Informal Seminar on Stanley-Reisner Theory, UMN, Fall 2002 21 November 2002

Minimal free resolutions of Stanley-Reisner rings Speaker: Vic Reiner

Scribe notes by Sandra Di Rocco, LaTeX’ed by Jeremy Martin

Outline: I. (Minimal) free resolutions and Tor_{·}^{S}(k[Δ], k)

II. Theorem for k[Δ] III. Proof

For more details, see especially section 5.5 of [1].

Let Δ be the simplicial complex

(1)  [figure: •; 1; •; 2; •; 3; •; 4]

so that

I_{Δ} = (x_{1}x_{2}x_{3}, x_{1}x_{4}, x_{2}x_{4})

and

k[Δ] = \underbrace{k[x_{1}, …, x_{4}]}_{S} /I_{Δ}.

The S-module k[Δ] is ℤ^{n}-graded; for α ∈ ℤ^{n}, the corresponding graded piece is the linear span of the monomial x^{α} = x_{1}^{α_{1}} … x_{n}^{α_{n}}. k[Δ] has the following ℤ^{n}-graded free resolution as an S-module:

(2)

0 → S^{1} \overrightarrow{\begin{aligned}& g \\ \begin{matrix}f_{1} \\ f_{2} \\ f_{3} \end{matrix} & [\begin{matrix}x_{3} \\ 1 \\ −1 \end{matrix}] \end{aligned}} S^{3} \overrightarrow{\begin{matrix}f_{1} & f_{2} & f_{3} \\ \begin{matrix}e_{1} \\ e_{2} \\ e_{3} \end{matrix} & [\begin{matrix}0 & x_{4} & x_{4} \\ x_{2} & −x_{2}x_{3} & 0 \\ −x_{1} & 0 & −x_{1}x_{3} \end{matrix}] \end{matrix}} S^{3} \overrightarrow{\begin{matrix}e_{1} & e_{2} & e_{3} \\ [x_{1}x_{2}x_{3} & x_{1}x_{4} & x_{2}x_{4}] \end{matrix}} S → S/I_{Δ} → 0.

Here the first copy of S^{1} has basis element g of degree x_{1}x_{2}x_{3}x_{4}; the first copy of S^{3} has basis {f_{1}, f_{2}, f_{3}} of degrees x_{1}x_{2}x_{4}, x_{1}x_{2}x_{3}x_{4}, x_{1}x_{2}x_{3}x_{4} respectively; and the second copy of S^{3} has basis {e_{1}, e_{2}, e_{3}} of degrees x_{1}x_{2}x_{3}, x_{1}x_{4}, x_{2}x_{4} respectively.

The resolution (2) is not minimal. In a \underline{minimal} free resolution (henceforth MFR), we want at each stage that the columns give a minimal generating set of the kernel. The following is an MFR of S/I_{Δ}:

(3)  0 → S^{2} \overrightarrow{\begin{matrix}f_{1} & f_{2} \\ \begin{matrix}e_{1} \\ e_{2} \\ e_{3} \end{matrix} & [\begin{matrix}0 & x_{4} \\ x_{2} & −x_{2}x_{3} \\ −x_{1} & 0 \end{matrix}] \end{matrix}} S^{3} \overrightarrow{\begin{matrix}e_{1} & e_{2} & e_{3} \\ [x_{1}x_{2}x_{3} & x_{1}x_{4} & x_{2}x_{4}] \end{matrix}} S → S/I_{Δ} → 0.

Proposition 1. Let R be a graded k-algebra and M a graded R-module. Then a graded R-free resolution

⋯ → R^{β_{1}} → R^{β_{0}} → M → 0

is minimal if and only if the matrices only contain elements of

𝔪 := ⨁_{n>0} R_{n}

where R_{n} denotes the nth graded piece of R. (Note: 𝔪 is also denoted R_{+} and often called the irrelevant ideal of R.)

For instance, the resolution (2) is not minimal because of the 1 and −1 appearing in the leftmost map.

Note that in this MFR, the free module S^{2} has S-basis {f_{1}, f_{2}} of degrees x_{1}x_{2}x_{4}, x_{1}x_{2}x_{3}x_{4}. It can be shown that the degrees of the basis elements in the terms of an MFR are uniquely determined, even though the maps are not unique. Corollary 2. The number of basis elements of R^{β_{i}} of degree x^{α} in any MFR is

dim_{k} Tor_{i}^{S}(M, k)_{x^{α}},

where k = R/𝔪 = R/R_{+}.

Proof. To compute Tor_{i}^{S}(M, k), we first write down an MFR of M as an S-module:

(4)  ⋯ → R^{β_{i}} ⟶_{ϕ} R^{β_{i−1}} → ⋯ → R^{β_{0}} → M → 0.

Here the matrix ϕ has all entries in 𝔪. Therefore, tensoring the complex (4) with k over R produces

(5)  ⋯ → k^{β_{i}} ⟶^{0} R^{β_{i−1}} ⟶^{0} … ⟶^{0} R^{β_{0}} → 0.

The ith homology of this complex is by definition Tor_{i}^{S}(M, k); since all the maps are zero, we have Tor_{i}^{S}(M, k) = k^{β_{i}}. Note that k^{β_{i}} still carries a grading, and the k-basis vectors for k^{β_{i}} = Tor_{i}^{S}(M, k) have the same degrees as the R-basis elements for R^{β_{i}}. □

Let Δ be a simplicial complex on vertices [n]. Recall that the link of a face is defined as

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

For S ⊂ [n], we define

Δ|_{S} := {F ∈ Δ : F ⊂ S}.

The dual complex of Δ is

Δ^{∨} := {F ∈ [n] : [n] − F ∉ Δ}

For instance, if Δ is the simplicial complex of (1), then Δ^{∨} is

(6)  [figure: •; 1; •; 2; •; 3; •; 4]

One way to picture this is as follows. Draw the full Boolean algebra on [4] (I’m not going to bother to put in all the edges):

(7)  1234 123 124 134 234 12 13 14 23 24 34 1  2  3  4 ∅

The boxes indicate faces of Δ. The unboxed faces are the complements of faces in the dual complex.

We will also need the notion of the barycentric subdivision Sd(Δ) of a simplicial complex Δ. Abstractly, this is the simplicial complex whose vertices are the nonempty faces of Δ and whose faces are the flags in Δ, that is, strictly increasing sequences

∅ ≠ F_{0} ⊊ F_{1} ⊊ ⋯ ⊊ F_{r}

where F_{i} ∈ Δ for all i. (For those familiar with posets, the barycentric subdivision of Δ is the same thing as the order complex of the poset of faces of Δ.)

For example, if Δ is the 2-dimensional simplex on vertices {a, b, c}, then Sd(Δ) has six facets, namely the flags

\begin{matrix}\{a\} ⊂ \{a, b\} ⊂ \{a, b, c\}, & \{b\} ⊂ \{a, b\} ⊂ \{a, b, c\}, & \{c\} ⊂ \{a, c\} ⊂ \{a, b, c\}, \\ \{a\} ⊂ \{a, c\} ⊂ \{a, b, c\}, & \{b\} ⊂ \{b, c\} ⊂ \{a, b, c\}, & \{c\} ⊂ \{b, c\} ⊂ \{a, b, c\}. \end{matrix}

[figure: Δ; •; a; •; b; •; c; Sd(Δ); •; a; •; ab; •; b; •; ac; •; bc; •; abc; •; c]

Proposition 3. Each of Δ and Δ^{∨} is homotopy equivalent to the complement of the other in ∂Δ^{n−1} ≅ 𝕊^{n−2}. Consequently, their (co-)homology groups are related by Alexander duality:

H̃_{i}(Δ^{∨}) ≅ H̃^{n−3−i}(Δ^{∨}).

Idea of proof: It is possible to embed both barycentric subdivisions Sd(Δ) and Sd(Δ^{∨}) simultaneously and disjointly inside Sd(∂Δ^{n−1}). For Sd(Δ^{∨}) one must apply the antipodal map on the barycentric subdivison before embedding it in the usual way. (This is unLaTeXable, but try it yourself with the complex Δ of (1).) □

For this reason, the complex Δ^{∨} may be called the canonical Alexander dual of Δ.

We now state and prove the main result. Theorem 4. Let Δ be a simplicial complex on vertices [n], S = k[x_{1}, …, x_{n}], and α ∈ ℕ^{n}. Then

Tor_{i}^{S}(k[Δ], k)_{x^{α}}  ≅ \begin{cases}H̃_{|S|−i−1}(Δ|_{S}; k) & if x^{α} = x^{S} for some S ⊂ [n] \\ 0 & otherwise \end{cases}

≅ \begin{cases}H̃_{i−2}(lk_{Δ^{∨}}(F); k) & if x^{α} = x^{[n]−F} for some F ∈ Δ^{∨} \\ 0 & otherwise. \end{cases}

The first characterization of Tor is due to Hochster [3] and the second to Eagon and Reiner [2]. The equivalence of the two comes from Alexander duality after one checks that (Δ|_{S})^{∨} = lk_{Δ^{∨}}(F) if S = [n] − F.

Proof. By general homological nonsense we have

(8)  Tor_{i}^{S}(k[Δ], k) = Tor_{i}^{S}(S/I_{Δ}, k) ≅ Tor_{i−1}^{S}(I_{Δ}, k) ≅ Tor_{i−1}^{S}(k, I_{Δ}).

We compute the last module via the Koszul resolution of k as an S-module:

(9)  0 → S ⊗^{⋀n} k^{n} → ⋯ → S ⊗^{⋀2} k^{n} → S ⊗^{⋀1} k^{n} → S → k → 0,

where the boundary maps are defined S-linearly by

(10)  e_{i_{1}} ∧ ⋯ ∧ e_{i_{r}} ↦ ∑_{j=1}^{r}(−1)^{j}x_{r}e_{i_{1}} ∧ ⋯ ∧ ê_{i_{1}} ∧ ⋯ ∧ e_{i_{r}}

(the hat denoting removal). Now, we tensor (9) with I_{Δ} over S, obtaining

(11)  … → I_{Δ} ⊗^{⋀r} k^{n} → ^{⋀r−1} k^{n} → …

where the boundary maps are given by

(12)  x^{β} ⊗ e_{i_{1}} ∧ ⋯ ∧ e_{i_{r}} ↦ ∑_{j=1}^{r}(−1)^{j}x_{r}e_{i_{1}} ∧ ⋯ ∧ ê_{i_{1}} ∧ ⋯ ∧ e_{i_{r}}.

Denoting e_{i_{1}} ∧ ⋯ ∧ e_{i_{r}} by e_{G} if G = {i_{1}, …, i_{r}}, we see that a k-basis in degree x^{α} for the complex (11) is

(13)  {x^{β} ⊗ e_{G} : G ⊂ [n], x^{β} ∈ I_{Δ}, x^{β}x^{G} = x^{α}} .

This shows that in degree x^{α}, this complex coincides (up to shift in homological degree by 1) with the usual augmented chain complex for the simplicial complex

(14)  Δ_{α} := \{G ⊂ [n] : \frac{x^{α}}{x^{G}} ∈ I_{Δ}\} = \{G ⊂ [n] : supp (\frac{x^{α}}{x^{G}} ) ∉ Δ\} .

Thus we have

(15)  Tor_{i}^{S}(k[Δ], k)_{x^{α}} ≅ Tor_{i−1}^{S}(k, I_{Δ})_{x^{α}} ≅ H̃_{i−2}(Δ_{α}; k).

Note that if x^{α} is divisible by x_{i}^{2}, then

supp (\frac{x^{α}}{x^{G}} ) = supp (\frac{x^{α}}{x^{G∪\{i\}}} )

so i will be a cone vertex for Δ_{α}. Hence without loss of generality x^{α} = x^{S} for some subset S ⊂ [n]. Let F = [n] − S; then

\frac{x^{S}}{x^{G}} ∈ I_{Δ} ⟺ G ⊂ S and S − G ∉ Δ
⟺ G ∩ F = ∅ and [n] − (G ∪ F) ∉ Δ
⟺ G ∈ lk_{Δ^{∨}}(F).

Therefore Δ_{α} = lk_{Δ^{∨}}(F). □

As an illustration, we can now go back and explain the degrees of the basis elements for the terms in the MFR:

Restricted complex  Homological observations

[figure: Δ|_{1234} =; •; 1; •; 2; •; 3; •; 4]  H̃_{1} ≠ 0

[figure: Δ|_{123} ≅; •; •; •]  H̃_{1} ≠ 0

Δ|_{14} ≅ Δ|_{23} ≅  • •  H̃_{0} ≠ 0

Δ|_{124} ≅  • • •  H̃_{0} ≠ 0

Link  Homological observations

[figure: Δ^{∨} = lk_{Δ^{∨}}(∅) =; •; 1; •; 2; •; 3; •; 4]  H̃_{0} ≠ 0

lk_{Δ^{∨}}(3) ≅  • •  H̃_{0} ≠ 0

lk_{Δ^{∨}}(4), lk_{Δ^{∨}}(23), lk_{Δ^{∨}}(13) = {∅}
H̃_{−1} ≠ 0

References

[1] W. Bruns and J. Herzog. Cohen-Macaulay rings. Cambridge U. Press, 1993

[2] J. Eagon and V. Reiner. Resolutions of Stanley-Reisner rings and Alexander duality. J. of Pure and Applied Algebra 130 (1998), 265–275.

[3] M. Hochster. Cohen-Macaulay rings, combintorics, and simplicial complexes. In B.R. McDonald and R.A. Morris, eds. Ring theory II, Lecture Notes in Pure and Applied Mathematics 26, M. Dekker, 1977, pp. 171–223.