MULTIPLIER IDEALS OF SUMS VIA CELLULAR RESOLUTIONS

SHIN-YAO JOW AND EZRA MILLER

Abstract. Fix nonzero ideal sheaves 𝔞_{1}, …, 𝔞_{r} and 𝔟 on a normal ℚ-Gorenstein complex variety X. For any positive real numbers α and β, we construct a resolution of the multiplier ideal J ((𝔞_{1} + ⋯ + 𝔞_{r})^{α}𝔟^{β}) by sheaves that are direct sums of multiplier ideals J (𝔞_{1}^{λ_{1}} ⋯ 𝔞_{r}^{λ_{r}}𝔟^{β}) for various λ ∈ ℝ_{≥0}^{r} satisfying ∑_{i=1}^{r} λ_{i} = α. The resolution is cellular, in the sense that its boundary maps are encoded by the algebraic chain complex of a regular CW-complex. The CW-complex is naturally expressed as a triangulation Δ of the simplex of nonnegative real vectors λ ∈ ℝ^{r} with ∑_{i=1}^{r} λ_{i} = α. The acyclicity of our resolution reduces to that of a cellular free resolution, supported on Δ, of a related monomial ideal. Our resolution implies the multiplier ideal sum formula

J (X, (𝔞_{1} + ⋯ + 𝔞_{r})^{α}𝔟^{β}) = ∑_{λ_{1}+···+λ_{r}=α} J (X, 𝔞_{1}^{λ_{1}} ⋯ 𝔞_{r}^{λ_{r}}𝔟^{β}),

generalizing Takagi’s formula for two summands [Tak05], and recovering Howald’s multiplier ideal formula for monomial ideals [How01] as a special case. Our resolution also yields a new exactness proof for the Skoda complex [Laz04, Section 9.6.C].

Introduction

Let X be a smooth complex algebraic variety and let 𝔞 ⊆ O_{X} be an ideal sheaf. Applications in algebraic geometry of the multiplier ideal sheaves

J (𝔞^{α}) = J (X, 𝔞^{α}) ⊆ O_{X}

for real numbers α > 0 have led to investigations of their behavior with respect to natural algebraic operations. For example, Demailly, Ein, and Lazarsfeld [DEL00] proved that given two ideal sheaves 𝔞_{1} and 𝔞_{2}, one has

J ((𝔞_{1}𝔞_{2})^{α}) ⊆ J (𝔞_{1}^{α})J (𝔞_{2}^{α}).

For the subtler case of sums, on the other hand, Mustaţǎ [Mus02] showed that

J ((𝔞_{1} + 𝔞_{2})^{α}) ⊆ ∑_{0≤t≤α} J (𝔞_{1}^{α−t})J (𝔞_{2}^{t}),

and Takagi [Tak05] later refined this to

(1)  J ((𝔞_{1} + 𝔞_{2})^{α}) = ∑_{0≤t≤α} J (𝔞_{1}^{α−t}𝔞_{2}^{t}),

Date: 4 September 2007. Key words and phrases. Multiplier ideal, summation formula, cellular resolution, monomial ideal,

homology-manifold-with-boundary.

where he proved it more generally when X is normal and ℚ-Gorenstein.

Takagi used characteristic p methods to deduce (1), which makes his work distinctly different from [DEL00] and [Mus02], where the arguments proceed by geometric techniques such as log resolutions and sheaf cohomology. Our purpose is to show that such geometric techniques, combined with combinatorial methods from topology and commutative algebra, can recover Takagi’s equality (1) and generalize it. As a consequence, we derive a new proof (Corollary 3) of Howald’s formula for multiplier ideals of monomial ideals [How01], and demonstrate how it can be reformulated to hold for all ideals. In addition, we obtain a new “cellular” exactness proof (Corollary 4.4) for the Skoda complex [Laz04, Section 9.6.C] via the contractibility of simplices. Our main results center around the following, which constitutes part of Theorem 3.6.

Theorem 1. Fix nonzero ideal sheaves 𝔞_{1}, …, 𝔞_{r}, 𝔟 on a normal ℚ-Gorenstein complex variety X and α, β > 0. There is a resolution 0 → J_{r−1} → ⋯ → J_{0} → 0 of the multiplier ideal J ((𝔞_{1} + ⋯ + 𝔞_{r})^{α}𝔟^{β}) by sheaves J_{i} that are finite direct sums of multiplier ideals of the form J (𝔞_{1}^{λ_{1}} ⋯ 𝔞_{r}^{λ_{r}}𝔟^{β}) for various nonnegative λ ∈ ℝ^{r} with ∑_{i=1}^{r} λ_{i} = α. Every distinct ideal sheaf of that form appears as a summand of J_{0}.

Part of the final claim of Theorem 1 is that there are only finitely many distinct multiplier ideals of the form J (X, 𝔞_{1}^{λ_{1}} ⋯ 𝔞_{r}^{λ_{r}}𝔟^{β}) for λ_{1} + ⋯ + λ_{r} = α. (This fact alone is not very hard to see from the definition of multiplier ideals.) In particular, the surjection J_{0} ↠ J (X, (𝔞_{1} + ⋯ + 𝔞_{r})^{α}𝔟^{β}) in our resolution implies the following (finite) summation formula; see also Section 4 for refinements.

Corollary 2. J (X, (𝔞_{1} + ⋯ + 𝔞_{r})^{α}𝔟^{β}) = ∑_{λ_{1}+···+λ_{r}=α} J (X, 𝔞_{1}^{λ_{1}} ⋯ 𝔞_{r}^{λ_{r}}𝔟^{β}).

Corollary 2 reduces the calculation of the multiplier ideals of arbitrary polynomial ideals to those of principal ideals. In the special case of a monomial ideal 𝔞 = ⟨x^{γ_{1}}, …, x^{γ_{r}}⟩, generated by the monomials in the polynomial ring ℂ[x_{1}, …, x_{d}] with exponent vectors γ_{1}, …, γ_{r} ∈ ℕ^{d}, the summation formula becomes particularly explicit. For a subset Γ ⊆ ℝ^{d}, let conv Γ denote its convex hull. By the integer part of a vector ν = (ν_{1}, …, ν_{d}) ∈ ℝ^{d}, we mean the vector (⌊ν_{1}⌋, …, ⌊ν_{d}⌋) ∈ ℤ^{d} whose entries are the greatest integers less than or equal to the coordinates of ν.

Corollary 3. If 𝔞 = ⟨x^{γ_{1}}, …, x^{γ_{r}}⟩ is a monomial ideal in ℂ[x_{1}, …, x_{d}], then J (𝔞^{α}) is generated by the monomials in ℂ[x_{1}, …, x_{d}] whose exponent vectors are the integer parts of the vectors in conv{α · γ_{1}, …, α · γ_{r}} ⊆ ℝ^{d}.

Proof. Using Corollary 2 with 𝔞_{j} = ⟨x^{γ_{j}}⟩, it suffices to note that the divisor of a single monomial has simple normal crossings, so no log resolution is necessary. □

It is easy to check that for α = 1, the vectors in the conclusion of Corollary 3 are precisely those from Howald’s result [How01], namely the vectors γ ∈ ℕ^{d} such that γ + (1, …, 1) lies in the interior of the convex hull of all exponents of monomials in 𝔞.

Our approach to Theorem 1 is to construct a specific resolution satisfying the hypotheses, including the part about J_{0}. The resolution we construct is cellular, in a sense generalizing the manner in which resolutions of monomial ideals can be cellular [BS98]; see [MS05, Chapter 4] for an introduction. In general, a complex in any abelian category could be called cellular if each homological degree is a direct sum indexed by the faces of a CW-complex, and the boundary maps are determined in a natural way from those of the CW-complex. An elementary way to phrase this in the present context is as follows. (Theorem 3.6 is more precise: a specific triangulation Δ is constructed in Section 2, and the sheaves J_{σ} are specified in Remark 3.3.)

Theorem 4. Resume the notation from Theorem 1. There is a triangulation Δ of the simplex {λ ∈ ℝ^{r} | ∑_{i=1}^{r} λ_{i} = α and λ ≥ 0} such that

J_{i} = ⨁_{σ∈Δ_{i}} J_{σ}

can be taken to be a direct sum indexed by the set Δ_{i} of i-dimensional faces σ ∈ Δ, and the differential of J. is induced by natural maps between ideal sheaves, using the signs from the boundary maps of Δ. If λ ∈ Δ_{0} is a vertex, then J_{λ} = J (X, 𝔞_{1}^{λ_{1}} ⋯ 𝔞_{r}^{λ_{r}}𝔟^{β}).

Comparing the final sentences of Theorems 1 and 4, a key point is that every possible multiplier ideal of the form J (X, 𝔞_{1}^{λ_{1}} ⋯ 𝔞_{r}^{λ_{r}}𝔟^{β}) occurs at some vertex λ ∈ Δ_{0}. Writing down what it means for two such multiplier ideals to coincide, this stipulation provides strong hints as to potential choices for triangulations; see Section 2.

The proof of exactness for the cellular resolution in Theorem 4 proceeds by lifting the problem to an appropriate log resolution X′ → X; see the proof of Theorem 3.6. Over X′, we resolve the lifted ideal sheaf by a complex (of locally principal ideal sheaves in O_{X′}) that is, analytically locally at each point of X′, a cellular free complex over a polynomial ring. This cellular free complex turns out to be a cellular free resolution of an appropriate monomial ideal; its construction and proof of acyclicity occupy Section 2, particularly Proposition 2.2. Having cellularly resolved the lifted sheaf over X′, the desired cellular resolution over O_{X} is obtained by pushing forward to X and using local vanishing; again, see the proof of Theorem 3.6. Thus Theorems 1 and 4 constitute a certain global version of cellular free resolutions of monomial ideals.

The acyclicity of the cellular free resolutions in Proposition 2.2 reduce to a simplicial homology vanishing statement, Corollary 1.7, for simplicial complexes obtained by deleting boundary faces from certain contractible manifolds-with-boundary. We deduce Corollary 1.7 from the following more general statement, which is of independent interest. Its rimmed hypothesis is satisfied by the barycentric subdivision of any polyhedral homology-manifold-with-boundary. In what follows, to delete a simplex σ from a simplicial complex M means to remove from M every simplex containing σ.

Proposition 5. Fix a simplicial complex M whose geometric realization |M| is a homology-manifold with boundary ∂M. Assume that M is rimmed, meaning that

(2)  if σ is a face of M, then σ ∩ |∂M| is a face of M.

Then deleting any collection of boundary simplices from M results in a simplicial subcomplex whose (reduced) homology is canonically isomorphic to that of M.

Acknowledgements. We are grateful to Rob Lazarsfeld for valuable discussions, and to Shunsuke Takagi for the suggestion to work with singular varieties. EM was supported by NSF CAREER grant DMS-0449102 and a University of Minnesota McKnight Land-Grant Professorship. The latter funded his visit to the University of Michigan, which he wishes to thank for its hospitality while this work was completed.

1. Combinatorial topological preliminaries

In this section we collect some elementary results from simplicial topology. For additional background, see [Mun84], especially §63 for homology-manifolds, and [Zie95, Section 5.1] for polyhedral complexes. The reader interested solely in the construction of resolutions for Theorems 1 and 4, as opposed to their acyclicity proofs, can proceed to Section 2 after Example 1.1. The goal for the remainder of this section is the proof of Proposition 5 and its immediate consequence, Corollary 1.7.

Our convention is to identify any abstract simplicial complex Δ with the underlying topological space of any geometric realization |Δ|. Thus, fixing a vertex set V, we view Δ as a collection of simplices (finite subsets of V), called faces of Δ, such that any subset of any face of Δ is a face of Δ. For example, we can express the link in Δ of a face σ as the subcomplex

link_{Δ}(σ) = {τ ∈ Δ | τ ∪ σ ∈ Δ and τ ∩ σ = ∅}.

In what follows, all links are taken inside of the ambient simplicial complex, unless otherwise noted. Our motivation is the following class of simplicial complexes.

Example 1.1. Let P be a polyhedral complex, such as a polyhedral subdivision of a polytope. (This example works just as well for any regular cell complex; see [Bjö84]). The barycentric subdivision of P is the simplicial complex Δ whose vertex set is the set of faces of P, and whose simplices are the chains P_{1} < ⋯ < P_{ℓ} of faces P_{j} ∈ P. Here P < Q means that P is a proper face of Q. The barycentric subdivision Δ is homeomorphic to (the space underlying) P; one way to realize Δ is to let each face P_{1} < ⋯ < P_{ℓ} be the convex hull of the barycenters of the faces P_{1}, …, P_{ℓ}.

The main idea in the proof of Proposition 5 is to delete the boundary simplices one by one, in a suitable order, so that at each step the homology remains unchanged, using the following.

Lemma 1.2. Fix a simplicial complex Δ and a chain of simplicial subcomplexes

Δ = Δ_{0} ⊇ Δ_{1} ⊇ ⋯ ⊇ Δ_{r} = Γ.

If the relative chain complex of each pair (Δ_{i−1}, Δ_{i}) has vanishing homology for all i = 1, …, r, then the homology of Γ is canonically isomorphic to that of Δ.

Proof. Repeatedly apply the long exact sequence of homology. □

Throughout the rest of this section, we use M to denote a simplicial homologymanifold-with-boundary of dimension dim(M) = d, defined as follows.

Definition 1.3. A simplicial homology-manifold-with-boundary of dimension d is a simplicial complex whose maximal faces all have dimension d, and such that the link of each i-face has the homology of either a ball or a sphere of dimension d − i − 1.

Remark 1.4. If the polyhedral complex P in Example 1.1 subdivides a homologymanifold-with-boundary, then the conditions of Proposition 5, including the rimmed condition (2), are satisfied by the barycentric subdivision of P.

Lemma 1.5. The link of any i-face in M is a dimension d−i−1 simplicial homologymanifold-with-boundary.

Proof. If σ ⊆ M is a face of dimension i, then Γ = link_{M}(σ) is pure of dimension d − i − 1 because M is pure of dimension d. The condition on the homology of the link in Γ of a face τ ∈ Γ holds simply because the condition holds for link_{M}(σ ∪ τ), which equals link_{Γ}(τ) by definition. □

The above lemma required no special hypotheses on M. Our final observation before the proof of Proposition 5 is that the condition (2) of being rimmed is inherited by links of boundary faces.

Lemma 1.6. Under the hypotheses of Proposition 5, the link of any boundary simplex of M satisfies all of the hypotheses on M in Proposition 5, including being rimmed.

Proof. Let σ be a boundary simplex of M. In view of Lemma 1.5, we need only show that link(σ) satisfies (2). This condition is a consequence of the equality

(3)  ∂(link(σ)) = ∂M ∩ link(σ),

because for any simplex τ in link(σ), the intersection τ ∩ ∂(link(σ)) is forced to equal τ ∩ ∂M, which is a face of τ by (2).

To prove (3), we first show that if τ ∈ ∂(link(σ)) then τ ∈ ∂M; note that for this implication, condition (2) is not necessary. Replacing τ by a face of ∂(link(σ)) containing it, if necessary, we may assume that dim(σ ∪ τ) = d − 1; equivalently, τ is a maximal face of ∂(link(σ)). In this case, there is only one maximal face of link(σ) containing τ. But the faces of link(σ) containing τ are in bijection with the faces of M containing σ ∪ τ. Hence there is only one maximal face of M containing σ ∪ τ.

Therefore σ ∪ τ must be a boundary face of M. We conclude that τ ∈ ∂M, because τ is a face of σ ∪ τ.

For the reverse containment, suppose now that τ ∈ ∂M ∩ link(σ), and let τ′ = τ ∪ σ ∈ M. The intersection τ′ ∩ ∂M is a face of M by condition (2), and it contains all of the vertices of σ and τ, because both σ and τ are boundary faces of M. The only face of τ′ containing all of the vertices of σ and τ is τ′ itself; hence τ′ = τ′ ∩ ∂M is a boundary face of M. It follows that link(τ′) has the homology of a ball (rather than a sphere). But link(τ′) = link_{M}(σ ∪ τ) = link_{link(σ)}(τ), so τ ∈ ∂(link(σ)). □

Proof of Proposition 5. Use induction on the dimension of M. For dim M = 1 the statement is an elementary claim concerning subdivided intervals, so assume that dim M = d ≥ 2. Let S be the collection of boundary faces to be deleted. Without loss of generality, we may assume that S is a cocomplex inside ∂M, which means that if σ ∈ S and τ ∈ ∂M such that τ ⊇ σ, then τ ∈ S.

Totally order the simplices in S in such a way that all of the simplices of maximal dimension d−1 come first, then those of dimension d−2, and so on. Let σ_{1}, σ_{2}, … be this totally ordered sequence of simplices. Let M_{i} be the result of deleting σ_{1}, …, σ_{i} from M, with M = M_{0}. The desired result will follow from Lemma 1.2, with M = Δ, as soon as we show that the relative chain complexes C.(M_{i−1}, M_{i}) are all acyclic.

Let Γ be the simplicial complex obtained by deleting the boundary from link_{M}(σ_{i}), which is acyclic by induction, via Lemma 1.6. We claim that C.(M_{i−1}, M_{i}) is (noncanonically) isomorphic to the reduced chain complex C̃.(Γ). Indeed, recall the bijective correspondence between the faces of link_{M}(σ_{i}) and the faces of M containing σ_{i}. Under this correspondence, Γ is mapped bijectively to the set

{τ ∈ M | τ ∩ ∂M = σ_{i}} = {τ ∈ M_{i−1} | τ ⊇ σ_{i}}.

The faces of this cocomplex inside M_{i−1} constitute a free basis of C.(M_{i−1}, M_{i}), and this induces an isomorphism from C̃.(Γ) to C.(M_{i−1}, M_{i}). □

For future reference, here is the special case of Proposition 5 that we apply later.

Corollary 1.7. If M is a contractible rimmed simplicial manifold-with-boundary, then deleting any collection of boundary faces from M results in a simplicial subcomplex with vanishing reduced homology.

Remark 1.8. It would be preferable to conclude in Corollary 1.7 that the subcomplex is contractible, and more generally that the subcomplex in Proposition 5 is homotopyequivalent to M, using [Bjö84, Lemma 10.3(i)]: if Δ_{i} = Δ_{i−1} ∪ Γ_{i} is a union of two simplicial subcomplexes such that Γ_{i} and Δ_{i−1} ∩ Γ_{i} are both contractible, then Δ_{i} is homotopy-equivalent to Δ_{i−1}. The notation here is consistent with Lemma 1.2; in our case Δ_{i−1} is the result of deleting a boundary simplex σ_{i} from Δ_{i}, and Γ_{i} is the cone from σ_{i} over Δ_{i−1} ∩ Γ_{i} (thus Γ_{i} is the closed star of σ_{i} in Δ_{i}). The problem is that, while Γ_{i} is always contractible (it is a cone), the subcomplex Δ_{i−1} ∩ Γ_{i} = link_{Δ_{i}}(σ_{i}) need not be simply-connected, even though it has vanishing homology.

2. A cellular free resolution

Let A ∈ ℤ^{n×r} be an integer matrix with n rows and r columns, and fix a real column vector b ∈ ℝ^{n} with coordinates b_{1}, …, b_{n}. (When applying these results in Section 3, all entries in the matrix A will be nonnegative.) Viewing the rows A^{1}, …, A^{n} as functionals on ℝ^{r}, we get an (infinite but locally finite) affine hyperplane arrangement

A = ⋃_{\begin{matrix}z∈ℤ^{n} \\ 1≤j≤n \end{matrix}}\{λ ∈ ℝ^{r} | A^{j} · λ + b_{j} = z_{j}\}.

The arrangement A induces a polyhedral subdivision of ℝ^{r}. Fixing a nonnegative real number α ∈ ℝ_{≥0}, the restriction of this polyhedral subdivision to the simplex

Δ^{α} = {(λ_{1}, …, λ_{r}) ∈ ℝ_{≥0}^{r} | λ_{1} + ⋯ + λ_{r} = α}

is a polyhedral subdivision A_{α} of Δ^{α}. Two points λ and μ lie in the same (relatively open) cell of A_{α} if and only if for each j = 1, …, n, there is an integer z_{j} such that either

• A^{j} · λ + b_{j} = A^{j} · μ + b_{j} = z_{j}, or else

• A^{j} · λ + b_{j} and A^{j} · μ + b_{j} both lie in the open interval (z_{j}, z_{j} + 1). For the duration of this paper, fix any polyhedral subdivision P of Δ^{α} that re-

fines A_{α}, meaning that every face of P is contained in a face of A_{α}. As in Example 1.1, denote by Δ the barycentric subdivision of P, realized so that the vertices of Δ are the barycenters of the cells of P. We write σ(P) for the vertex of Δ that is the barycenter of the polytope P ∈ P.

Construct a module M ⊆ ℂ[x_{1}^{±1}, …, x_{n}^{±1}] generated by (Laurent) monomials over the polynomial ring ℂ[x_{1}, …, x_{n}], so M is a Laurent monomial module [MS05, Definition 9.6], as follows. For each polytope P ∈ P, set

m_{P} = x_{1}^{⌊A^{1}·λ+b_{1}⌋} ⋯ x_{n}^{⌊A^{n}·λ+b_{n}⌋} for any λ ∈ P^{∘},

where ⌊z⌋ is the greatest integer less than or equal to z, and the cell P^{∘} is the relative interior of P; the monomial m_{P} is well-defined because the greatest integer parts defining it are equal for all vectors in the same cell of P. Now define

M = ⟨m_{P} | P ∈ P⟩.

We can label each simplex in Δ with a monomial as follows. By virtue of the fact that each polytope in P indexes a monomial m_{P} in the Laurent polynomial ring, the vertices of the simplicial complex Δ, which are the barycenters σ(P) of the polytopes P ∈ P, are labeled with monomials. For each such vertex σ(P), let us set m_{σ(P)} = m_{P}. More generally, for each simplex σ ∈ Δ,

if σ = (P_{1} < ⋯ < P_{ℓ}) then set m_{σ} = m_{P_{1}};

i.e., we label σ by the monomial of the smallest face in the chain. For (Laurent) monomials m and m′, we say that m divides m′ if m′/m is a monomial in ℂ[x_{1}, …, x_{n}].

Lemma 2.1. If P < Q are polytopes in P, then m_{Q} divides m_{P}. Consequently, if σ ∈ Δ is a simplex, then m_{σ} is the least common multiple of {m_{τ} | τ is a vertex of σ}.

Proof. We prove only the first sentence, as the second follows easily. From the perspective of the greatest integer parts used in constructing m_{P} and m_{Q}, the only difference between the barycenters of P and Q is that more of the affine functions λ ↦ A^{j} ·λ+b_{j} can take integer values on σ(P) than on σ(Q). In particular, if ⌊A^{j} · σ(Q) + b_{j}⌋ = z_{j}, then ⌊A^{j} · σ(P) + b_{j}⌋ equals either z_{j} or z_{j} + 1. This being true for all j, we conclude that m_{Q} divides m_{P} (in fact, m_{P} /m_{Q} is a squarefree monomial in ℂ[x_{1}, …, x_{n}]). □

Lemma 2.1 is precisely the condition under which the monomial labels m_{σ} for σ ∈ Δ define a cellular free complex on Δ [BS98, Section 1] (see also Chapters 4 and 9 in [MS05] for background). Briefly, this is the complex

0 → ⨁_{σ∈Δ_{r−1}}⟨m_{σ}⟩ → ⋯ → ⨁_{σ∈Δ_{i}}⟨m_{σ}⟩ → ⋯ → ⨁_{σ∈Δ_{0}}⟨m_{σ}⟩ → 0

of free ℂ[x_{1}, …, x_{n}]-modules where the morphism ⟨m_{σ}⟩ → ⟨m_{τ}⟩ of principal Laurent monomial modules is, for each codimension 1 face τ ⊂ σ, the natural inclusion times ±1. The sign is determined by arbitrary but fixed orientations for the simplices in Δ.

Proposition 2.2. The cellular free complex supported on Δ, in which σ ∈ Δ is labeled by m_{σ}, is a cellular free resolution of the Laurent monomial module M.

Proof. By [BS98, Proposition 1.2], we need to show that the simplicial subcomplex

Δ_{⪯m} = {σ ∈ Δ | m_{σ} divides m}

is acyclic for all Laurent monomials m. Let m = x_{1}^{e_{1}} ⋯ x_{n}^{e_{n}}. For σ = (P_{1} < ⋯ < P_{ℓ}), the monomial m_{σ} divides m if and only if the barycenter τ = σ(P_{1}) satisfies the conditions A^{j} · τ + b_{j} < e_{j} + 1 for all j.

Compare the subcomplex Γ_{⪯m} ⊆ Δ consisting of those points λ ∈ |Δ| such that A^{j} · λ + b_{j} ≤ e_{j} + 1 for all j. This subcomplex is a (compact convex) polytope, and hence Γ_{⪯m} is a contractible simplicial manifold-with-boundary. But Δ_{⪯m} is obtained from Γ_{⪯m} by deleting those vertices τ, necessarily on the boundary of Γ_{⪯m}, having A^{j} · τ + b_{j} = e_{j} + 1 for some j. The desired result is therefore a consequence of Corollary 1.7, which applies by Remark 1.4. □

3. Log resolution to cellular resolution

For the duration of this section, fix a normal ℚ-Gorenstein complex variety X and ideal sheaves 𝔞_{1}, …, 𝔞_{r}, 𝔟 on X. In addition, fix a log resolution π: X′ → X of 𝔟 and 𝔞_{i} +𝔞_{j} for all i, j ∈ {1, …, r}. By definition, this means that 𝔟·O_{X′} and (𝔞_{i} +𝔞_{j})·O_{X′} are locally principal ideal sheaves on X′. The special case 𝔞_{i} = 𝔞_{j} implies that

𝔞_{i} · O_{X′} = O_{X′}(−A_{i}) for i ∈ {1, …, r} and
𝔟 · O_{X′} = O_{X′}(−B)

are the ideal sheaves associated to effective divisors A_{i} and B on X′.

We wish to calculate the multiplier ideals J ((𝔞_{1} +⋯+𝔞_{r})^{α}𝔟^{β}) for nonnegative real numbers α and β. This requires a log resolution of 𝔞 = 𝔞_{1} + ⋯ + 𝔞_{r}. Fortunately, the morphism π : X′ → X is one, as can be seen by applying the following locally on X′.

Lemma 3.1. Let R be a local integral domain.

1. If ⟨f_{1}, …, f_{n}⟩ is a nonzero principal ideal of R, then there is an index i ∈ {1, …, n} such that f_{i} divides f_{j} for all j ∈ {1, …, n}.

2. If f_{1}, …, f_{r} ∈ R are elements such that ⟨f_{i}, f_{j}⟩ is a principal ideal for all i, j ∈ {1, …, r}, then there exists a permutation ξ ∈ S_{r} such that f_{ξ(1)}|f_{ξ(2)}| ⋯ |f_{ξ(r)}. In particular ⟨f_{1}, …, f_{r}⟩ = ⟨f_{ξ(1)}⟩ is a principal ideal.

Proof. Suppose ⟨f_{1}, …, f_{n}⟩ is generated by f ∈ R. Then each f_{j} = f′_{j} · f is a multiple of f. On the other hand, f = ∑ c_{j}f_{j} = f · ∑ c_{j}f′_{j} with c_{j} ∈ R. Since R is a domain, this implies that 1 = ∑ c_{j}f′_{j}. Since R is local, we conclude that some f′_{i} is a unit. This proves the first claim. The second is an immediate consequence of the first. □

Similar to 𝔞_{i} and 𝔟 from before, the ideal sheaf 𝔞 · O_{X′} = O_{X′}(−C) is determined by an effective divisor C on X′. Let D_{1}, …, D_{n} be the set of prime divisors appearing in A_{1}, …, A_{r}, B, C, and the relative canonical divisor K_{X′/X}. Thus

A_{j} = ∑_{i=1}^{n} a_{ij}D_{i} for j ∈ {1, …, r},

βB − K_{X′/X} = ∑_{i=1}^{n} b_{i}D_{i}, and

C = ∑_{i=1}^{n} γ_{i}D_{i}

in terms of D_{1}, …, D_{n}, where the coefficients a_{ij} and γ_{i} are integers, but b_{i} can be real numbers. Recall that the multiplier ideal of 𝔞 with weighting coefficient α is

J (𝔞^{α}) = π_{∗}O_{X′}(−⌊αC − K_{X′/X}⌋) ⊆ O_{X},

where ⌊∑_{i=1}^{n} δ_{i}D_{i}⌋ = ∑_{i=1}^{n} ⌊δ_{i}⌋D_{i} is the greatest integer divisor less than or equal to ∑_{i=1}^{n} δ_{i}D_{i}. More generally, for λ ∈ ℝ_{≥0}^{r} and β ≥ 0, set

J (𝔞_{1}^{λ_{1}} ⋯ 𝔞_{r}^{λ_{r}}𝔟^{β}) = π_{∗}O_{X′}(−⌊λ_{1}A_{1} + ⋯ + λ_{r}A_{r} + βB − K_{X′/X}⌋) ⊆ O_{X}.

How do these multiplier ideals depend on the choices of λ ∈ ℝ^{r} and β ≥ 0? Given the decompositions into prime divisors D_{1}, …, D_{n} with integer coefficient matrix A = (a_{ij}) and real vector b = (b_{i}), we are verbatim in the situation from Section 2, whose notation we assume henceforth.

Lemma 3.2. For each polytope P ∈ P, there is a single ideal sheaf J_{P} ⊆ O_{X} that coincides with the multiplier ideals J (𝔞_{1}^{λ_{1}} ⋯ 𝔞_{r}^{λ_{r}}𝔟^{β}) for all λ ∈ P^{∘} in the interior of P.

Proof. In fact, the greatest integer divisors ⌊λ_{1}A_{1} + ⋯ + λ_{r}A_{r} + βB − K_{X′/X}⌋ on X′ all coincide for λ ∈ P^{∘}, by construction. Therefore we can define

I_{P} = O_{X′}(−⌊λ_{1}A_{1} + ⋯ + λ_{r}A_{r} + βB − K_{X′/X}⌋)

for any λ ∈ P^{∘}, and J_{P} = π_{∗}I_{P} does not depend on λ ∈ P^{∘}. □

As we did for monomials in Section 2 (before Lemma 2.1), having defined objects I_{P} and J_{P} indexed by polytopes in P, we can define objects I_{σ} and J_{σ} indexed by simplices in Δ:

if σ = (P_{1} < ⋯ < P_{ℓ}) then set I_{σ} = I_{P_{1}},

and set J_{σ} = π_{∗}I_{σ}.

Remark 3.3. Unwinding the definitions, we find that

J_{σ} = J (𝔞_{1}^{λ_{1}} ⋯ 𝔞_{r}^{λ_{r}}𝔟^{β}),

where λ is the barycenter of the smallest polytope P_{1} in the chain P_{1} < ⋯ < P_{ℓ} corresponding to the simplex σ ∈ Δ. (Recall that Δ is the barycentric subdivision of the polyhedral complex P from the beginning of Section 2.)

The analogue of Lemma 2.1 holds here, as well.

Lemma 3.4. If P < Q are polytopes in P, then I_{Q} ⊇ I_{P}. Consequently, if σ ∈ Δ, then I_{σ} = ⋂{I_{τ} | τ is a vertex of σ}. The same is true with J in place of I.

Proof. The argument is essentially the same as the proof of Lemma 2.1. □

Lemma 3.5. Resume the notation involving 𝔞 = 𝔞_{1} + ⋯ + 𝔞_{r} from after Lemma 3.1, so 𝔞 · O_{X′} = O_{X′}(−C). Then, as ideal sheaves over X′,

O_{X′}(−⌊αC + βB − K_{X′/X}⌋) = ∑_{P ∈P} I_{P} .

Proof. Let p ∈ X′ be an arbitrary point. By Lemma 3.1, there is an open neighborhood U of p and a permutation ξ ∈ S_{r} such that after restricting to U, we have A_{ξ(1)} ⪯ ⋯ ⪯ A_{ξ(r)}, where E ⪯ F for divisors E and F means that F − E is effective. It follows that on U we have C = A_{ξ(1)}, and also

αA_{ξ(1)} ⪯ λ_{1}A_{1} + ⋯ + λ_{r}A_{r}

whenever λ_{1} + … + λ_{r} = α. Hence both sides of the desired equality are equal to O_{X′}(−⌊αA_{ξ(1)} + βB − K_{X′/X}⌋) on U. Equality holds on X′ because p is arbitrary. □

We have arrived at our main result. For notational purposes, choose an arbitrary but fixed collection of orientations for the simplices in Δ. There results an incidence function that assigns to each codimension 1 face τ of each simplex σ a sign (−1)^{σ,τ}.

Theorem 3.6. With J_{σ} as in Remark 3.3, there is an exact sequence

0 → ⨁_{σ∈Δ_{r−1}} J_{σ} → ⋯ → ⨁_{σ∈Δ_{i}} J_{σ} → ⋯ → ⨁_{σ∈Δ_{0}} J_{σ} → J ((𝔞_{1} + ⋯ + 𝔞_{r})^{α}𝔟^{β}) → 0

of sheaves on X, in which the morphism J_{σ} → J_{τ} is inclusion times (−1)^{σ,τ}. Here, Δ_{i} = {σ ∈ Δ | dim(σ) = i} is the set of i-dimensional simplices in Δ.

Proof. We shall first verify the exactness of the complex

0 → ⨁_{σ∈Δ_{r−1}} I_{σ} → ⋯ → ⨁_{σ∈Δ_{i}} I_{σ} → ⋯ → ⨁_{σ∈Δ_{0}} I_{σ} → ∑_{σ∈Δ_{0}} I_{σ} → 0

on X′, where it should be noted that the sum at the far right is not a direct sum. It is enough to verify exactness at the stalk of an arbitrary point p ∈ X′. Reordering the prime divisors D_{1}, …, D_{n} appearing in A_{1}, …, A_{r}, B, and K_{X′/X} if necessary, we assume that only D_{1}, …, D_{k} pass through p. Since D_{1}, …, D_{n} intersect transversally, if we pass to an analytic neighborhood or completion at p, then we can choose local coordinates x_{1}, …, x_{k} so that D_{i} is given by the vanishing of x_{i} for i = 1, …, k. In these coordinates, I_{σ} becomes the principal Laurent monomial module in ℂ[x_{1}^{±1}, …, x_{k}^{±1}] generated by the monomial m_{σ} from before Lemma 2.1, except that all of the variables x_{k+1}, …, x_{n} have been set equal to 1 in m_{σ}. The exactness thus follows from Proposition 2.2, given that our polyhedral complex P refines the subdivision induced by the linear functionals A^{j} · λ + b_{j} for j = 1, …, k, instead of j = 1, …, n. (Another reason the exactness follows from Proposition 2.2 is that setting the variables equal to 1 is the exact operation of “homogeneous localization”; see [Mil00, Section 3.6], particularly Proposition 3.31.2 there.)

Pushing forward under π completes the proof. Indeed, Lemma 3.5 implies that

∑_{σ∈Δ_{0}} I_{σ} = ∑_{P ∈P} I_{P} = O_{X′}(−⌊αC + βB − K_{X′/X}⌋)

pushes forward to yield J ((𝔞_{1} + ⋯ + 𝔞_{r})^{α}𝔟^{β}), while local vanishing [Laz04, Theorem 9.4.17(i)] guarantees that the higher direct images vanish on all I_{σ}. □

Remark 3.7. In the special case where X = Spec ℂ[x_{1}, …, x_{n}] and 𝔞_{1}, …, 𝔞_{r}, 𝔟 are all principal monomial ideals, all of the J_{σ} are principal monomial ideals as well, so the exact sequence in Theorem 3.6 becomes a cellular resolution of the monomial ideal J ((𝔞_{1} + ⋯ + 𝔞_{r})^{α}𝔟^{β}) in the sense of [MS05, Definition 4.3].

Example 3.8. Let us illustrate Theorem 3.6 by an example. Take X = Spec ℂ[x, y], r = α = 2, 𝔞_{1} = ⟨xy⟩, 𝔞_{2} = ⟨x + y⟩, and 𝔟 = O_{X}. A direct computation shows that on the doubled 1-simplex {(λ_{1}, λ_{2}) ∈ ℝ_{≥0} | λ_{1} + λ_{2} = 2}, we have

J (𝔞_{1}^{λ_{1}}𝔞_{2}^{λ_{2}}) = \begin{cases}⟨x^{2}y^{2}⟩ & if λ_{2} = 0 \\ ⟨xy⟩ & if 0 < λ_{2} < 1 \\ ⟨xy(x + y)⟩ & if λ_{2} = 1 \\ ⟨x + y⟩ & if 1 < λ_{2} < 2 \\ ⟨(x + y)^{2}⟩ & if λ_{2} = 2. \end{cases}

These regions where J (𝔞_{1}^{λ_{1}}𝔞_{2}^{λ_{2}}) stays constant yield a polyhedral subdivision P of the doubled 1-simplex, and we let Δ be the barycentric subdivision of P. Explicitly, Δ consists of five vertices

v_{1} = (2, 0), v_{2} = (\frac{3}{2}, \frac{1}{2}), v_{3} = (1, 1), v_{4} = (\frac{1}{2}, \frac{3}{2}), v_{5} = (0, 2)

and four edges

l_{i} = the line segment between v_{i} and v_{i+1}, for i = 1, 2, 3, 4.

The vertices in Δ are naturally labeled with the multiplier ideals

J_{v_{1}} = J (𝔞_{1}^{2}𝔞_{2}^{0}) = ⟨x^{2}y^{2}⟩,
J_{v_{2}} = J (𝔞_{1}^{3/2}𝔞_{2}^{1/2}) = ⟨xy⟩,
J_{v_{3}} = J (𝔞_{1}^{1}𝔞_{2}^{1}) = ⟨xy(x + y)⟩,
J_{v_{4}} = J (𝔞_{1}^{1/2}𝔞_{2}^{3/2}) = ⟨x + y⟩,
J_{v_{5}} = J (𝔞_{1}^{0}𝔞_{2}^{2}) = ⟨(x + y)^{2}⟩.

The edges of Δ are labeled by J_{l_{i}}, which for i = 1, 2, 3, 4 is defined to be the least common multiple of J_{v_{i}} and J_{v_{i+1}}. It is easily verified that J_{l_{i}} equals the smaller one of J_{v_{i}} and J_{v_{i+1}}. The exact sequence of Theorem 3.6 is

0 → ⨁_{i=1}^{4} J_{l_{i}} → ⨁_{i=1}^{5} J_{v_{i}} → J ((𝔞_{1} + 𝔞_{2})^{2}) → 0,

or more explicitly,

0 → \begin{aligned}& ⟨x^{2}y^{2}⟩ \\ ⊕ & ⟨xy(x + y)⟩ \\ ⊕ & ⟨xy(x + y)⟩ \\ ⊕ & ⟨(x + y)^{2}⟩ \end{aligned} → \begin{aligned}& ⟨x^{2}y^{2}⟩ \\ ⊕ & ⟨xy⟩ \\ ⊕ & ⟨xy(x + y)⟩ \\ ⊕ & ⟨x + y⟩ \\ ⊕ & ⟨(x + y)^{2}⟩ \end{aligned} → J (⟨xy, x + y⟩^{2}) → 0.

After canceling redundant terms, this is essentially a Koszul resolution

0 → ⟨xy(x + y)⟩ → ⟨xy⟩ ⊕ ⟨x + y⟩ → J (⟨xy, x + y⟩^{2}) → 0.

4. Applications

The Introduction already contains some immediate applications of Theorem 3.6, namely the Takagi-style summation formula in Corollary 2 and the derivation of Howald’s monomial multiplier ideal formula in Corollary 3. In this section, we collect further applications: a new exactness proof for the Skoda complex, and additional summation formulas for multiplier ideals, including the graded system case.

We begin with the summation formulas. First, we note the following.

Remark 4.1. Theorem 3.6 still holds when 𝔟^{β} is replaced by 𝔟_{1}^{β_{1}} ⋯ 𝔟_{s}^{β_{s}}, since the contants β_{k} merely translate the affine hyperplane arrangement A in Section 2.

Corollary 4.2. Let 𝔞_{1}, …, 𝔞_{r}, 𝔟_{1}, …, 𝔟_{s} ⊆ O_{X} be nonzero ideal sheaves on a normal ℚ-Gorenstein variety X and let α, β_{1}, …, β_{s} be positive real numbers. Then

J (X, (𝔞_{1} + ⋯ + 𝔞_{r})^{α}𝔟_{1}^{β_{1}} ⋯ 𝔟_{s}^{β_{s}}) = ∑_{λ_{1}+···+λ_{r}=α} J (X, 𝔞_{1}^{λ_{1}} ⋯ 𝔞_{r}^{λ_{r}}𝔟_{1}^{β_{1}} ⋯ 𝔟_{s}^{β_{s}}).

Proof. This follows immediately from the surjectivity on the right of the exact sequence in Theorem 3.6. □

Our next result concerns the notion of graded system of ideals. For the definition of this and the multiplier ideal associated to it, see [Laz04, Section 11.1.B]. A special case has already appeared in [Tak05, Proposition 4.10].

Corollary 4.3. Corollary 4.2 still holds when the ideal sheaves 𝔞_{i} and 𝔟_{j} are all replaced by graded systems of ideals.

Proof. The formula for r > 2 can be obtained by repeatedly applying the formula for r = 2, so it suffices to prove this case, i.e.,

J (X, (𝔞. + 𝔟.)^{α}𝔠_{1,.}^{β_{1}} ⋯ 𝔠_{s,.}^{β_{s}}) = ∑_{0≤t≤α} J (X, 𝔞.^{α−t}𝔟.^{t} · 𝔠_{1,.}^{β_{1}} ⋯ 𝔠_{s,.}^{β_{s}}).

First let us verify ⊇, i.e., J (X, 𝔞.^{α−t}𝔟.^{t}·𝔠_{1,.}^{β_{1}} ⋯ 𝔠_{s,.}^{β_{s}}) ⊆ J (X, (𝔞.+𝔟.)^{α}𝔠_{1,.}^{β_{1}} ⋯ 𝔠_{s,.}^{β_{s}}) for any

fixed t ∈ [0, α]. By definition, the left-hand side is equal to J (X, 𝔞p^{\frac{α−t}{p}} 𝔟p^{\frac{t}{p}} · 𝔠1,p^{\frac{β_{1}}{p}} ⋯ 𝔠s,p^{\frac{βs}{p}}) for some large p, and we see that

J (X, 𝔞p^{\frac{α−t}{p}} 𝔟p^{\frac{t}{p}} · 𝔠1,p^{\frac{β_{1}}{p}} ⋯ 𝔠s,p^{\frac{βs}{p}}) ⊆ J (X, (𝔞_{p} + 𝔟_{p})^{\frac{α}{p}} 𝔠1,p^{\frac{β_{1}}{p}} ⋯ 𝔠s,p^{\frac{βs}{p}}) (by Corollary 4.2)

⊆ J (X, (𝔞. + 𝔟.)p^{\frac{α}{p}} 𝔠1,p^{\frac{β_{1}}{p}} ⋯ 𝔠s,p^{\frac{βs}{p}})

⊆ J (X, (𝔞. + 𝔟.)^{α}𝔠_{1,.}^{β_{1}} ⋯ 𝔠_{s,.}^{β_{s}}) (by definition).

To prove the reverse inclusion, first by definition there exists some large p such that

J (X, (𝔞. + 𝔟.)^{α}𝔠_{1,.}^{β_{1}} ⋯ 𝔠_{s,.}^{β_{s}}) = J (X, (∑_{i=0}^{p} 𝔞_{i}𝔟_{p−i})^{\frac{α}{p}} 𝔠1,p^{\frac{β_{1}}{p}} ⋯ 𝔠s,p^{\frac{βs}{p}}),

and by Corollary 4.2 this right-hand side can be expressed as

∑_{λ_{0}+···+λ_{p}= \frac{α}{p}} J (X, (∏_{i=1}^{p} 𝔞_{i}^{λ_{i}})(∏_{i=0}^{p−1} 𝔟_{p−i}^{λ_{i}}) · 𝔠1,p^{\frac{β_{1}}{p}} ⋯ 𝔠s,p^{\frac{βs}{p}}).

Now take a positive integer m which is divisible by 1, 2, …, p. Since 𝔞., 𝔟. and 𝔠_{j,.} are graded systems, we have

𝔞_{i}^{λ_{i}} = 𝔞i^{\frac{m}{i} \frac{iλ_{i}}{m}} ⊆ 𝔞m^{\frac{iλ_{i}}{m}} , 𝔟_{p−i}^{λ_{i}} = 𝔟p−i^{\frac{m}{p−i} \frac{(p−i)λ_{i}}{m}} ⊆ 𝔟m^{\frac{(p−i)λ_{i}}{m}} , 𝔠j,p^{\frac{β_{j}}{p}} = 𝔠j,p^{\frac{m}{p} \frac{β_{j}}{m}} ⊆ 𝔠j,m^{\frac{β_{j}}{m}},

so

(∏_{i=1}^{p} 𝔞_{i}^{λ_{i}})(∏_{i=0}^{p−1} 𝔟_{p−i}^{λ_{i}}) · 𝔠1,p^{\frac{β_{1}}{p}} ⋯ 𝔠s,p^{\frac{βs}{p}} ⊆ 𝔞m^{∑_{i=1}^{p} \frac{iλ_{i}}{m}} 𝔟m^{∑_{i=0}^{p−1} \frac{(p−i)λ_{i}}{m}} · 𝔠1,m^{\frac{β_{1}}{m}} ⋯ 𝔠s,m^{\frac{βs}{m}},

and since λ_{0} + ⋯ + λ_{p} = \frac{α}{p}, if we let t := ∑_{i=0}^{p−1}(p − i)λ_{i}, then ∑_{i=1}^{p} iλ_{i} = α − t, hence

J (X, (∏_{i=1}^{p} 𝔞_{i}^{λ_{i}})(∏_{i=0}^{p−1} 𝔟_{p−i}^{λ_{i}}) · 𝔠1,p^{\frac{β_{1}}{p}} ⋯ 𝔠s,p^{\frac{βs}{p}}) ⊆ J (X, 𝔞m^{\frac{α−t}{m}} 𝔟m^{\frac{t}{m}} · 𝔠1,m^{\frac{β_{1}}{m}} ⋯ 𝔠s,m^{\frac{βs}{m}})
⊆ J (X, 𝔞.^{α−t}𝔟.^{t} · 𝔠_{1,.}^{β_{1}} ⋯ 𝔠_{s,.}^{β_{s}}).  □

All of the corollaries of Theorem 3.6 we have given so far are Takagi-style summation formulas which may also be obtained by straightforwardly generalizing the proof of Takagi’s original formula (1) (see [Tak05, Theorem 3.2]). However, in our final application we want to demonstrate that Theorem 3.6 may be potentially more useful than (1) by deriving the exactness of the Skoda complex from the former. For this we assume that the ideal sheaves 𝔞_{1}, …, 𝔞_{r} are locally principal on the ℚ-Gorenstein variety X. For every nonnegative integer α < r, the inclusion 𝔞_{i} → O_{X} induces a natural map 𝔞_{i}J (𝔞^{α}𝔟^{β}) → J (𝔞^{α+1}𝔟^{β}). Writing 𝔞_{τ} = ∏_{i∈τ} 𝔞_{i} for τ ∈ {0, 1}^{r} ⊆ ℝ^{r}, the Skoda complex is the cellular complex

0 → 𝔞_{1} ⋯ 𝔞_{r}J (𝔟^{β}) →⋯→ ⨁_{|τ|=r−α}𝔞_{τ}J (𝔞^{α}𝔟^{β}) →⋯→ ⨁_{i=1}^{r} 𝔞_{i}J (𝔞^{r−1}𝔟^{β}) → J (𝔞^{r}𝔟^{β}) → 0

supported on a simplex whose faces are in bijection with the vectors τ ∈ {0, 1}^{r}.

Corollary 4.4. The Skoda complex is exact.

A proof for smooth X appears in [Laz04, Section 9.6.C]. Here, we provide an alternate proof, which works with no extra effort in the ℚ-Gorenstein setting.

Proof. Consider the subdivisions A_{α} of the simplices Δ^{α} for α = 0, …, r, defined at the beginning of Section 2. Every vector τ ∈ {0, 1}^{r} ⊆ ℝ^{r} induces an embedding Δ^{α} ↪ Δ^{r}, where α = r − |τ|, by adding τ. Choose a refinement P of A_{r} so fine that the subdivision induced on Δ^{α} under every one of these inclusions, for all τ and all α = 0, …, r, refines A_{α}. For example, in the notation from the beginning of Section 2, let P be the subdivision induced by the affine hyperplane arrangement

A + \{0, 1\}^{r} = ⋃_{\begin{matrix}z∈ℤ^{n} \\ 1≤j≤n \end{matrix}} ⋃_{τ∈\{0,1\}^{r}}\{λ + τ ∈ ℝ^{r} | A^{j} · λ + b_{j} = z_{j}\}

obtained by translating the arrangement A up by every vector τ ∈ {0, 1}^{r}.

As in Theorem 3.6, let Δ be the barycentric subdivision of P. Recall the definition of J_{σ} = J (𝔞_{1}^{λ_{1}} ⋯ 𝔞_{r}^{λ_{r}}𝔟^{β}) from Remark 3.3, and write λ(σ) = λ = (λ_{1}, …, λ_{r}). Define a double complex J_{.,.} in which

J_{p,q} = ⨁_{dim(σ)=q} ⨁_{\begin{matrix}|τ|=p \\ τ⪯λ(σ) \end{matrix}} J_{σ},

where τ ⪯ λ(σ) means that λ(σ) − τ ∈ Δ^{r−|τ|} (equivalently, λ(σ) − τ has nonnegative entries). Each horizontal complex J_{.,q} is a direct sum, over σ ∈ Δ with dim(σ) = q, of complexes J_{σ} ⊗_{ℂ} C̃.(Γ_{τ}), where τ ⪯ λ(σ) is maximal and C̃.(Γ_{τ}) is the reduced chain complex of a simplex Γ_{τ} with |τ| many vertices. The vertical differentials of J_{.,.} are induced by the differentials of Theorem 3.6, for each fixed τ. In particular, the vertical complex on the summands J_{σ} indexed by a fixed τ ⪯ λ = λ(σ) is a resolution of 𝔞_{τ}J (𝔞^{r−|τ|}𝔟^{β}) by Theorem 3.6, because

J (𝔞_{1}^{λ_{1}} ⋯ 𝔞_{r}^{λ_{r}}𝔟^{β}) = 𝔞_{τ}J (𝔞_{1}^{λ_{1}−τ_{1}} ⋯ 𝔞_{r}^{λ_{r}−τ_{r}}𝔟^{β}).

It follows that the vertical homology of J_{.,.} is the Skoda complex. On the other hand, the horizontal homology of J_{.,.} is identically zero because every simplex Γ_{τ} is contractible. The standard spectral sequence argument shows the desired result. □

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