define

F (α) = {x_{i} | α_{i} < 0},
G(α) = {x_{i} | α_{i} > 0},
supp(α) = F (α) ∪ G(α) = {x_{i} | α_{i} ≠ 0}.

Theorem 8 (Hochster). We have

Hilb(H^{i}(k[Δ]); q) = ∑_{F ∈Δ} dim_{k} H̃_{i−|F |−1}(lk_{Δ} F; k) ∏_{x_{i}∈F} \frac{q_{i}^{−1}}{1 − q_{i}^{−1}} .

Proof. We compute H^{i}(R) explicitly as the ith cohomology of the Čech complex Č^{•} = Č^{•}(x_{1}, …, x_{n}; M). If F ∉ Δ, then the ring R_{F} is zero, because x_{F} = 0 in R. On the other hand, if F ∈ Δ, then the variables in F become units in R_{F}, and those not in st_{Δ} F get killed (since they annihilate the unit x_{F}). That is,

R_{F} = k [{x_{i}, x_{i}^{−1} : i ∈ F } ∪ {x_{j} : x_{j} ∈ lk F }] ⊗ R.

Let α ∈ ℤ^{n}. We will compute the αth graded piece Č_{α}^{•} of the Čech complex. If supp(α) ∉ Δ, then Č_{α}^{•} = 0, because R_{α} = 0 and adjoining inverses doesn’t change this. So suppose that supp(α) ∈ Δ. Let F = F (α), j = |F |, and G = G(α). A priori, we have

(5)  Č_{α}^{r} = [ ⨁_{|F′|=r} R_{F′} ]_{α} = ⨁_{|F′|=r}[R_{F′}]_{α}.

A whole bunch of these summands are zero. Specifically, for R_{F′} to be nonzero, we must have F′ ∈ Δ (as previously noted), F′ ⊃ F (since the variables in F must be units in the αth graded piece of the Čech complex), and F′ ∪ G ∈ Δ (so that x^{α} itself is nonzero). This is all equivalent to the condition that F′′ = F′ \ F belong to lk_{st G} F, so we may write

(6)  Č_{α}^{r} = [ ⨁_{\begin{matrix}F′′∈lk_{st G} F \\ |F′′|=r−j \end{matrix}} R_{F ∪F′′} ]_{α} .

The maps in Č_{α}^{r} correspond to the usual coboundary maps of the simplicial cochain complex of lk_{st G} F, shifted by j + 1. That is,

(7)  [H^{i}(R)]_{α} = H̃^{i−j−1}(lk_{st G} F; k)
(8)  ≅ H̃_{i−j−1}(lk_{st G} F; k)

since this is a finite-dimensional k-vector space, hence isomorphic to its dual. (The isomorphism is not canonical, but we don’t care because we’re really only interested in its dimension.)

If G ≠ ∅, then lk_{st G} F is a cone over G. In particular it is contractible, so H̃_{•}(lk_{st G} F; k) = 0. Therefore we only have nonzero terms when G = ∅, so lk_{st G} F = lk_{Δ} F and the last equation becomes

(9)  [H^{i}(R)]_{α} ≅ H̃_{i−j−1}(lk F; k).

Therefore

(10)  Hilb(H^{i}(k[Δ]); q) = ∑_{F ∈Δ} ∑_{\begin{matrix}α: \\ supp(α)=F (α)=F \end{matrix}} dim_{k} H̃_{i−|F |−1}(lk F; k)q^{α}

(11)  = ∑_{F ∈Δ} dim_{k} H̃_{i−|F |−1}(lk F; k) ∑_{\begin{matrix}α: \\ supp(α)=F (α)=F \end{matrix}} q^{α}

(12)  = ∑_{F ∈Δ} dim_{k} H̃_{i−|F |−1}(lk F; k) ∏_{x_{i}∈F} \frac{q_{i}^{−1}}{1 − q_{i}^{−1}}

as desired. □

Here’s an example (from Stanley) of computing a Čech complex. Let Δ be the complex ⟨12, 13, 23, 4⟩ =

(13)  [figure: •; 1; •; 2; •; 3; •; 4]

and R = k[Δ] = k[x_{1}, x_{2}, x_{3}, x_{4}]/(x_{1}x_{4}, x_{2}x_{4}, x_{3}x_{4}, x_{1}x_{2}x_{3}). Then the Čech complex is

0 → \underbrace{R}_{Č^{0}} → \underbrace{R_{1} ⊕ R_{2} ⊕ R_{2} ⊕ R_{3} ⊕ R_{4}}_{Č^{1}} → \underbrace{R_{12} ⊕ R_{13} ⊕ R_{23}}_{Č^{2}} → 0,

where R_{1} = R[x_{1}^{−1}], R_{12} = R[x_{1}^{−1}, x_{2}^{−1}], etc.

• For α = (0, 0, 0, 0), so F (α) = G(α) = ∅, we have

lk_{st G} F = lk_{Δ} ∅ = Δ,

so [H^{i}(R)]_{α} = H̃_{i−1}(Δ; k).

• For α = (−2, 3, 0, 0), we have

F = {x_{1}},  G = {x_{2}},  lk_{st G} F = ⟨x_{2}⟩ (i.e., a point)

so [H^{i}(R)]_{α} = H̃_{i−2}(point) = 0.

4. Reisner’s Theorem

Let Δ be a simplicial complex and R = −k[Delta]. Let d = dim R = 1 + dim Δ. We will say that Δ satisfies Reisner’s criterion if for all F ∈ Δ, and i < dim(lk F). we have

H̃_{i}(lk F; k) = 0.

Theorem 9. Δ is Cohen-Macaulay if and only if it satisfies Reisner’s criterion.

Remark: Δ is Gorenstein (a stronger condition than Cohen-Macaulayness) if in addition H̃_{i}(lk F; k) ≅ k for i = dim(lk F).

Proof. First, we show that a Cohen-Macaulay complex is pure (i.e., all maximal faces have the same dimension). Indeed, if Δ is Cohen-Macaulay of dimension d − 1 and dim F < d − 1, then H̃_{−1}(lk F) = 0 by Hochster’s theorem, so lk F ≠ 0 and F is not maximal. (This can also be shown without Hochster’s theorem; see Bruns and Herzog, p. 210.)

Next, we show that a complex Δ satisfying Reisner’s criterion is pure. If dim F = 0 then there is nothing to show. Otherwise, we induct on dimension. Reisner’s criterion gives H̃_{0}(Δ) = H̃_{0}(lk ∅) = 0, so Δ is connected. Moreover, for every vertex v, the subcomplex lk{v} of Δ satisfies Reisner’s criterion and has dimension less than that of Δ, so it is pure by induction. Now, for any maximal face F, let v, w ∈ F; we have

dim lk{v} = |F − v| − 1 = |F − w| − 1 = dim lk{w};

by connectedness all links of vertices must have the same dimension, and the same equation implies that Δ is pure.

By these two observations together, we may assume that Δ is pure, so |F | = d for all maximal faces and dim(lk F) = d − |F | − 1 for all faces. So Cohen-Macaulayness

F is Cohen-Macaulay ⟺ k[Δ] is Cohen-Macaulay
⟺ H̃_{j−|F |−1}(lk F; k) = 0 for j < d, F ∈ Δ

which is exactly Reisner’s criterion (set j = i + |F | + 1). □

By the way, a pure connected simplicial complex Δ certainly need not be Cohen-Macaulay (unless dim Δ ≤ 1). The “minimal” example is the complex ⟨123, 345⟩ =

[figure: •; 1; •; 2; •; 3; •; 4; •; 5]

which is not Cohen-Macaulay because lk(3) = ⟨12, 45⟩ is disconnected, so has H̃_{0} ≠ 0. (Also, the h-vector of this complex can be computed as (1, 2, −1). In a Cohen-Macaulay complex, every entry of the h-vector is nonnegative.)