Higher Cohen-Macaulay property of squarefree modules and simplicial posets

Recently, G. Floystad studied"higher Cohen-Macaulay property"of certain finite regular cell complexes. In this paper, we partially extend his results to squarefree modules, toric face rings, and simplicial posets. For example, we show that if (the corresponding cell complex of) a simplicial poset is $l$-Cohen-Macaulay then its codimension one skeleton is $(l+1)$-Cohen-Macaulay.

Theorem 1.2 (Fløystad [8]). Let X be a finite regular cell complex with the intersection property.
(1) 2-Cohen-Macaulay property of X is a topological property of the underlying space.
Motivated by the above result, we study the higher Cohen-Macaulay property (especially, 2-Cohen-Macaulay property) of relatively new notions of Combinatorial Commutative Algebra, such as squarefree modules, toric face rings, and (the face rings of) simplicial posets.
For example, in Theorem 4.5, we show that Theorem 1.2 (2) holds for the corresponding regular cell complex Γ(P ) of a simplicial poset P , while Γ(P ) does not satisfy the intersection property and Theorem 1.2 (1) is no longer true.
The notion of toric face rings, which generalizes both Stanley-Reisner rings and affine semigroup rings, is studied in (for example) [3,5,9]. A toric face ring is supported by a finite regular cell complex satisfying the intersection property. In Theorem 3.3, we show that, under the assumption that a toric face ring R is "conewise normal", the supporting cell complex X of R is 2-Cohen-Macaulay if and only if X (equivalently R) is Cohen-Macaulay and the canonical module ω R of R is generated by its degree 0 part. The corresponding statement does not hold for simplicial posets (at least, certain modification is required).

Higher Cohen-Macaulay property of squarefree modules
Let S = k[x 1 , . . . , x n ] be a polynomial ring, and regard it as a Z n -graded ring. For a = (a 1 , . . . , a n ) ∈ N n , set supp Let Sq S be the full subcategory of gr S consisting of squarefree S-modules.
For a simplicial complex ∆ with the vertex set [n], the Stanley-Reisner ring k[∆] = S/I ∆ is a squarefree S-modules. As shown in [8,15,16,17], the notion of squarefree modules is useful in the study of the Stanley-Reisner rings. We have that Sq S is an abelian category with enough projectives and injectives, and an indecomposable injective is of the form k We say a ∈ N n is squarefree if a i = 0, 1 for all i. We sometimes identify a squarefree vector a ∈ N n with its support supp(a) ⊂ [n]. For example, in this situation, M supp(a) denotes the homogeneous component M a of M ∈ gr S. If M ∈ Sq S, the essential information of M appears in its squarefree part F ⊂[n] M F .
. We see that ω F is l-Cohen-Macaulay for all l.  . This is a squarefree module. In the following sections, we will study a few generalizations of Stanley-Reisner rings. Their canonical modules are defined in a similar way.
As shown in [10,12], we can also define the Alexander duality functor A : Sq S → (Sq S) op as follows: For M ∈ Sq S, A(M) F is the dual k-vector space of M [n]\F , and the multiplication map The next result is just a module version of [8, Theorem 2.5]. However, it has an application to (the face ring of) a simplicial poset.
implies that a is a squarefree vector. Hence, M| W is either the 0 module or a Cohen-Macaulay module of dimension d if and only if β Hence the assertion follows from Lemma 2.4 and easy computation.
. Now the equivalence is clear.
Note that if an S-module N satisfies Serre's condition (S i ) for i ≥ 1 then Ass(N) = Ass(S) = { (0) } and dim N = n. Hence we can prove the equivalence (iii) ⇔ (iv) by the same way as [15,Corollary 3.7].
The equivalence (iv) ⇔ (v) is nothing other than (a special case of) [7, Theorem 3.8], which is a classical result essentially due to Auslander and Bridger.
The following is a special case of the equivalence (i) ⇔ (iii) of Theorem 2.5. However, we remark it here, since this fact will be mentioned repeatedly in the following sections.
. Hence the assertion follows from the equivalence between (i) and (v) of Theorem 2.5.

Toric face rings and 2-Cohen-Macaulay cell complexes
While we discuss toric face rings in this section, we only give "casual" definition/construction of this ring and some related notions. See [11] for precise information. The original construction found in [5] is equivalent to that of [11], but does not mention the regular cell complex supporting a toric face ring.
A toric face ring is constructed from a monoidal complex M supported by a finite regular cell complex X with the intersection property. Here M is a collection {M σ } σ∈X of an affine semigroup M σ ⊂ Z dim σ+1 (i.e., M σ is a finitely generated additive submonoid of Z dim σ+1 ) with ZM σ = Z dim σ+1 and M σ ∩ (−M σ ) = {0}. Of course, we require several conditions on M σ 's (not all finite regular cell complexes with the intersection property can support a monoidal complex). We assume that the boundary complex of the cross section P σ of the polyhedral cone P σ := R ≥0 M σ ⊂ R dim σ+1 can be identified with the subcomplex {τ | τ ⊂ σ} of X (note that P σ is a convex polytope of dimension dim σ). The face C τ of P σ corresponding to τ ⊂ σ is isomorphic to P τ as a polyhedral cone. Moreover, the monoid M τ is isomorphic to C τ ∩ M σ .
Let |M| be the set given by glueing all M σ 's along with X . We can regard M σ ⊂ |M| for all σ ∈ X . For a, b ∈ |M|, we can not define their sum in general, that is, |M| is no longer a monoid. However, if there is some σ ∈ X with a, b ∈ M σ , then we have their sum a + b ∈ M σ ⊂ |M|.
Then the toric face ring R := k[M] of M over k is the vector space a∈|M| k t a with the k-linear multiplication defined by Note that dim R = dim X + 1.
For each σ ∈ X , p σ := ( t a | a ∈ M σ ) is a prime ideal of R with R/p σ ∼ = k[M σ ], where k[M σ ] is the semigroup ring of M σ . Clearly, k[M σ ] can be seen as a subring of R. In R, k[M σ ] for σ ∈ X are "glued" along with X . We say M is cone-wise . , x n ] be the semigroup ring of an affine semigroup M ⊂ N n . An easy example of a toric face ring is the quotient ring A/I of A by a radical Z n -graded ideal I. Let P := R ≥0 M ⊂ R n be the polyhedral cone spanned by M, and L its face lattice. For the ideal I, there is a subset Σ of L such that A/I = a∈C∩Z n C∈Σ k x a .
(Note that C ∈ Σ is a face of P, and it is also a cone in R n .) Clearly, {C} C∈Σ forms a polyhedral fan, that is, C ′ ⊂ C ∈ Σ and C ′ ∈ L imply that C ′ ∈ Σ. The monoidal complex giving A/I is M := { M ∩ C | C ∈ Σ }. Let H ⊂ R n be a hyperplane intersecting P transversely. The cell complex supporting M is given by { rel-int(H ∩ C) | C ∈ Σ }. If A is normal, then A/I is cone-wise normal. If A is the polynomial ring k[x 1 , . . . , x n ], then A/I can be attained as the Stanley-Reisner ring k[∆] of a simplicial complex ∆. In this case, the supporting cell complex is nothing other than ∆.
Clearly, A/I of the above Example has a Z n -grading inherited from that of A. However, a toric face ring does not admit a nice multi-grading in its most general setting, while the decomposition R = a∈|M| k t a plays a similar role to the graded structure.
Known Results. Let M be a cone-wise normal monoidal complex supported by a cell complex X . For R := k[M], we have the following. See [11] for detail.
(1) R is Cohen-Macaulay (resp. Buchsbaum) if and only if X is Cohen-Macaulay (resp. Buchsbaum) in the sense of §1. (2) We can naturally define squarefree modules over R. For example, R itself is squarefree. If R is Buchsbaum, then the canonical module ω R of R is also. A squarefree R-module M has the decomposition M = a∈|M| M a as a k-vector space. Note that |M| has the 0 element. In the sequel, the homogeneous component M 0 plays a role.
. If X is a manifold, then R is Buchsbaum and W X is the orientation sheaf of X with the coefficients in k. Moreover, (ω R ) + ∼ = W X . Proof. For each ∅ = σ ∈ X , take an element a(σ) ∈ M σ ⊂ |M| contained in the interior of the cone R ≥0 M σ in R dim σ+1 . The stalk (M + ) p at any point p ∈ σ is isomorphic to M a(σ) . Since depth R M ≥ 2, we have Γ(X, M + ) ∼ = M 0 by [11, Theorem 6.2 (a)], and the natural map Γ(X, M + ) → (M + ) p corresponds to the map ϕ σ : M 0 ∋ x −→ t a(σ) x ∈ M a(σ) . Hence M + is generated by global sections if and only if the map ϕ σ is surjective for all σ = ∅. Since M is squarefree, the latter condition states that M is generated by M 0 as an R-module. (i) X is 2-Cohen-Macaulay.
(ii) R is Cohen-Macaulay, and the canonical module ω R is generated by its degree 0 part. If dim R ≥ 2, the above conditions are also equivalent to the following. (iii) X is Cohen-Macaulay and the sheaf W X is generated by global sections.
Proof. A toric face rings of dimension 1 (i.e., the case when dim X = 0) is always a Stanley-Reisner ring. Moreover, in this case, X is 2-Cohen-Macaulay unless X consists of a single point. Hence the assertion is easy if dim R = 1.
So it suffices to show the equivalence of (i), (ii), and (iii) under the assumption that dim R ≥ 2. Now depth R ω R ≥ 2. Since W X ∼ = (ω R ) + , the equivalence (ii) ⇔ (iii) follows from Lemma 3.2. It remains to prove the equivalence (i) ⇔ (iii). Let ∆ be the barycentric subdivision of X . Since X satisfies the intersection property, X Remark 3.4. Let X be a finite regular cell complex with the intersection property, and set V := { σ ∈ X | dim σ = 0 } and d := dim X . Fløystad ([8]) constructed the ith enriched cohomology H i (X ; k) (or just H i (X ), since we have fixed the base field k) of X , which is a squarefree module over the polynomial ring S : , ω S ) for all i. Even for general X , it is Cohen-Macaulay if and only if H i (X ) = 0 for all i = d.
From now, we assume that X is Cohen-Macaulay. [8,Theorem 2.4] states that X is l-Cohen-Macaulay if and only if H d (X ) is the (l − 1)-st syzygy module of some S-module. Hence X is 2-Cohen-Macaulay if and only if A(H d (X )) is generated by its degree 0 part. Clearly, this is analogous to Theorem 3.3. However, the relation between A(H d (X )) and ω R is not direct unless X is a simplicial complex.

Simplicial Poset
A finite partially ordered set (poset, for short) P is called simplicial, if it admits the smallest element0, and the interval [0, x] := { y ∈ P | y ≤ x } is isomorphic to a boolean algebra for all x ∈ P . For the simplicity, we denote rank(x) of x ∈ P by ρ(x). If P is simplicial and ρ(x) = m, then [0, x] is isomorphic to the boolean algebra 2 {1,...,m} .
Let ∆ be a finite simplicial complex (with ∅ ∈ ∆). Its face poset (i.e., the set of the faces of ∆ with the order given by inclusion) is a simplicial poset. Any simplicial poset P is the face (cell) poset of a regular cell complex, which we denote by Γ(P ). Unless Γ(P ) is a simplicial complex, P does not satisfy the intersection property. For example, if two d-simplices are glued along their boundaries, then it is not a simplicial complex, but gives a simplicial poset.
From now on, let P be a simplicial poset. For x, y ∈ P , set Clearly, P | W is simplicial again. We simply denote P | V \W by P | −W .
Stanley [13] defined the face ring A P of a simplicial poset P . For the definition, we remark that if [x ∨ y] = ∅ then { z ∈ P | z ≤ x, y } has the largest element, which is denoted by x ∧ y. Let S := k[ t x | x ∈ P ] be the polynomial ring in the variables t x . Consider the ideal of S (if [x ∨ y] = ∅, we interpret that t x∧y · z∈[x∨y] t z = 0), and set A P := S/I P .
For a rank 1 element y i ∈ V , we simply denote t y i by t i . If {x} = [U] for some U ⊂ V with #U ≥ 2, then t x = i∈U t i in A P , and t x is a "dummy". Since I P is a homogeneous ideal under the grading given by deg(t x ) = ρ(x), A P is a graded ring. The algebra A P is generated by degree 1 elements if and only if Γ(P ) is a simplicial complex. In this case, A P coincides with the Stanley-Reisner ring of Γ(P ).
We say P is Cohen-Macaulay (resp. Buchsbaum), if the cell complex Γ(P ) is Cohen-Macaulay (resp. Buchsbaum) in the sense of §1. Duval [6] showed that P is Cohen-Macaulay if and only if A P is a Cohen-Macaulay ring. The same is true for the Buchsbaum property (c.f. [17]). Proof. We can define squarefree modules over A in a natural way, and a squarefree A-module M gives the constructible sheaf M + on Γ(P ) (see [17]). The theory of squarefree modules over A is quite parallel to that over a toric face ring. Hence we can prove the assertion by the same way as Theorem 3.3.
Definition 4.2. We say a simplicial poset P is l-Cohen-Macaulay if P | −W is Cohen-Macaulay and rank(P | −W ) = rank P for all W ⊂ V with #W < l. Clearly, P is l-Cohen-Macaulay if and only if the cell complex Γ(P ) is l-Cohen-Macaulay.
Recall our convention that V = { y ∈ P | ρ(y) = 1 } = {y 1 , . . . , y n }. Clearly, A := A P has a Z n -grading such that deg t i ∈ N n is the ith unit vector. Consider the polynomial ring T := Sym(A 1 ) ∼ = k[t 1 , . . . , t n ]. Then A is a finitely generated Z n -graded T -module, moreover A is a squarefree T -module. If A is Buchsbaum, then ω A is a squarefree T -module.
Let W ⊂ V . If we regard the face ring A P | W of P | W as a module over the polynomial ring T | W = k[t i | i ∈ W ], it coincides with the restriction (A P )| W of A P as a squarefree T -module (see §2). As Remark 2.3, P is l-Cohen-Macaulay if and only if A P is l-Cohen-Macaulay as a squarefree T -module. Proposition 4.3. Let the notation be as above. A simplicial poset P is 2-Cohen-Macaulay if and only if A := A P is Cohen-Macaulay and the canonical module ω A is generated by its degree 0 part as a T -module.
Proof. The first assertion follows from Corollary 2.6. The second follows from the first and Proposition 4.1.
Remark 4.4. Even if a finite simplicial decomposition of Γ(P ) is 2-Cohen-Macaulay, P itself is not so in general. For example, let P be the simplicial poset given by two d-simplices glued along their boundaries. Since the underlying space of Γ(P ) is a ddimensional sphere, its simplicial decompositions are 2-Cohen-Macaulay. However rank(P | −{y} ) < rank P for all y ∈ V , and P is not 2-Cohen-Macaulay.
Consider the induced subposet P i := { x ∈ P | ρ(x) ≤ i } of P . This is a simplicial poset again. Proof. Recall that A P is a squarefree module over T = k[ t 1 , . . . , t n ]. The face ring A P i of P i coincides with the skeleton (A P ) i of A P as a squarefree T -module. Hence the assertion follows from Theorem 2.7.
The next statement immediately follows from Theorem 4.5: Let P be a Cohen-Macaulay simplicial poset with rank P = d. Then the edge graph of P (i.e., the skeleton P 2 ) is (d−1)-connected. However, this also follows from [6,Theorem 4.5].
Remark 4.6 (Buchsbaum* complex). Athanasiadis and Welker ([1]) call a finite simplicial complex ∆ Buchsbaum* (over k), if k[∆] is Buchsbaum and the canonical module ω k[∆] is generated by its degree 0 part. (Their original definition is different, but equivalent to the above one by [1, Proposition 2.8]). Since the Buchsbaum* property is topological, we say a finite regular cell complex X is Buchsbaum*, if some (equivalently, all) finite simplicial decomposition of X is so. By the same argument as the proof of Theorem 3.3, we have the following.
(1) Let R be a cone-wise normal toric face ring. The supporting cell complex of R is Buchsbaum* if and only if R is Buchsbaum and the canonical module ω R is generated by its degree 0 part. (2) Let P be a simplicial poset, and A its face ring. Then Γ(P ) is Buchsbaum* if and only if A is Buchsbaum and ω A is generated by its degree 0 part.