Resolutions of local face modules, functoriality, and vanishing of local h -vectors

We study the local face modules of triangulations of simplices, i


Introduction
In this paper, we study the modules over face rings, introduced by Athanasiadis and Stanley, whose Hilbert functions are the relative local h-vectors of quasi-geometric homology triangulations of simplices, a broad class of formal subdivisions that includes all geometric triangulations and is natural from the point of view of combinatorial commutative algebra.See Section 2.1 for the precise definition and further references.
Fix an infinite field k.Let σ : Γ → 2 V be a quasi-geometric homology triangulation of a simplex, and let E be a face of Γ. Say that a face G ∈ Γ is interior if σ(G) = V , and let I be the ideal in the face ring k[lk Γ (E)] generated by the faces that are interior relative to E, i.e.I = (x F : F ⊔ E is interior).Let d = |V | − |E|, which is the Krull dimension of k[lk Γ (E)], and let θ 1 , . . ., θ d be a special l.s.o.p., as in [14,1].See also Section 2.2, where we recall the definition and construction of special l.s.o.p.s.Note that L(Γ, E) is a finite dimensional graded k-vector space.The local h-vector is its Hilbert function: where The local face module L(Γ, E) depends on the choice of a special l.s.o.p., but ℓ(Γ, E) is an invariant of the triangulation with the symmetry ℓ i = ℓ d−i .See Section 2.1 activity on the combinatorics of local h-vectors and relations to intersection homology [3,10,16,4].Recent advances include a proof that every non-negative integer vector satisfying ℓ 0 = 0 and ℓ i = ℓ d−i is the local h-vector of a quasi-geometric triangulation for E = ∅ [8], and a relative hard Lefschetz theorem that yields unimodality of local h-vectors for regular subdivisions in a more general setting (for regular nonsimplicial polyhedral subdivisions that are not necessarily rational) [9].
Here, we investigate the local face modules L(Γ, E) using methods of combinatorial commutative algebra.In particular, we describe natural combinatorial resolutions of these modules as well as natural maps of k[lk Γ (E)]-modules, L(Γ, E) → L(Γ, E ′ ), for E ⊂ E ′ .Our first theorem gives explicit generators for the kernel of the natural map I → k[lk Γ (E)]/(θ 1 , . . ., θ d ).Moreover, we extend this to an exact sequence of graded k[lk Γ (E)]-modules in which each term is a direct sum of degree-shifted monomial ideals.
Label the vertices of the simplex V = {v 1 , . . ., v n }.For a subset U ⊂ V , let U c := V ∖ U .After relabeling, we may assume that σ(E) c = {v 1 , . . ., v b }.Given S ⊂ {v 1 , . . ., v d }, we define the ideal I S ⊂ k[lk Γ (E)] by Note that I S ′ ⊂ I S for S ′ ⊂ S, and I S depends only on S ∩ {v 1 , . . ., v b }.For instance, I ∅ = I and I S = k[lk Γ (E)] if {v 1 , . . ., v b } ⊂ S. By the definition of a special l.s.o.p. (Definition 2.3), after reordering, we may assume Corollary 1.3.The kernel of the surjection I → L(Γ, E) is the ideal J generated by We also construct maps between local face modules, as follows.For faces and, up to reordering, we have it generates does not, nor does the map ϕ : L(Γ, E) → L(Γ, E ′ ).Moreover, for E ′′ ⊃ E ′ , one readily checks that the maps ϕ ′ : L(Γ, E ′ ) → L(Γ, E ′′ ) and ϕ ′′ : L(Γ, E) → L(Γ, E ′′ ) are independent of all choices and satisfy ϕ ′′ = ϕ ′ • ϕ.Thus one obtains a functor from the poset of faces of Γ that contain E to graded vector spaces, given by E ′ → L(Γ, E ′ ).
We now give two applications of the above theorems.The first is a monotonicity property for local h-vectors.
The inequality in Theorem 1.6 is term by term, i.e. dim L(Γ, E) i ⩾ dim L(Γ, E ′ ) i for all i.The proof is by showing that the map ϕ : L(Γ, E) → L(Γ, E ′ ) given by Theorem 1.4 is surjective.
Our second application of the above theorems is to a decades old problem posed by Stanley, who introduced and studied local h-vectors in the special case where E = ∅ and asked for a characterization of triangulations for which they vanish [14,Problem 4.13].This problem remains open, and is of enduring interest [3,Problem 2.12].The extension to the case where E is not empty is particularly relevant for applications to the monodromy conjecture [7,6,16].In [11], we prove a theorem on the structure of geometric triangulations with vanishing local h-vectors that is tailored to this purpose, and we use it to prove the monodromy conjectures for all singularities that are nondegenerate with respect to a simplicial Newton polyhedron.See Theorems 1.1.1,1.4.5, and 4.1.3in loc.cit..
Here, we apply Theorem 1.2 to prove another theorem on the structure of faces in triangulations with vanishing local h-vectors.Let F ∈ lk Γ (E) be a face such that F ⊔ E is interior.Following terminology from the monodromy conjecture literature (see, e.g.[12]), we say that F is a pyramid with apex w ∈ F if (F ⊔ E) ∖ w is not interior.Let A F := {w ∈ F : F is a pyramid with apex w}, and The elements of V w correspond to the base directions of F , i.e. the facets of 2 V that contain the base of F , when viewed as a pyramid with apex w.We say F is a Upyramid if there is an apex w ∈ A F such that |V w | = 1.In other words, a U -pyramid is a pyramid with a unique base direction, for some choice of apex.
See Remark 3.2 for a short proof in a special case that illustrates the naturality of the U -pyramid condition.The method of proof breaks down when |F i | ⩾ 3. See Example 5.3.
Remark 1.9.The analogous theorem in [11] requires that the triangulation be geometric and that the interior partition satisfies the additional condition σ(F 2 ⊔ E) c = w∈A F V w .But then the hypothesis that |F 1 | ⩽ 2 is dropped entirely.So, even for geometric triangulations, there are cases of Theorem 1.8 that are not necessarily covered by [11,Theorem 4.1.3].It should be interesting to look for a common generalization of these vanishing results, and to pursue further progress on Stanley's problem of characterizing triangulations with vanishing local h-vector more generally.
Algebraic Combinatorics, Vol. 6 #4 (2023) Remark 1.10.To the best of our knowledge, all of the theorems stated in the introduction are new even for regular triangulations.The reader who prefers to do so may safely restrict attention to geometric or even regular triangulations.However, while the structure results for triangulations with vanishing local h-vectors in [5] and [11] rely on special properties of geometric triangulations, the proofs presented here work equally well for quasi-geometric homology triangulations, and we find it natural to work in this level of generality.
We conclude the introduction with an example illustrating the above theorems.
Example 1.11.Let Γ be the triforce triangulation, which figures prominently in [5] and in the adventures of hero protagonist Link in the video game series The Legend of Zelda.
and its ideal of interior faces is subject to the condition that the restrictions (of the corresponding affine linear functions) to the face {a, b, c} are linearly independent.Our resolution of the local face module L(Γ, E) also involves the monomial ideals The resolution given by Theorem 1.2 is then In particular, we have L(Γ, E) ∼ = I/J, where Since θ 1 , θ 2 , and θ 3 restrict to linearly independent functions on {a, b, c}, the elements A special l.s.o.p. is any l.s.o.p. of the form ζ 1 , ζ 2 , where supp(ζ 1 ) ⊂ {a, b}.The ideal of interior faces in this case is I ′ = (x a , x b ), and the resolution given by Theorem 1.2 is Let us now consider Theorem 1.4 in this example.Let θ ′ i denote the restriction of ), e.g. by choosing ζ 2 to be a linear combination of θ ′ 1 and θ ′ 2 in which the coefficient of and the map ϕ in Theorem 1.4 is given as follows.First, we set Then, writing θ 2 = λ c x c + λ a x a + λ v x v , with all three coefficients nonzero, we set Note that there is no subset of {θ 1 , θ 2 , θ 3 } whose restrictions to k[lk Γ (E ′ )] form an l.s.o.p.This explains and motivates our two-step process for constructing the map: first restricting to Star(E ′ ∖ E) and then intersecting with k[lk Γ (E ′ )] to produce the special l.s.o.p. that yields the functorial map ϕ : Let also describe how Theorems 1.6 and 1.8 manifest in this example.For Theorem 1.8, observe that the face F = {a, b} in lk Γ (E ′ ) has an interior partition F = {a} ⊔ {b}.The proof in this case shows that the classes of both x a and x b are nonzero in L(Γ, E ′ ), for any choice of special l.s.o.p.

Preliminaries
We begin by recalling definitions and background results that will be used throughout, following [15, Chapter III] and [3].We work over a field k.In particular, all rings are commutative k-algebras and singular homology is computed with coefficients in k.
2.1.Triangulations of simplices.In this section only, for the purposes of providing context, we allow that the field k may be finite, and the triangulation σ : Γ → 2 V is not necessarily quasi-geometric.
We recall the notion of a homology triangulation, following [2].A d-dimensional simplicial complex Γ with trivial reduced homology is a homology ball of dimension d if there is a subcomplex ∂Γ ⊂ Γ such that • ∂Γ is a homology sphere of dimension d − 1, The interior faces of a homology ball Γ are the faces not contained in ∂Γ.A homology triangulation of the simplex 2 V is a finite simplicial complex Γ and a map σ : Γ → 2 V such that for every non-empty U ⊂ V , • the simplicial complex Γ is the set of interior faces of the homology ball σ −1 (2 U ).Note that the Betti numbers of a simplicial complex, and hence the property of being a homology ball, depend only on the characteristic of the field k.Homology triangulations are a special case of the (strong) formal subdivisions of Eulerian posets considered in [14, Section 7] and [10,Section 3].
The carrier of a face F ∈ Γ is σ(F ).A homology triangulation σ : Γ → 2 V is quasi-geometric if there is no face F ∈ Γ and U ⊂ V such that the dimension of Γ U is strictly smaller than the dimension of F and the carrier of every vertex in F Algebraic Combinatorics, Vol. 6 #4 (2023) is contained in U .A homology triangulation is geometric if it can be realized in R n as the subdivision of a geometric simplex into geometric simplices.Every geometric homology triangulation is quasi-geometric.
The local h-vector, which we have defined in the introduction as the Hilbert function of the local face module, can be expressed in terms of h-vectors of subcomplexes of links of faces in the homology balls Γ U : (1) Note that (1) makes sense even when k is finite or σ : Γ → 2 V is not quasi-geometric, and should be taken as the definition of the local h-vector in this broader context.
Note that the proof of non-negativity for quasi-geometric triangulations, due to Stanley and Athanasiadis, is via the identification with the Hilbert function of the local face module.It suffices to consider the case where k is infinite, since (1) is invariant under field extensions.
2.2.Face rings and special l.s.o.p.s.Here, and for the remainder of the paper, the field k is fixed and infinite, and all triangulations are quasi-geometric homology triangulations.
Given a finite simplicial complex Γ with vertex set V = {v 1 , . . ., v n }, let k[Γ] denote the face ring.In other words, for each subset F ⊂ V , let x F be the corresponding squarefree monomial in the polynomial ring k[x 1 , . . ., x n ], i.e. x F := vi∈F x i .Then the face ring is In particular, each F in Γ may be viewed as a subcomplex, and we write θ| F for the restriction of θ to this subcomplex.
Note that k[Γ] is graded by degree.By definition, a linear system of parameters (l.s.o.p.) for a finitely generated graded This characterization provides flexibility in constructing l.s.o.p.s in which the linear functions have specified support, where the support of θ = a i x i is supp(θ) := {v i : Let σ : Γ → 2 V be a quasi-geometric homology triangulation, and let E ∈ Γ be a face.

Definition 2.3 ([14, 1]). A linear system of parameters
, there is an element θ v of the l.s.o.p. such that supp(θ v ) consists of vertices in lk Γ (E) whose carrier contains v, and such that In other words, after reordering so that σ(E The existence of special l.s.o.p.s is well-known to experts and the proof is similar to Stanley's argument in the case E = ∅.For completeness, we provide a short proof. Proposition 2.4.Suppose k is infinite.Let σ : Γ → 2 V be a quasi-geometric homology triangulation of a simplex, and let E be a face of Γ.Then there is a special l.s.o.p. for k[lk Γ (E)].
Proof.Let V = {v 1 , . . ., v n }.After renumbering, we may assume that σ(E) c = {v 1 , . . ., v b }.Fix d = n − |E|.Note that b ⩽ d.We define subsets S 1 , S 2 , . . ., S d of the vertices in lk Γ (E), as follows.For i ⩽ b, let S i be the set of vertices w such that v i ∈ σ(w).For i > b, let S i be the set of all vertices of lk Γ (E).Because σ is quasi-geometric, for each face F of lk Γ (E), the union of the sets σ(w) ⊂ V , as w ranges over vertices of E ⊔ F , has size at least

A resolution of the local face module
In this section, we prove Theorem 1. Replacing each term in (5) with its corresponding Koszul resolution, gives a complex of complexes ( 6) which may be expanded as the commuting double complex shown in Figure 1.The The double complex obtained by taking the Koszul resolution of (5).
columns of this complex are exact by construction.We claim that the rows are also exact, and prove this using ideas from [14,Theorem 4.6].First, we show that all rows except for the top row are exact.Choose a subset S of {v 1 , . . ., v d }, and consider the piece of the complex indexed by S: When S = ∅, we obtain (4).Observe that the complex ( 7) is multigraded by N m , where m is the number of vertices of lk Γ (E).Explicitly, deg , α m ).Therefore it suffices to show exactness on graded pieces.Fix α = (α 1 , . . ., α m ).By the definition of the face ring, every term of (7) will have 0 in the graded piece corresponding to α unless the set of vertices with α i ̸ = 0 forms a face F , in which case the α-graded part can be identified with the augmented cochain Algebraic Combinatorics, Vol. 6 #4 (2023) complex of a simplex, indexed by all U that contain σ(E) ∪ σ(F ) ∪ S, and hence is exact.
We now recall the proof that the top row of the double complex, (5), is exact.
The proof involves showing that the quotients of ( 4) by (θ d , . . ., θ d−(r−1) ) is exact by induction on r.The case of r = 0 is the exactness of the second row.Now assume that (4) remains exact after quotienting by (θ d , . . ., θ d−(r−1) ).Let C i denote the ith term of (4) tensored with k[lk Γ (E)]/(θ d , . . ., θ d−(r−1) ).By the induction hypothesis, we have an exact sequence ∈ U .Hence, we have an exact sequence where For example, when m > b, v m ∈ σ(E) and B • = 0. Up to signs and a degree shift, we can then identify B • with the complex (4) for Γ| {vm} c quotiented by (θ d , . . ., θ m+1 ).Then B • is exact by the induction hypothesis applied to Γ| {vm} c .By breaking (8) up into two short exact sequences we see that Now that we know the exactness of (6), let Then, by construction, we have an exact sequence of complexes As above, we repeatedly apply the long exact sequence on cohomology to see that A • is exact.We may then identify A • with the exact sequence Let σ : Γ → 2 V be a quasi-geometric homology triangulation of a simplex, and let E be a face of Γ.Let F ∈ lk Γ (E) such that F ⊔ E is interior, and suppose that F = A F is an interior partition of F , i.e. with F 1 = F 2 = ∅.Suppose that F is not a U -pyramid.By Corollary 1.3, J is generated by elements of the form Because F is not a U -pyramid, no monomial appearing in any of these generators divides x F , so x F is nonzero in L(Γ, E).This proves Theorem 1.8 in the special case when

Functorial properties of local face modules
In this section, we prove Theorem 1. Proof.Consider the exact sequence of k-linear maps where the right hand map takes r + i α i x i to i α i x i , for any r ∈ R 1 and α i ∈ k.This restricts to an exact sequence of k-linear maps where the surjectivity of the right-hand map follows from the fact that θ 1 , . . ., θ n is an l.s.o.p.Hence, for 1 ⩽ i ⩽ m, we can write x i = r i + s i , for some r i ∈ R 1 and s i ∈ (θ 1 , . . ., θ n ) 1 .For any R-algebra map ϕ : R[x 1 , . . ., x m ]/(θ 1 , . . ., θ n ) → R/R ∩ (θ 1 , . . ., θ n ), we must have that ϕ(x i ) = r i , so there is a unique such map.On the other hand, the R-algebra homomorphism defined by ϕ(x i ) = r i is well-defined, since if Since ϕ is an isomorphism and factors through R/(R 1 ∩ (θ 1 , . . ., θ n ) 1 ), we conclude that the R-ideal R ∩ (θ 1 , . . ., θ n ) is generated in degree 1 and hence any k-basis for R 1 ∩ (θ , which lifts to the unique homomorphism ϕ in the statement of the theorem.It remains to construct a special l.s.o.p. for k[lk Γ (E ′ )] with the specified properties.
The relation θ 1 θ 2 − θ 2 θ 1 = 0 expands into a relation between the generators of the right-hand side.But the left-hand side is 4-dimensional, a contradiction.
Proof.By considering the degree 1 part of ( 9), as the codimension of σ(E) is 1, we get the following exact sequence.
and the result follows.□ Proof of Theorem 5.2.We must show that L(Γ, E)| F is non-zero in degree |F 1 |.Recall that L(Γ, E)| F is isomorphic to I| F /J| F , where I| F and J| F are described in (10) and ( 11) respectively.First we handle the cases when |F 1 | ⩽ 1.If F 1 = ∅, then E is interior and x ∅ = 1, but J| F is a proper ideal as it is generated by elements of positive degree, so x F1 ̸ ∈ J| F .If F 1 = {v}, then we assume that E is not an interior face.Then J| F is generated by elements of degree at least 2, so x F1 ̸ ∈ J| F .Suppose |F 1 | = 2.We assume that there are no vertices v with {v} ⊔ E interior and E is not interior.If σ(E) has codimension 1, then both F 1 and F 2 must have a vertex v with {v} ⊔ E interior.Then by Lemma 5.6, we see that dim L(Γ, E)| F ⩾ 1. Hence we may assume that σ(E) has codimension at least 2.
Let F 1 = {u, t} and assume that L(Γ, E)| F has no non-zero elements in degree 2. Consider the connected component of the internal edge graph containing F 1 .By Lemma 5.4, we may assume that σ({u} ⊔ E) = {v 1 } c .Note that v 1 ∈ σ(t).There is a vertex t ′ ∈ F 2 such that v 1 ∈ σ(t ′ ), so {u, t ′ } ⊔ E is interior.Therefore either {t} ⊔ E or {t ′ } ⊔ E has carrier codimension 1.
If u ′ = t ′ , then the internal edge graph contains a cycle and hence every vertex w in it (including t) has {w} ⊔ E of carrier codimension 1.As F 2 is interior and {u ′ } ⊔ E has carrier codimension 1, there is a vertex w ∈ F 2 such that {u ′ , w} ⊔ E is interior.But then either {u, w} ⊔ E or {t, w} ⊔ E is interior, contradicting the uniqueness of the cycle in Lemma 5.4.

2 ,
giving an explicit resolution of the local face module L(Γ, E) by a subcomplex of the Koszul resolution of k[lk Γ (E)]/(θ 1 , . . ., θ d ).We continue to use the notation established above.In particular, σ : Γ → 2 V is a quasigeometric homology triangulation of the simplex with vertex set V = {v 1 , . . ., v n }.We consider a face E ∈ Γ with d = n−|E| and b = n−|σ(E)|.After reordering, we assume is exact.Here, for a graded module M and a finite set S, we write M S := M [−|S|].
1 , ..., θ n ) is an l.s.o.p. for R.□ Proof of Theorem 1.4.Note that Star(E ′ ∖ E) is the join of E ′ ∖ E with lk Γ (E ′ ).The face ring k[Star(E ′ ∖E)] is therefore a polynomial ring over k[lk Γ (E ′ )].Its Krull dimension is equal to d = dim k[lk Γ (E)],and hence the restrictions θ ′ 1 , . . ., θ ′ d form an l.s.o.p., where θ ′ i := θ i | Star(E ′ ∖E) .By Lemma 4.1, there is a unique graded k[lk Γ and note that {ζ 1 , . . ., ζ b ′ } is linearly independent.Extending this independent set to a basis produces a special l.s.o.p. for k[lk Γ (12)equired.In particular, both vector spaces in(12)naturally decompose into a direct sum of vector spaces indexed by the connected components of the internal edge graph.Therefore, in each connected component of the internal edge graph, the number of edges e with e ⊔ E interior is less than or equal to the number of vertices v with {v} ⊔ E of carrier codimension 1.As the only connected graphs (V, E) where |E| ⩽ |V | are either trees or contain a unique cycle, the result follows.□ Lemma 5.5.Assume σ(E) has codimension at least 2. Let F ⊂ lk Γ (E) be a face.Assume F has no vertices v with {v} ⊔ E interior.If L(Γ, E)| F is zero in degree 2, then no component of the internal edge graph of F contains a cycle of length 4. Proof.Suppose a component of the internal edge graph contains a 4-cycle of vertices F = {t 1 , t 2 , u 1 , u 2 }.By Lemma 5.4, this is the unique cycle in this component and every vertex w ∈ F has {w} ⊔ E of carrier codimension 1.Because F is a face and there are no 3-cycles in this component of the internal edge graph, we may assume that σ({t i □Proof ofTheorem 1.6.Let E ⊂ E ′ be faces of a quasi-geometric homology triangulation Γ of a simplex, and assume that σ(E) = σ(E ′ ).It is enough to show that the induced map ϕ : L(Γ, E) → L(Γ, E ′ ) given by Theorem 1.4 is surjective.Note that