Infinitesimal Deformations and Obstructions for Toric Singularities

Klaus Altmann^{∗}

Institut für reine Mathematik, Humboldt-Universität zu Berlin Ziegelstr. 13a, D-10099 Berlin, Germany. E-mail: altmann@mathematik.hu-berlin.de

Abstract The obstruction space T^{2} and the cup product T^{1}×T^{1} → T^{2} are computed

for toric singularities.

Contents

1 Introduction 1 2 T^{1}, T^{2}, and the cup product (in general) 2 3 T^{1}, T^{2}, and the cup product (for toric varieties) 4 4 Proof of the T^{2}-formula 9 5 Proof of the cup product formula 16 6 An alternative to the complex L(E^{R})_{•} 20 7 3-dimensional Gorenstein singularities 21

1 Introduction

(1.1) For an affine scheme Y = Spec A, there are two important A-modules, T_{Y}^{1} and T_{Y}^{2}, carrying information about its deformation theory: T_{Y}^{1} describes the infinitesimal deformations, and T_{Y}^{2} contains the obstructions for extending deformations of Y to larger base spaces (cf. [KPR], [LiS], or [Pa]).

∗This paper was written at M.I.T. and supported by a DAAD-fellowship.

In case Y admits a versal deformation, T_{Y}^{1} is the tangent space of the versal base space S. Moreover, if J denotes the ideal defining S as a closed subscheme of the affine space T_{Y}^{1}, the module (J/m_{T^{1}} J )^{∗} can be canonically embedded into T_{Y}^{2}, i.e. (T_{Y}^{2})^{∗}-elements induce the equations defining S in T_{Y}^{1}.

The vector spaces T_{Y}^{i} come with a cup product T_{Y}^{1} × T_{Y}^{1} → T_{Y}^{2}. The associated quadratic form T_{Y}^{1} → T_{Y}^{2} describes the quadratic part of the elements of J, i.e. it can be used to get a better approximation of the versal base space S as regarding its tangent space only.

(1.2) In [Al 1] we have determined the vector space T_{Y}^{1} for affine toric varieties. The present paper can be regarded as its continuation - we will compute T_{Y}^{2} and the cup product. These modules T_{Y}^{i} are canonically graded (induced from the character group of the torus). We will describe their homogeneous pieces as cohomology groups of certain complexes that are directly induced from the combinatorial structure of the rational, polyhedral cone defining our variety Y. (The paper [LdSl] has suggested that this should be possible at all.) The results can be found in §3.

Switching to another, quasiisomorphic complex provides a second formula for the vector spaces T_{Y}^{i} (cf. §6). We will use this particular version for describing these spaces and the cup product in the special case of three-dimensional toric Gorenstein singularities (cf. §7).

2 T^{1}, T^{2}, and the cup product (in general)

In this section we will give a brief reminder to the well known definitions of these objects. Moreover, we will use this opportunity to fix some notations.

(2.1) Let Y ⊆ I C^{w+1} be given by equations f_{1},…, f_{m}, i.e. its ring of regular functions equals

A = P/I with \begin{matrix}P = I C[z_{0},…, z_{w}] \\ I = (f_{1},…, f_{m}) . \end{matrix}

Then, using d :I /I^{2} → A^{w+1} (d(f_{i}) := (\frac{∂f_{i}}{∂z_{0}} ,… \frac{∂f_{i}}{∂z_{w}})), the vector space T_{Y}^{1} equals

T_{Y}^{1} = Hom_{A}(I/I^{2}, A)/ ; Hom_{A}(A^{w+1}, A) .

(2.2) Let R ⊆ P^{m} denote the P -module of relations between the equations f_{1},…, f_{m}. It contains the so-called Koszul relations R_{0} := ⟨f_{i} e^{j} − f_{j} e^{i}⟩ as a submodule. Now, T_{Y}^{2} can be obtained as

T_{Y}^{2} = Hom_{P} (R/R_{0}, A)/ ; Hom_{P} (P^{m}, A) .

(2.3) Finally, it is well known (cf. [Ld], (5.1.5)) that the cup product T^{1}×T^{1} → T^{2} can be defined in the following way:

(i) Starting with an φ ∈ Hom_{A}(I/I^{2}, A), we lift the images of the f_{i} obtaining elements φ̃(f_{i}) ∈ P.

(ii) Given a relation r ∈ R, the linear combination ∑_{i} r_{i} φ̃(f_{i}) vanishes in A, i.e. it is contained in the ideal I ⊆ P. Denote by λ(φ) ∈ P^{m} any set of coefficients such that

∑_{i} r_{i} φ̃(f_{i}) + ∑_{i} λ_{i}(φ) f_{i} = 0 in P.

(Of course, λ depends on r also.)

(iii) If φ, ψ ∈ Hom_{A}(I/I^{2}, A) represent two elements of T_{Y}^{1}, then we define for each relation r ∈ R

(φ ∪ ψ)(r) := ∑_{i} λ_{i}(φ) ψ(f_{i}) + ∑_{i} φ(f_{i}) λ_{i}(ψ) ∈ A .

Remark: The definition of the cup product does not depend on the choices we made:

(a) Choosing a λ′(φ) instead of λ(φ) yields λ′(φ) − λ(φ) ∈ R, i.e. in A we obtain the same result.

(b) Let φ̃′(f_{i}) be different liftings to P. Then, the difference φ̃′(f_{i}) − φ̃(f_{i}) is contained in I, i.e. it can be written as some linear combination

φ̃′(f_{i}) − φ̃(f_{i}) = ∑_{j} t_{ij} f_{j} .

Hence,

∑_{i} r_{i} φ̃′(f_{i}) = ∑_{i} r_{i} φ̃(f_{i}) + ∑_{i,j} t_{ij} r_{i} f_{j} ,

and we can define λ′_{j}(φ) := λ_{j}(φ) − ∑_{i} t_{ij} r_{i} (corresponding to φ̃′ instead of φ̃). Then, we obtain for the cup product

(φ ∪ ψ)′(r) − (φ ∪ ψ)(r) = − ∑_{i} r_{i} · (∑_{j} t_{ij} ψ(f_{j})) ,

but this expression comes from some map P^{m} → A.

3 T^{1}, T^{2}, and the cup product (for toric varieties)

(3.1) We start with fixing the usual notations when dealing with affine toric varieties (cf. [Ke] or [Od]):

• Let M, N be mutually dual free Abelian groups, we denote by M_{IR}, N_{IR} the associated real vector spaces obtained by base change with IR.

• Let σ = ⟨a^{1},…, a^{N}⟩ ⊆ N_{IR} be a rational, polyhedral cone with apex - given by its fundamental generators. σ^{∨} := {r ∈ M_{IR} | ⟨σ, r⟩ ≥ 0} ⊆ M_{IR} is called the dual cone. It induces a partial order on the lattice M via [ a ≥ b iff a − b ∈ σ^{∨}].

• A := I C[σ^{∨} ∩ M] denotes the semigroup algebra. It is the ring of regular functions of the toric variety Y = Spec A associated to σ.

• Denote by E ⊂ σ^{∨} ∩ M the minimal set of generators of this semigroup (”the Hilbert basis”). E equals the set of all primitive (i.e. non-splittable) elements of σ^{∨}∩M. In particular, there is a surjection of semigroups π : IN^{E} ⟶ ⟶ σ^{∨}∩M, and this fact translates into a closed embedding Y ↪ I C^{E}. To make the notations coherent with §2, assume that E = {r^{0},…, r^{w}} consists of w + 1 elements.

(3.2) To a fixed degree R ∈ M we associate “thick facets” K_{i}^{R} of the dual cone

K_{i}^{R} := {r ∈ σ^{∨} ∩ M | ⟨a^{i}, r⟩ < ⟨a^{i}, R⟩} (i = 1,…, N).

Lemma:

(1) ∪_{i}K_{i}^{R} = (σ^{∨} ∩ M) \ (R + σ^{∨}).

(2) For each r, s ∈ K_{i}^{R} there exists an ℓ ∈ K_{i}^{R} such that ℓ ≥ r, s. Moreover, if Y is smooth in codimension 2, the intersections K_{i}^{R} ∩ K_{j}^{R} (for 2-faces ⟨a^{i}, a^{j}⟩ < σ) have the same property.

Proof: Part (i) is trivial; for (ii) cf. (3.7) of [Al 1]. □

Intersecting these sets with E ⊆ σ^{∨} ∩ M, we obtain the basic objects for describing the modules T_{Y}^{i}:

E_{i}^{R} := K_{i}^{R} ∩ E = {r ∈ E | ⟨a^{i}, r⟩ < ⟨a^{i}, R⟩} ,
E_{0}^{R} := ⋃_{i=1}^{N} E_{i}^{R} , and
E_{τ}^{R} := ⋂_{a^{i}∈τ} E_{i}^{R} for faces τ < σ .

We obtain a complex L(E^{R})_{•} of free Abelian groups via

L(E^{R})_{−k} := ⨁ ; \begin{matrix}τ < σ \\ dim τ = k \end{matrix} L(E_{τ}^{R})

with the usual differentials. (L(…) denotes the free Abelian group of integral, linear dependencies.) The most interesting part (k ≤ 2) can be written explicitely as

L(E^{R})_{•} : ⋯ → ⊕_{⟨a^{i},a^{j}⟩<σ}L(E_{i}^{R} ∩ E_{j}^{R}) ⟶ ⊕_{i}L(E_{i}^{R}) ⟶ L(E_{0}^{R}) → 0 .

(3.3) Theorem:

(1) T_{Y}^{1}(−R) = H^{0} (L(E^{R})_{•}^{∗} ⊗_{ZZ} I C)

(2) T_{Y}^{2}(−R) ⊇ H^{1} (L(E^{R})_{•}^{∗} ⊗_{ZZ} I C)

(3) Moreover, if Y is smooth in codimension 2 (i.e. if the 2-faces ⟨a^{i}, a^{j}⟩ < σ are spanned by a part of a ZZ-basis of the lattice N), then

T_{Y}^{2}(−R) = H^{1} (L(E^{R})_{•}^{∗} ⊗_{ZZ} I C) .

(4) Module structure: If x^{s} ∈ A (i.e. s ∈ σ^{∨} ∩ M), then E_{i}^{R−s} ⊆ E_{i}^{R}, hence L(E^{R})_{•}^{∗} ⊆ L(E^{R−s})_{•}^{∗}. The induced map in cohomology corresponds to the multiplication with x^{s} in the A-modules T_{Y}^{1} and T_{Y}^{2}.

The first part was shown in [Al 1]; the formula for T^{2} will be proved in §4. Then, the claim concerning the module structure will become clear automatically.

Remark: The assumption made in (3) can not be dropped: Taking for Y a 2-dimensional cyclic quotient singularity given by some 2-dimensional cone σ, there are only two different sets E_{1}^{R} and E_{2}^{R} (for each R ∈ M). In particular, H^{1} (L(E^{R})_{•}^{∗} ⊗_{ZZ} I C) = 0.

(3.4) We want to describe the isomorphisms connecting the general T^{i}-formulas of (2.1) and (2.2) with the toric ones given in (3.3).

Y ↪ I C^{w+1} is given by the equations

f_{ab} := \underline{z}^{a} − \underline{z}^{b} (a, b ∈ IN^{w+1} with π(a) = π(b) in σ^{∨} ∩ M),

and it is easier to deal with this infinite set of equations (which generates the ideal I as a I C-vector space) instead of selecting a finite number of them in some noncanonical way. In particular, for m of (2.1) and (2.2) we take

m := {(a, b) ∈ IN^{w+1} × IN^{w+1} | π(a) = π(b)} .

The general T^{i}-formulas mentioned in (2.1) and (2.2) remain true.

Theorem: For a fixed element R ∈ M let φ : L(E)_{I C} → I C represent an element of

(^{L(ER 0)/}∑_{i} L(E_{i}^{R}))^{∗} ⊗_{ZZ} I C ≅ T_{Y}^{1}(−R) (cf. Theorem (3.3)(1)).

Then, the A-linear map

I/I^{2} ⟶ A
\underline{z}^{a} − \underline{z}^{b} ↦ φ(a − b) · x^{π(a)−R}

provides the same element via the formula (2.1).

Again, this Theorem follows from the paper [Al 1] - together with the commutative diagram of (4.3) in the present paper. (Cf. Remark (4.4).)

Remark: A simple, but nevertheless important check shows that the map (\underline{z}^{a} − \underline{z}^{b}) ↦ φ(a − b) · x^{π(a)−R} goes into A, indeed: Assume π(a) − R ∉ σ^{∨}. Then, there exists an index i such that ⟨a^{i}, π(a) − R⟩ < 0. Denoting by “supp q” (for a q ∈ IR^{E}) the set of those r ∈ E providing a nonvanishing entry q_{r}, we obtain

supp (a − b) ⊆ supp a ∪ supp b ⊆ E_{i}^{R} ,

i.e. φ(a − b) = 0.

(3.5) The P -module R ⊆ P^{m} is generated by relations of two different types:

r(a, b; c) := e^{a+c, b+c} − \underline{z}^{c} e^{a,b} (a, b, c ∈ IN^{w+1}; π(a) = π(b)) and
s(a, b, c) := e^{b,c} − e^{a,c} + e^{a,b} (a, b, c ∈ IN^{w+1}; π(a) = π(b) = π(c)) .

(e^{•,•} denote the standard basis vectors of P^{m}.)

Theorem: For a fixed element R ∈ M let ψ_{i} : L(E_{i}^{R})_{I C} → I C represent an element of

(\begin{matrix}Ker & (⊕_{i}L(E_{i}^{R}) ⟶ L(E^{′R})) \\ Im & (⊕_{⟨a^{i},a^{j}⟩<σ}L(E_{i}^{R} ∩ E_{j}^{R}) → ⊕_{i}L(E_{i}^{R})) \end{matrix})^{∗} ⊗_{ZZ} I C ⊆ T_{Y}^{2}(−R) (cf. (3.3)(2)).

Then, the P -linear map

R/R_{0} ⟶ A
r(a, b; c) ↦ \{ \begin{matrix}ψ_{i}(a − b) x^{π(a+c)−R} & for π(a) ∈ K_{i}^{R}; π(a + c) ≥ R \\ 0 & for π(a) ≥ R or π(a + c) ∈ ⋃_{i} K_{i}^{R} \end{matrix}
s(a, b, c) ↦ 0

is correct defined, and via the formula (2.2) it induces the same element of T_{Y}^{2}.

For the proof we refer to §4. However, we may easily check the correctness of the definition of the P -linear map R/R_{0} → A:

(i) If π(a) is contained in two different sets K_{i}^{R} and K_{j}^{R}, then the two fundamental generators a^{i} and a^{j} can be connected by a sequence a^{i_{0}},…, a^{i_{p}}, such that

• a^{i_{0}} = a^{i}, a^{i_{p}} = a^{j},

• a^{i_{v−1}} and a^{i_{v}} are the edges of some 2-face of σ (v = 1,…, p), and

• π(a) ∈ K_{i_{v}}^{R} for v = 0,…, p.

Hence, supp (a − b) ⊆ E_{i_{v−1}}^{R} ∩ E_{i_{v}}^{R} (v = 1,…, p), and we obtain

ψ_{i}(a − b) = ψ_{i_{1}}(a − b) = … = ψ_{i_{p−1}}(a − b) = ψ_{j}(a − b) .

(ii) There are three types of P -linear relations between the generators r(…) and s(…) of R:

0 = \underline{z}^{d} r(a, b; c) − r(a, b; c + d) + r(a + c, b + c; d) ,
0 = r(b, c; d) − r(a, c; d) + r(a, b; d) − s(a + d, b + d, c + d) + \underline{z}^{d} s(a, b, c) ,
0 = s(b, c, d) − s(a, c, d) + s(a, b, d) − s(a, b, c) .

Our map respects them all.

(iii) Finally, the typical element (\underline{z}^{a} − \underline{z}^{b})e^{cd} − (\underline{z}^{c} − \underline{z}^{d})e^{ab} ∈ R_{0} equals

−r(c, d; a)+r(c, d; b)+r(a, b; c)−r(a, b; d)−s(a+c, b+c, a+d)+s(b+c, a+d, b+d) .

It will be sent to 0, too.

(3.6) The cup product T_{Y}^{1} × T_{Y}^{1} → T_{Y}^{2} respects the grading, i.e. it splits into pieces

T_{Y}^{1}(−R) × T_{Y}^{1}(−S) ⟶ T_{Y}^{2}(−R − S) (R, S ∈ M) .

To describe these maps in our combinatorial language, we choose a set-theoretical section Φ : M → ZZ^{w+1} of the ZZ-linear map

π : ZZ^{w+1} ⟶ M
a ↦ ∑_{v} a_{v} r^{v}

with the additional property Φ(σ^{∨} ∩ M) ⊆ IN^{w+1}.

Let q ∈ L(E) ⊆ ZZ^{w+1} be an integral relation between the generators of the semigroup σ^{∨} ∩ M. We introduce the following notations:

• q^{+}, q^{−} ∈ IN^{w+1} denote the positive and the negative part of q, respectively. (With other words: q = q^{+} − q^{−} and ∑_{v} q_{v}^{−} q_{v}^{+} = 0.)

• q̄:= π(q^{+}) = ∑_{v} q_{v}^{+} r^{v} = ∑_{v} q_{v}^{−} r^{v} = π(q^{−}) ∈ M.

• If φ, ψ : L(E) → ZZ are linear maps and R, S ∈ M, then we define

t_{φ,ψ,R,S}(q) := φ(q)·ψ (Φ(q̄ − R) + Φ(R) − q^{−})+ψ(q)·φ (Φ(q̄ − S) + Φ(S) − q^{+}) .

(For an easier description in the most important special cases see the upcoming Proposition (3.7).)

Theorem: Assume that Y is smooth in codimension 2. Let R, S ∈ M, and let φ, ψ : L(E)_{I C} → I C be linear maps vanishing on ∑_{i} L(E_{i}^{R})_{I C} and ∑_{i} L(E_{i}^{S})_{I C}, respectively. In particular, they define elements φ ∈ T_{Y}^{1}(−R), ψ ∈ T_{Y}^{1}(−S) (which involves a slight abuse of notations). Then, the cup product φ ∪ ψ ∈ T_{Y}^{2}(−R − S) is given (via (3.3)(3)) by the linear maps (φ ∪ ψ)_{i} : L(E_{i}^{R+S})_{I C} → I C defined as follows:

(i) If q ∈ L(E_{i}^{R+S}) (i.e. ⟨a^{i}, supp q⟩ < ⟨a^{i}, R + S⟩) is an integral relation, then there exists a decomposition q = ∑_{k} q^{k} such that

– q^{k} ∈ L(E_{i}^{R+S}), and moreover

– ⟨a^{i}, q̄^{k}⟩ < ⟨a^{i}, R + S⟩.

(ii) (φ ∪ ψ)_{i} (q ∈ L(E_{i}^{R+S})) := ∑_{k} t_{φ,ψ,R,S}(q^{k}).

First, we must show that the map q ↦ ∑_{k} t(q^{k})

• does not depend on the representation of q as a particular sum of q^{k}’s (which implies linearity on L(E_{i}^{R+S})), and

• yields the same result on L(E_{i}^{R+S} ∩E_{j}^{R+S}) for i, j corresponding to edges a^{i}, a^{j} of some 2-face of σ.

The proof of these facts (cf. (5.4)) and of the entire theorem is contained in §5.

Remark: Replacing all the terms Φ(•) in the t’s of the previous formula for (φ ∪ ψ)_{i}(q) by arbitrary liftings from M to ZZ^{w+1}, the result in T_{Y}^{2}(−R − S) will be unchanged as long as we obey the following two rules:

(i) Use always (for all q, q^{k}, and i) the same liftings of R and S to ZZ^{w+1} (at the places of Φ(R) and Φ(S), respectively).

(ii) Elements of σ^{∨} ∩ M always have to be lifted to IN^{w+1}.

Proof: Replacing Φ(R) by Φ(R)+d (d ∈ L(E)) at each occurence changes all maps (φ ∪ ψ)_{i} by the summand ψ(d) · φ(•). However, this additional linear map comes from L(E)^{∗}, hence it is trivial on Ker (⊕_{i}L(E_{i}^{R+S}) → L(E_{0}^{R+S}) ⊆ L(E)).

Let us look at the terms Φ(q̄−R) in t(q) now: Unless q̄ ≥ R, the factor φ(q) vanishes (cf. Remark (3.4)). On the other hand, the expression t(q) is never used for those relations q satisfying q̄ ≥ R + S (cf. conditions for the q^{k}’s). Hence, we may assume that

(q̄ − R) ≥ 0 and, moreover, (q̄ − R) ∈ ⋃_{i} K_{i}^{S} .

Now, each two liftings of (q̄− R) to IN^{w+1} differ by an element of Ker ψ only (apply the method of Remark (3.4) again), in particular, they cause the same result for t(q). □

(3.7) Proposition: In the special case of R ≥ S ≥ 0 we can choose liftings Φ(R) ≥ Φ(S) ≥ 0 in IN^{w+1}. Then, there exists an easier description for t(q):

(i) Unless q̄ ≥ R, we have t(q) = 0.

(ii) In case of q̄ ≥ R we may assume that q^{+} ≥ Φ(R) is true in IN^{w+1}. (The general q’s are differences of those ones.) Then, t can be computed as the product t(q) = φ(q) ψ(q).

Proof: (i) As used many times, the property q̄ ∈ ⋃_{i} E_{i}^{R} implies φ(q) = 0. Now, we can distinguish between two cases: Case 1: q̄ ∈ ⋃_{i} E_{i}^{S}. We obtain ψ(q) = 0, in particular, both summands of t(q) vanish. Case 2: q̄ ≥ S. Then, q̄− S, S ∈ σ^{∨} ∩ M, and Φ lifts these elements to IN^{w+1}. Now, the condition q̄ ∈ ⋃_{i} E_{i}^{R} implies that φ (Φ(q̄ − S) + Φ(S) − q^{+}) = 0.

(ii) We can choose Φ(q̄ − R) := q^{+} − Φ(R) and Φ(q̄ − S) := q^{+} − Φ(S). Then, the claim follows straight forward. □

4 Proof of the T^{2}-formula

(4.1) We will use the sheaf Ω_{Y}^{1} = Ω_{A| I C}^{1} of Kähler differentials for computing the modules T_{Y}^{i}. The maps

α_{i} : Ext_{A}^{i} ( Ω1 Y/ ; tors (Ω_{Y}^{1} ) , A) ↪ Ext_{A}^{i} (Ω_{Y}^{1} , A) ≅ T_{Y}^{i} (i = 1, 2)

are injective. Moreover, they are isomorphisms for • i = 1, since Y is normal, and for

• i = 2, if Y is smooth in codimension 2.

(4.2) As in [Al 1], we build a special A-free resolution (one step further now)

E ⟶^{d_{E}} D ⟶^{d_{D}} C ⟶^{d_{C}} B ⟶^{d_{B}} Ω1 Y/ ; tors (Ω_{Y}^{1} ) → 0 .

With L^{2}(E) := L(L(E)), L^{3}(E) := L(L^{2}(E)), and

supp^{2}ξ := ⋃_{q∈supp ξ} supp q (ξ ∈ L^{2}(E)), supp^{3}ω := ⋃_{ξ∈supp ω} supp^{2}ξ (ω ∈ L^{3}(E)),

the A-modules involved in this resolution are defined as follows:

\begin{aligned}B & := & ⊕_{r∈E} A · B(r), C := ⊕_{\begin{matrix}q∈L(E) \\ ℓ≥supp q \end{matrix}}A · C(q; ℓ), \\ D & := & (⊕_{\begin{matrix}q∈L(E) \\ η≥ℓ≥supp q \end{matrix}}A · D(q; ℓ, η)) ⊕ (⊕_{\begin{matrix}ξ∈L^{2}(E) \\ η≥supp^{2}ξ \end{matrix}}A · D(ξ; η)) , and \\ E & := & \begin{matrix}(⊕_{\begin{matrix}q∈L(E) \\ μ≥η≥ℓ≥supp q \end{matrix}}A · E(q; ℓ, η, μ)) ⊕ (⊕_{\begin{matrix}ξ∈L^{2}(E) \\ μ≥η≥supp^{2}ξ \end{matrix}}A · E(ξ; η, μ)) ⊕ \\ ⊕ (⊕_{\begin{matrix}ω∈L^{3}(E) \\ μ≥supp^{3}ω \end{matrix}}A · E(ω; μ)) \end{matrix} \end{aligned}

(B, C, D, and E are just symbols). The differentials equal

\begin{matrix}d_{B} : & B(r) & ↦ & d x^{r} ; \\ d_{C} : & C(q; ℓ) & ↦ & ∑_{r∈E} q_{r} x^{ℓ−r} · B(r) ; \\ d_{D} : & D(q; ℓ, η) & ↦ & C(q; η) − x^{η−ℓ} · C(q; ℓ) , \\ d_{D} : & D(ξ; η) & ↦ & ∑_{q∈L(E)} ξ_{q} · C(q; η) ; and \\ d_{E} : & E(q; ℓ, η, μ) & ↦ & D(q; η, μ) − D(q; ℓ, μ) + x^{μ−η} · D(q; ℓ, η) , \\ d_{E} : & E(ξ; η, μ) & ↦ & D(ξ; μ) − x^{μ−η} · D(ξ; η) − ∑_{q∈L(E)} ξ_{q} · D(q; η, μ) , \\ d_{E} : & E(ω; μ) & ↦ & ∑_{ξ∈L^{2}(E)} ω_{ξ} · D(ξ; μ) . \end{matrix}

Looking at these maps, we see that the complex is M-graded: The degree of each of the elements B, C, D, or E can be obtained by taking the last of its parameters (r, ℓ, η, or μ, respectively).

Remark: If one prefered a resolution with free A-modules of finite rank (as it was used in [Al 1]), the following replacements would be necessary:

(i) Define succesively F ⊆ L(E), G ⊆ L(F) ⊆ L^{2}(E), and H ⊆ L(G) ⊆ L^{2}(F) ⊆ L^{3}(E) as the finite sets of normalized, minimal relations between elements of E, F, or G, respectively. Then, use them instead of L^{i}(E) (i = 1, 2, 3).

(ii) Let ℓ, η, and μ run through finite systems generating (via the (σ^{∨} ∩M)-action) all possible elements meeting the desired inequalities.

The disadvantages of this treatment are a more comlplicated description of the resolution, on the one hand, and difficulties to obtain the commutative diagram (4.3), on the other hand.

(4.3) Combining the two exact sequences

R/I R ⟶ A^{m} ⟶I/I^{2} → 0 and I/I^{2} ⟶ Ω_{I C^{w+1}}^{1} ⊗ A ⟶ Ω_{Y}^{1} → 0 ,

we get a complex (not exact at the place of A^{m}) involving Ω_{Y}^{1} . In the following commutative diagram, we will compare this complex with the previous resolution of

Ω1 Y/ ; tors (Ω_{Y}^{1} ) : [figure: I/I^{2}; R/I R; A^{m}; Ω_{I C^{w+1}} ⊗A; Ω_{Y}; 0; Im d_{D}; E; D; C; B; Ω1 Y/ ; tors (Ω_{Y}^{1} ); 0; d; p_{D}; p_{C}; p_{B} ∼; d_{E}; d_{D}; d_{C}]

Let us explain the three labeled vertical maps:

(B) p_{B} : dz_{r} ↦ B(r) is an isomorphism between two free A-modules of rank w + 1.

(C) p_{C} : e^{ab} ↦ C(a − b; π(a)). In particular, the image of this map is spanned by those C(q, ℓ) meeting ℓ ≥ q̄ (which is stronger than just ℓ ≥ supp q).

(D) Finally, p_{D} arises as pull back of p_{C} to R/I R. It can be described by r(a, b; c) ↦ D(a − b; π(a), π(a + c)) and s(a, b, c) ↦ D(ξ; π(a)) (ξ denotes the relation ξ = [(b − c) − (a − c) + (a − b) = 0]).

Remark: Starting with the typical R_{0}-element mentioned in (3.5)(iii), the previous description of the map p_{D} yields 0 (even in D).

(4.4) By (4.1) we get the A-modules T_{Y}^{i} by computing the cohomology of the complex dual to that of (4.2).

As in [Al 1], denote by G one of the capital letters B, C, D, or E. Then, an element ψ of the dual free module (⨁_{G} I C[σ̌ ∩ M] · G)^{∗} can be described by giving elements

g(x) ∈ I C[σ̌ ∩ M] to be the images of the generators G (g stands for b, c, d, or e, respectively).

For ψ to be homogeneous of degree −R ∈ M, g(x) has to be a monomial of degree

deg g(x) = −R + deg G.

In particular, the corresponding complex coefficient g ∈ I C (i.e. g(x) = g · x^{−R+deg G}) admits the property that

g ≠ 0 implies − R + deg G ≥ 0 (i.e. − R + deg G ∈ σ̌).

Remark: Using these notations, Theorem (3.3)(1) was proved in [Al 1] by showing that

(^{L(ER 0)/}∑_{i} L(E_{i}^{R}))^{∗} ⊗_{ZZ} I C ⟶ Ker(C{∗ \atop −R} → D{∗ \atop −R})/ ; Im(B_{−R}^{∗} → C_{−R}^{∗})

φ ↦ […, c(q; ℓ) := φ(q),…]

is an isomorphism of vector spaces. Moreover, looking at the diagram of (4.3), e^{ab} ∈ A^{m} maps to both \underline{z}^{a} − \underline{z}^{b} ∈I/I^{2} and C(a−b; π(a)) ∈ C. In particular, we can verify Theorem (3.4): Each φ : L(E)_{I C} → I C induces the same element of T_{Y}^{1} as its associated A-linear map

I/I^{2} ⟶ A
\underline{z}^{a} − \underline{z}^{b} ↦ φ(a − b) · x^{π(a)−R},

on the other hand, induce the same element of T_{Y}^{1}(−R).

(4.5) For computing T_{Y}^{2}(−R), the interesting part of the dualized complex (4.2)^{∗} in degree −R equals the complex of I C-vector spaces

C_{−R}^{∗} ⟶^{d_{D}^{∗}} D_{−R}^{∗} ⟶^{d_{E}^{∗}} E_{−R}^{∗}

with coordinates \underline{c}, \underline{d}, and \underline{e}, respectively:

C_{−R}^{∗} = \{\underline{c(q; ℓ)} | c(q; ℓ) = 0 for ℓ − R ∉ σ̌\}
D_{−R}^{∗} = \{[\underline{d(q; ℓ, η)}, \underline{d(ξ; η)}] | \begin{matrix}d(q; ℓ, η) & = & 0 & for & η − R ∉ σ̌, and \\ d(ξ; η) & = & 0 & for & η − R ∉ σ̌\} \end{matrix}
E_{−R}^{∗} = \{[\underline{e(q; ℓ, η, μ)}, \underline{e(ξ; η, μ)}, \underline{e(ω; μ)}] | each coordinate vanishes for μ − R ∉ σ̌\}.

The differentials d_{D}^{∗} and d_{E}^{∗} can be described by

d(q; ℓ, η) = c(q; η) − c(q; ℓ),
d(ξ; η) = ∑_{q∈L(E)} ξ_{q} · c(q; η), and
e(q; ℓ, η, μ) = d(q; η, μ) − d(q; ℓ, μ) + d(q; ℓ, η),
e(ξ; η, μ) = d(ξ; μ) − d(ξ; η) − ∑_{q∈F} ξ_{q} · d(q; η, μ),
e(ω; μ)  = ∑_{ξ∈L^{2}(E)} ω_{ξ} · d(ξ; μ).

Denote V := Ker d_{E}^{∗} ⊆ D_{−R}^{∗} and W := Im d_{D}^{∗} ⊆ V, i.e.

V = \{[\underline{d(q; ℓ, η)}; \underline{d(ξ; η)}] | \begin{matrix}q ∈ L(E), η ≥ ℓ ≥ supp q in M; \\ ξ ∈ L^{2}(E), η ≥ supp^{2}ξ; \\ d(q; ℓ, η) = d(ξ; η) = 0 for η − R ∉ σ̌, \\ d(q; ℓ, μ) = d(q; ℓ, η) + d(q; η, μ) (μ ≥ η ≥ ℓ ≥ supp q), \\ d(ξ; μ) = d(ξ; η) + ∑_{q} ξ_{q} · d(q; η, μ) (μ ≥ η ≥ supp^{2}ξ), \\ ∑_{ξ∈L^{2}(E)} ω_{ξ} d(ξ; μ) = 0 for ω ∈ L^{3}(E) with μ ≥ supp^{3}ω \}, \end{matrix}

W = \{[\underline{d(q; ℓ, η)}; \underline{d(ξ; η)}] | there are c(q; ℓ)’s with \begin{matrix}c(q, ℓ) = 0 for ℓ − R ∉ σ̌, \\ d(q; ℓ, η) = c(q; η) − c(q; ℓ), \\ d(ξ; η) = ∑_{q} ξ_{q} · c(q; η) \}. \end{matrix}

By construction, we obtain

V / ; W = Ext_{A}^{i} ( Ω1 Y/ ; tors (Ω_{Y}^{1} ) , A) (−R) ⊆ T_{Y}^{2}(−R)

(which is an isomorphism, if Y is smooth in codimension 2). (4.6) Let us define the much easier vector spaces

V_{1} := \{[\underline{x_{i}(q)}_{(q∈L(E_{i}^{R}))}] | \begin{matrix}x_{i}(q) = x_{j}(q) for • ⟨a^{i}, a^{j}⟩ < σ is a 2-face and • q ∈ L(E_{i}^{R} ∩ E_{j}^{R}) , \\ ξ ∈ L^{2}(E_{i}^{R}) implies ∑_{q} ξ_{q} · x_{i}(q) = 0 \} and \end{matrix}
W_{1} := \{[\underline{x(q)}_{(q∈∪_{i}L(E_{i}^{R}))}] | ξ ∈ L(⋃_{i} L(E_{i}^{R})) implies ∑_{q} ξ_{q} · x(q) = 0 \}.

Lemma: The linear map V_{1} → V defined by

d(q; ℓ, η) := \{ \begin{matrix}x_{i}(q) & for & ℓ ∈ K_{i}^{R}, η ≥ R \\ 0 & for & ℓ ≥ R or η ∈ ⋃_{i} K_{i}^{R}; \end{matrix}
d(ξ; η) := 0

induces an injective map

V_{1}/ ; W_{1} ↪ V / ; W .

If Y is smooth in codimension 2, it will be an isomorphism.

Proof: 1) The map V_{1} → V is correct defined: On the one hand, an argument as used in (3.5)(i) shows that ℓ ∈ K_{i}^{R} ∩ K_{j}^{R} would imply x_{i}(q) = x_{j}(q). On the other hand, the image of [x_{i}(q)]_{q∈L(E_{i}^{R})} meets all conditions in the definition of V.

2) W_{1} maps to W (take c(q, ℓ) := x(q) for ℓ ≥ R and c(q, ℓ) := 0 otherwise).

3) The map between the two factor spaces is injective: Assume for [x_{i}(q)]_{q∈L(E_{i}^{R})} that there exist elements c(q, ℓ), such that

c(q; ℓ) = 0 for ℓ ∈ ⋃_{i} K_{i}^{R} ,
x_{i}(q) = c(q; η) − c(q; ℓ) for η ≥ ℓ, ℓ ∈ K_{i}^{R}, η ≥ R ,
0 = c(q; η) − c(q; ℓ) for η ≥ ℓ and [ℓ ≥ R or η ∈ ⋃_{i} K_{i}^{R}] , and
0 = ∑_{q} ξ_{q} · c(q; η) for η ≥ supp^{2}ξ .

In particular, x_{i}(q) do not depend on i, and these elements have the property

∑_{q} ξ_{q} · x_{•}(q) = 0 for ξ ∈ L(⋃_{i} L(E_{i}^{R})).

4) If Y is smooth in codimension 2, the map is surjective : Given an element [d(q; ℓ, η), d(ξ; η)] ∈ V, there exist complex numbers c(q; η) such that:

(i) d(ξ; η) = ∑_{q} ξ_{q} · c(q; η) ,

(ii) c(q; η) = 0 for η ∉ R + σ^{∨} (i.e. η ∈ ⋃_{i} K_{i}^{R}) .

(Do this separately for each η and distinguish between the cases η ∈ R + σ^{∨} and η ∉ R + σ^{∨}.) In particular, [c(q; η) − c(q; ℓ), d(ξ; η)] ∈ W. Hence, we have seen that we may assume d(ξ; η) = 0.

Let us choose some sufficiently high degree ℓ^{∗} ≥ E. Then,

x_{i}(q) := d(q; ℓ, η) − d(q; ℓ^{∗}, η)

(with ℓ ∈ K_{i}^{R}, ℓ ≥ supp q (cf. Lemma (3.2)(2)), and η ≥ ℓ, ℓ^{∗}, R) defines some preimage:

(i) It is independent from the choice of η: Using a different η′ generates the difference d(q; η, η′) − d(q; η, η′).

(ii) It is independent from ℓ ∈ K_{i}^{R}: Choosing another ℓ′ ∈ K_{i}^{R} with ℓ′ ≥ ℓ would add the summand d(q; ℓ, ℓ′), which is 0; for the general case use Lemma (3.2)(2).

(iii) If ⟨a^{i}, a^{j}⟩ < σ is a 2-face with supp q ⊆ L(E_{i}^{R}) ∩ L(E_{j}^{R}), then by Lemma (3.2)(2) we can choose an ℓ ∈ K_{i}^{R} ∩ K_{j}^{R} achieving x_{i}(q) = x_{j}(q).

(iv) For ξ ∈ L^{2}(E_{i}^{R}) we have

∑_{q} ξ_{q} · d(q; ℓ, η) = ∑_{q} ξ_{q} · d(q; ℓ^{∗}, η) = 0 ,

and this gives the corresponding relation for the x_{i}(q)′s.

(v) Finally, if we apply to [\underline{x_{i}(q)}] ∈ V_{1} the linear map V_{1} → V, the result differs from [d(q; ℓ, η), 0] ∈ V by the W -element built from

c(q; ℓ) := \{ \begin{matrix}d(q; ℓ, η) − d(q; ℓ^{∗}, η) & if ℓ ≥ R \\ 0 & otherwise . \end{matrix}

□

(4.7) Now, it is easy to complete the proofs for Theorem (3.3) (part 2 and 3) and Theorem (3.5):

First, for a tuple [\underline{x_{i}(q)}]_{q∈L(E_{i}^{R})}, the condition

ξ ∈ L^{2}(E_{i}^{R}) implies ∑_{q} ξ_{q} · x_{i}(q) = 0

is equivalent to the fact the components x_{i}(q) are induced by elements x_{i} ∈ L(E_{i}^{R})_{I C}^{∗}. The other condition for elements of V_{1} just says that for 2-faces ⟨a^{i}, a^{j}⟩ < σ there is x_{i} = x_{j} on L(E_{i}^{R} ∩ E_{j}^{R})_{I C} = L(E_{i}^{R})_{I C} ∩ L(E_{j}^{R})_{I C}. In particular, we obtain

V_{1} = Ker (⊕_{i}L(E_{i}^{R})_{I C}^{∗} → ⊕_{⟨a^{i},a^{j}⟩<σ}L(E_{i}^{R} ∩ E_{j}^{R})_{I C}^{∗}) .

In the same way we get

W_{1} = (∑_{i} L(E_{i}^{R})_{I C})^{∗} ,

and our T^{2}-formula is proven.

Finally, if ψ_{i} : L(E_{i}^{R})_{I C} → I C are linear maps defining an element of V_{1}, they induce the following A-linear map on D (even on Im d_{D}):

D(q; ℓ, η) ↦ \{ \begin{matrix}ψ_{i}(q) · x^{η−R} & for & ℓ ∈ K_{i}^{R}, η ≥ R \\ 0 & for & ℓ ≥ R or η ∈ ⋃_{i} K_{i}^{R} \end{matrix}
D(ξ; η) ↦ 0 .

Now, looking at the diagram of (4.3), this translates exactly into the claim of Theorem (3.5).

5 Proof of the cup product formula

(5.1) Fix an R ∈ M, and let φ ∈ L(E)_{I C}^{∗} represent an element (also denoted by φ) of T_{Y}^{1}(−R). Using the notations of (2.3), (3.4), and (3.6) we can take

φ̃(f_{αβ}) := φ(α − β) · \underline{z}^{Φ(π(α)−R)}

for the auxiliary P -elements needed to compute the λ(φ)’s (cf. Theorem (3.4)).

Now, we have to distinguish between the two several types of relations generating the P -module R ⊆ P^{m}:

(r) Regarding the relation r(a, b; c) we obtain

∑_{(α,β)∈m} r(a, b; c)_{αβ} · φ̃(f_{αβ}) = φ̃(f_{a+c,b+c}) − \underline{z}^{c} φ̃(f_{ab})
= φ(a − b) · (\underline{z}^{Φ(π(a+c)−R)} − \underline{z}^{c+Φ(π(a)−R)})
= φ(a − b) · f_{Φ(π(a+c)−R), c+Φ(π(a)−R)} .

In particular,

λ_{αβ}^{r(a,b;c)}(φ) = \{ \begin{matrix}φ(a − b) & for [α, β] = [c + Φ(π(a) − R), Φ(π(a + c) − R)] \\ 0 & otherwise . \end{matrix}

(s) The corresponding result for the relation s(a, b, c) is much nicer:

∑_{(α,β)∈m} s(a, b, c)_{αβ} · φ̃(f_{αβ}) = φ̃(f_{bc}) − φ̃(f_{ac}) + φ̃(f_{ab})
= [φ(b − c) − φ(a − c) + φ(a − b)] · \underline{z}^{Φ(π(a)−R)}
= 0 .

In particular, λ^{s(a,b,c)}(φ) = 0.

(5.2) Now, let R, S, φ, and ψ as in the assumption of Theorem (3.6). Using formula (2.3)(iii), our previous computations yield (φ ∪ ψ)(s(a, b, c)) = 0 and

(φ ∪ ψ)(r(a, b; c)) = ∑_{α,β} λ_{αβ}^{r(a,b;c)}(φ) · ψ(f_{αβ}) + ∑_{α,β} φ(f_{αβ}) · λ_{αβ}^{r(a,b;c)}(ψ)

= \begin{matrix}φ(a − b) · ψ (c + Φ(π(a) − R) − Φ(π(a + c) − R)) · x^{π(c+Φ(π(a)−R))−S}+ \\ +ψ(a − b) · φ (c + Φ(π(a) − S) − Φ(π(a + c) − S)) · x^{π(c+Φ(π(a)−S))−R} \end{matrix}

= \begin{matrix}[φ(a − b) · ψ (c + Φ(π(a) − R) − Φ(π(a + c) − R)) + \\ +ψ(a − b) · φ (c + Φ(π(a) − S) − Φ(π(a + c) − S))] · x^{π(a+c)−R−S} . \end{matrix}

Remark: Unless π(a + c) ≥ R + S, both summand in the brackets will vanish. For instance, on the one hand, π(a) ∈ ⋃_{i} K_{i}^{R} would cause φ(a − b) = 0, and, on the other hand, π(a) − R ≥ 0 and π(c + Φ(π(a) − R)) ∈ ⋃_{i} K_{i}^{S} imply ψ(c + Φ(π(a) − R) − Φ(π(a + c) − R)) = 0.

To apply Theorem (3.5) we would like to remove the argument c from the big coefficient. This will be done by adding a suitable coboundary T.

(5.3) Let us start with defining for (α, β) ∈ m

t(α, β) := \begin{matrix}φ(α − β) · ψ (Φ(π(α) − R) + Φ(R) − β) + \\ +ψ(α − β) · φ (Φ(π(α) − S) + Φ(S) − α) . \end{matrix}

(This expression is related to t_{φ,ψ,R,S} from (3.6) by t(q) = t(q^{+}, q^{−}).)

Lemma:Let α, β, γ ∈ IN^{E} with π(α) = π(β) = π(γ).

(1) t(α, β) = t(α − β) as long as π(α) ∈ ⋃_{i} K_{i}^{R+S}.

(2) t(β, γ) − t(α, γ) + t(α, β) = 0.

Proof: (1) It is enough to show that t(α + r, β + r) = t(α, β) for r ∈ IN^{E}, π(α + r) ∈ ⋃_{i} K_{i}^{R+S}. But the difference of these two terms has exactly the shape of the coefficient of x^{π(a+c)−R−S} in (5.2). In particular, the argument given in the previous remark applies again.

(2) By extending φ and ψ to linear maps I C^{E} → I C, we obtain

t(α, β) = \begin{matrix}[φ(α − β) ψ (Φ(π(α) − R) + Φ(R)) + ψ(α − β) φ (Φ(π(α) − S) + Φ(S))]+ \\ +[φ(β) ψ(β) − φ(α) ψ(α)]. \end{matrix}

Now, since π(α) = π(β) = π(γ), both types of summands add up to 0 separately in t(β, γ) − t(α, γ) + t(α, β). □

Remark: The previous lemma does not imply that t(q) is ZZ-linear in q. The assumption for π(α) made in (1) is really essential.

Now, we obtain a P -linear map T ∈ Hom(P^{m}, A) by

T : e^{αβ} ↦ \{ \begin{matrix}t(α, β) x^{π(α)−R−S} & for π(α) ≥ R + S \\ 0 & otherwise . \end{matrix}

Pulling back T to R ⊆ P^{m} yields (in case of π(a + c) ≥ R + S)

T (r(a, b; c)) = \{ \begin{matrix}[t(a + c, b + c) − t(a, b)] · x^{π(a+c)−R−S} & for π(a) ≥ R + S \\ t(a + c, b + c) · x^{π(a+c)−R−S} & otherwise \end{matrix}
= \{ \begin{matrix}−(φ ∪ ψ)(r(a, b; c)) & for π(a) ≥ R + S \\ t(a, b) x^{π(a+c)−R−S} − (φ ∪ ψ)(r(a, b; c)) & otherwise \end{matrix}

and T (s(a, b, c)) = 0 (by (2) of the previous lemma).

On the other hand, T yields a trivial element of T_{Y}^{2}(−R − S), i.e. inside this group we may replace φ ∪ ψ by (φ ∪ ψ) + T to obtain

(φ ∪ ψ)(r(a, b; c)) = \{ \begin{matrix}t(a, b) · x^{π(a+c)−R−S} & for π(a) ∈ ⋃_{i} K_{i}^{R+S}; π(a + c) ≥ R + S \\ 0 & otherwise , \end{matrix}
(φ ∪ ψ)(s(a, b, c)) = 0 .

Having Theorem (3.5) in mind, this formula for φ∪ψ is exactly what we were looking for: Given an r(a, b; c) with π(a) ∈ K_{i}^{R+S}, let us compute (φ ∪ ψ)_{i}(q := a − b) following the recipe of (i), (ii) of Theorem (3.6): We do not need to split q = a − b into a sum q = ∑_{k} q^{k} - the element q itself already satisfies the condition

⟨a^{i}, q̄⟩ ≤ ⟨a^{i}, π(a)⟩ < ⟨a^{i}, R + S⟩.

In particular, with (φ ∪ ψ)_{i}(a − b) = t(a − b) = t(a, b) we obtain the right result - if the recipe is assumed to be correct.

(5.4) We will fill this remaining gap now, i.e. we will show that

(a) each q ∈ L(E_{i}^{R+S}) admits a decomposition q = ∑_{k} q^{k} with the desired properties,

(b) ∑_{k} q^{k} = 0 (with q̄^{k} ∈ K_{i}^{R+S}) implies ∑_{k} t(q^{k}) = 0, and

(c) for adjacent a^{i}, a^{j} the relations q ∈ L(E_{i}^{R+S} ∩ E_{j}^{R+S}) admit a decomposition q = ∑_{k} q^{k} that works for both i and j.

(In particular, this answers the questions rised right after stating the theorem in (3.6).)

Let us fix an element i ∈ {1,…, N}. Since σ^{∨} ∩ M contains elements r with ⟨a^{i}, r⟩ = 1, some of them must be contained in the generating set E, too. We choose one of these elements and call it r(i). Now, to each r ∈ E we associate some relation p(r) ∈ L(E) via

p(r) := e^{r} − ⟨a^{i}, r⟩ · e^{r(i)} + [suitable element of ZZ^{E∩(a^{i})^{⊥}}] .

The two essential properties of these special relations are

(i) ⟨a^{i}, p̄(r)⟩ = ⟨a^{i}, r⟩, and

(ii) if q ∈ L(E) is any relation, then q and ∑_{r∈E} q_{r} · p(r) differ by some element of L(E ∩ (a^{i})^{⊥}) only.

In particular, this proves (a). For (b) we start with the following

Claim: Let q^{k} ∈ L(E) be relations such that ∑_{k}⟨a^{i}, q̄^{k}⟩ < ⟨a^{i}, R + S⟩. Then, ∑_{k} t(q^{k}) = t(∑_{k} q^{k}).

Proof: We can restrict ourselves to the case of two summands, q^{1} and q^{2}. Then, by Lemma (5.3),

t(q^{1}) + t(q^{2}) = t ((q^{1})^{+}, (q^{1})^{−}) + t ((q^{2})^{+}, (q^{2})^{−})
= t ((q^{1})^{+} + (q^{2})^{+}, (q^{1})^{−} + (q^{2})^{+}) + t ((q^{2})^{+} + (q^{1})^{−}, (q^{2})^{−} + (q^{1})^{−})
= t ((q^{1})^{+} + (q^{2})^{+}, (q^{2})^{−} + (q^{1})^{−})
= t(q^{1} + q^{2}) .  □

In particular, if ∑_{k} q^{k} = 0 (with q̄^{k} ∈ K_{i}^{R+S}), then this applies for the special decompositions

q^{k} = ∑_{r} q_{r}^{k} · p(r) + q^{0,k} (q^{0,k} ∈ L(E ∩ (a^{i})^{⊥}))

of the summands q^{k} themselves. We obtain

∑_{q_{r}^{k}>0} q_{r}^{k} · t (p(r)) + t(q^{0,k}) = t (∑_{q_{r}^{k}>0} q_{r}^{k} p(r) + q^{0,k}) =: t(q^{1,k})

and

∑_{q_{r}^{k}<0} q_{r}^{k} · t (p(r)) = t (∑_{q_{r}^{k}<0} q_{r}^{k} p(r)) =: t(q^{2,k}) .

Up to elements of E ∩(a^{i})^{⊥}, the relations q^{1,k} and q^{2,k} are connected by the common

(q^{1,k})^{−} = −q_{r(i)}^{1,k} · e^{r(i)} = ⟨a^{i}, q̄^{k}⟩ · e^{r(i)} = q_{r(i)}^{2,k} · e^{r(i)} = (q^{2,k})^{+} .

Hence, Lemma (5.3) yields

∑_{r} q_{r}^{k} · t (p(r)) + t(q^{0,k}) = t(q^{1,k}) + t(q^{2,k}) = t (q^{1,k} + q^{2,k}) = t(q^{k}) ,

and we conclude

∑_{k} t(q^{k}) = ∑_{k} (∑_{r} q_{r}^{k} · t (p(r)) + t(q^{0,k}))

= ∑_{r} (∑_{k} q_{r}^{k}) t (p(r)) + t (∑_{k} q^{0,k}) (cf. previous claim)

= 0 + t (∑_{k} q^{k} − ∑_{k,r} q_{r}^{k} p(r))
= 0 .

Finally, only (c) is left. Let a^{i}, a^{j} be two adjacent edges of σ. We adapt the construction of the elementary relations p(r). Instead of the r(i)’s, we will use elements r(i, j) ∈ E characterized by the property

⟨a^{i}, r(i, j)⟩ = 1 , ⟨a^{j}, r(i, j)⟩ = 0 .

(Those elements exist, since Y is assumed to be smooth in codimension 2.) Now, we define

p(r) := e^{r} − ⟨a^{i}, r⟩ · e^{r(i,j)} − ⟨a^{j}, r⟩ · e^{r(j,i)} + [suitable element of ZZ^{E∩(a^{i})^{⊥}∩(a^{j})^{⊥}}] .

These special p(r)’s meet the usual properties (i) and (ii) - but for the two different indices i and j at the same time. In particular, if q ∈ L(E) is any relation, then q and ∑_{r∈E} q_{r} · p(r) differ by some element of L(E ∩ (a^{i})^{⊥} ∩ (a^{j})^{⊥}) only.

6 An alternative to the complex L(E^{R})_{•}

(6.1) Let R ∈ M be fixed for the whole §6. The complex L(E^{R})_{•} introduced in (3.2) fits naturally into the exact sequence

0 → L(E^{R})_{•} ⟶ (ZZ^{E^{R}})_{•} ⟶ span(E^{R})_{•} → 0

of complexes built in the same way as L(E^{R})_{•}, i.e.

(ZZ^{E^{R}})_{−k} := ⊕ \begin{matrix}τ<σ \\ dim τ=k \end{matrix}ZZ^{E_{τ}^{R}} and span(E^{R})_{−k} := ⊕ \begin{matrix}τ<σ \\ dim τ=k \end{matrix}span(E_{τ}^{R}) .

Lemma: The complex (ZZ^{E^{R}})_{•} is exact.

Proof: The complex (ZZ^{E^{R}})_{•} can be decomposed into a direct sum

(ZZ^{E^{R}})_{•} = ⨁_{r∈M}(ZZ^{E^{R}})(r)_{•}

showing the contribution of each r ∈ M. The complexes occuring as summands are defined as

(ZZ^{E^{R}})(r)_{−k} := ⊕ \begin{matrix}τ<σ \\ dim τ=k \end{matrix}\{ \begin{matrix}ZZ = ZZ^{\{r\}} & for r ∈ E_{τ}^{R} \\ 0 & otherwise \end{matrix} \}
= ZZ^{#{τ | dim τ=k; r∈E_{τ}^{R}}} .

Denote by H^{+} the halfspace

H^{+} := {a ∈ N_{IR} | ⟨a, r⟩ < ⟨a, R⟩} ⊆ N_{IR}.

Then, for τ ≠ 0, the fact that r ∈ E_{τ}^{R} is equivalent to τ \ {0} ⊆ H^{+}. On the other hand, r ∈ E_{0}^{R} corresponds to the condition σ ∩ H^{+} ≠ ∅. In particular, (ZZ^{E^{R}})(r)_{•}, shifted by one place, equals the complex for computing the reduced homolgy of the topological space ∪{τ | τ \ {0} ⊆ H^{+}} ⊆ σ cut by some affine hyperplane. Since this space is contractable, the complex is exact. □

Corollary: The complexes L(E^{R})_{•}^{∗} and span(E^{R})_{•}^{∗}[1] are quasiisomorphic. In particular, under the usual assumptions (cf. Theorem (3.3)), we obtain

T_{Y}^{i} (−R) = H^{i} (span(E^{R})_{•}^{∗} ⊗_{ZZ} I C) .

(6.2) We define the IR-vector spaces

V_{i}^{R} := span_{IR}(E_{i}^{R}) = \{ \begin{matrix}0  for & ⟨a^{i}, R⟩ ≤ 0 \\ [a^{i} = 0] ⊆ M_{IR} for & ⟨a^{i}, R⟩ = 1 \\ M_{IR} = IR^{n}  for & ⟨a^{i}, R⟩ ≥ 2 \end{matrix}
(i = 1,…, N), and
V_{τ}^{R} := ∩_{a^{i}∈τ}V_{i}^{R} ⊇ span_{IR}(E_{τ}^{R}) (for faces τ < σ) .

Proposition: With V_{−k}^{R} := ⊕ \begin{matrix}τ<σ \\ dim τ=k \end{matrix}V_{τ}^{R} we obtain a complex V_{•}^{R} ⊇ span_{IR}(E^{R})_{•}.

Moreover, if Y is smooth in codimension k, then both complexes are equal at ≥(−k).

Proof: V_{τ}^{R} = span_{IR}(E_{τ}^{R}) is true for smooth cones τ < σ (cf.(3.7) of [Al 1]). □

Corollary:

(1) If Y is smooth in codimension 2, then T_{Y}^{1}(−R) = H^{1} ((V_{•}^{R})^{∗} ⊗_{IR} I C).

(2) If Y is smooth in codimension 3, then T_{Y}^{2}(−R) = H^{2} ((V_{•}^{R})^{∗} ⊗_{IR} I C).

The formula (1) for T_{Y}^{1} (with a more boring proof) was already obtained in (4.4) of [Al 1].

7 3-dimensional Gorenstein singularities

(7.1) We want to apply the previous results for the special case of an isolated, 3-dimensional, toric Gorenstein singularity. We start with fixing the notations.

Let Q = conv(a^{1},…, a^{N}) ⊆ IR^{2} be a lattice polygon with primitive edges

d^{i} := a^{i+1} − a^{i} ∈ ZZ^{2} .

Embedding IR^{2} as the affine hyperplane [a_{3} = 1] into N_{IR} := IR^{3}, we can define the cone

σ := Cone(Q) ⊆ N_{IR} .

The fundamental generators of σ equal the vectors (a^{1}, 1),…, (a^{N}, 1), which we will also denote by a^{1},…, a^{N}, respectively.

The vector space M_{IR} contains a special element R^{∗} := [0, 0; 1]:

• ⟨•, R^{∗}⟩ = 1 defines the affine hyperplane containing Q,

• ⟨•, R^{∗}⟩ = 0 describes the vectorspace containing the edges d^{i} of Q.

The structure of the dual cone σ^{∨} can be described as follows:

• [c; η] ∈ M_{IR} is contained in σ^{∨}, iff ⟨Q, −c⟩ ≤ η.

• [c; η] ∈ ∂σ^{∨} iff there exists some i with ⟨a^{i}, −c⟩ = η.

• The set E contains R^{∗}. However, E \ {R^{∗}} ⊆ ∂σ^{∨}.

Remark: The toric variety Y built by the cone σ is 3-dimensional, Gorenstein, and regular outside its 0-dimensional orbit. Moreover, all those singularities can be obtained that way.

(7.2) Let V denote the (N − 2)-dimensional IR-vector space

V := {(t_{1},…, t_{N}) | ∑_{i} t_{i} d^{i} = 0} ⊆ IR^{N} .

The non-negative tuples among the \underline{t} ∈ V describe the set of Minkowski summands Q\underline{t} of positive multiples of the polygon Q. (t_{i} is the scalar by which d^{i} has to be multiplied to get the i-th edge of Q\underline{t}.)

We consider the bilinear map

\begin{matrix}V & × & IR^{E} & ⟶^{Ψ} & IR \\ \underline{t} & , & [c; η] ∈ E & ↦ & \{ \begin{matrix}0 & if c = 0 (i.e. [c; η] = R^{∗}) \\ ∑_{v=1}^{i−1} t_{v} · ⟨d^{v}, −c⟩ & if ⟨a^{i}, −c⟩ = η . \end{matrix} \end{matrix}

Assuming both a^{1} and the associated vertices of all Minkowski summands Q\underline{t} to coincide with 0 ∈ IR^{2}, the map Ψ detects the maximal values of the linear functions c on these summands

Ψ(\underline{t}, [c; η]) = Max (⟨a, −c⟩ | a ∈ Q\underline{t}) .

In particular, Ψ(\underline{1}, [c; η]) = η, i.e. Ψ induces a map

Ψ : V/IR · \underline{1} × L_{IR}(E) ⟶ IR .

The results of [Al 2] and [Sm] imply that Ψ provides an isomorphism

V^{I C}/I C · \underline{1} ⟶^{∼} (^{L(E{R^{∗} \atop 0})/}∑_{i} L(E_{i}^{R^{∗}}))^{∗} ⊗_{ZZ} I C ≅ T_{Y}^{1}(−R^{∗}) = T_{Y}^{1} .

In particular, dim T_{Y}^{1} = N − 3.

(7.3) Let R ∈ M. Combining the general results of §6 with the fact

⋂_{i} V_{i}^{R} = Ker [⊕_{i}(V_{i}^{R} ∩ V_{i+1}^{R}) ⟶ ⊕_{i}V_{i}^{R}] ,

we obtain the handsome formula

T_{Y}^{2}(−R) = [⋂_{i}(span_{I C}E_{i}^{R}) ; /span_{I C}(⋂_{i} E_{i}^{R})]^{∗} .

T_{Y}^{1} is concentrated in the degree −R^{∗}. Hence, for computing T_{Y}^{2}, the degrees −kR^{∗} (k ≥ 2) are the most interesting (but not only) ones. In this special case, the vector spaces V_{i}^{kR^{∗}} equal M_{IR}, i.e.

T_{Y}^{2}(−kR^{∗}) = [M^{I C}/span_{I C}(⋂_{i} E_{i}^{kR^{∗}})]^{∗} ⊆ [M^{I C}/I C · R^{∗}]^{∗} = span_{I C}(d^{1},…, d^{N}) ⊆ N_{I C} .

Proposition: For c ∈ IR^{2} denote by

d(c) := Max (⟨a^{i}, c⟩ | i = 1,…, N) − Min (⟨a^{i}, c⟩ | i = 1,…, N)

the diameter of Q in c-direction. If

k_{1} := Min_{c∈ZZ^{2}\0}d(c) and k_{2} := \begin{matrix}Min \\ c,c′∈ZZ^{2} \\ lin. indept. \end{matrix}Max [d(c), d(c′)] ,

then dim T_{Y}^{2}(−kR^{∗}) = 2 for 2 ≤ k ≤ k_{1} , dim T_{Y}^{2}(−kR^{∗}) = 1 for k_{1} + 1 ≤ k ≤ k_{2} , and dim T_{Y}^{2}(−kR^{∗}) = 0 for k_{2} + 1 ≤ k .

Proof: We have to determine the dimension of span_{I C} (⋂_{i} E_{i}^{kR^{∗}})/ ; I C · R^{∗}. Computing modulo R^{∗} simply means to forget the η in [c; η] ∈ M. Hence, we are done by the following observation for each c ∈ ZZ^{2} \ 0:

\begin{aligned}∃η ∈ ZZ : [c, η] ∈ ⋂_{i} K_{i}^{kR^{∗}} & ⟺ & ∃η ∈ ZZ : (k − 1)R^{∗} ≥ [c; η] ≥ 0 \\ & ⟺ & d(c) ≤ k − 1 . \end{aligned}

□

Corollary: Unless Y = I C^{3} or Y = cone over IP^{1} × IP^{1}, we have

T_{Y}^{2}(−2R^{∗}) = span_{I C}(d^{1},…, d^{N}),

i.e. dim T_{Y}^{2}(−2R^{∗}) = 2.

(7.4) Proposition: Using both the isomorphism V_{I C}/ ; I C · \underline{1} →^{∼} T_{Y}^{1} and the injection T_{Y}^{2}(−2R^{∗}) ↪ span_{I C}(d^{1},…, d^{N}), the cup product T_{Y}^{1} × T_{Y}^{1} → T_{Y}^{2} equals the bilinear map

\begin{matrix}V_{I C}/ ; I C · \underline{1} & × & V_{I C}/ ; I C · \underline{1} & ⟶ & span_{I C}(d^{1},…, d^{N}) \\ \underline{s} & , & \underline{t} & ↦ & ∑_{i} s_{i} t_{i} d^{i} . \end{matrix}

Proof: Step 1: To apply Theorem (3.6) we will combine the isomorphisms for T_{Y}^{2} presented in §6 and (7.3). Actually, we will describe the dual map by associating to each r ∈ M an element [q^{1}(r),…, q^{N}(r)] ∈ ⊕_{i}L(E_{i}^{2R^{∗}}). First, for every i = 1,…, N, we have to write r ∈ M = (span E_{i}^{2R^{∗}}) ∩ (span E_{i+1}^{2R^{∗}}) as a linear combination of elements from E_{i}^{2R^{∗}} ∩ E_{i+1}^{2R^{∗}}. This set contains a ZZ-basis for M consisting of

• r^{i} := primitive element of σ^{∨} ∩ (a^{i})^{⊥} ∩ (a^{i+1})^{⊥},

• R^{∗}, and

• r(i) := r(i, i + 1) (cf. notation at the end of (5.4)), i.e. \begin{matrix}⟨a^{i}, r(i)⟩ = 1 and \\ ⟨a^{i+1}, r(i)⟩ = 0 . \end{matrix}

In particular, we can write

r = g^{i}(r) · r^{i} + ⟨a^{i+1}, r⟩ · R^{∗} + (⟨a^{i}, r⟩ − ⟨a^{i+1}, r⟩) · r(i)

with some integer g^{i}(r) ∈ ZZ.

Now, we have to apply the differential in the complex (ZZ^{E^{2R^{∗}}})_{•}, i.e. we map the previous expression via the map

⊕_{i}ZZ^{E_{i}^{2R^{∗}}∩E_{i+1}^{2R^{∗}}} ⟶ ⊕_{i}ZZ^{E_{i}^{2R^{∗}}} .

The result is (for every i) the element of L(E_{i}^{2R^{∗}})

g^{i}(r) e^{r^{i}} − g^{i−1}(r) e^{r^{i−1}} + ⟨a^{i} − a^{i+1}, r⟩ · e^{r(i)} − ⟨a^{i−1} − a^{i}, r⟩ · e^{r(i−1)}+

+⟨a^{i+1} − a^{i}, r⟩ · e^{R^{∗}} = ⟨d^{i}, r⟩ · (e^{R^{∗}} − e^{r(i)}) + [(a^{i})^{⊥}-summands] =: q^{i}(r) .

Step 2: Defining

q^{i} := e^{R^{∗}} − e^{r(i)} + [(a^{i})^{⊥}-summands] ∈ L(E_{i}^{2R^{∗}}) (i = 1,…, N) ,

we use Theorem (3.6) and Proposition (3.7) to obtain

(\underline{s} ∪ \underline{t})_{i} (q^{i}(r)) = ⟨d^{i}, r⟩ · t_{Ψ(\underline{s},•),Ψ(\underline{t},•),R^{∗},R^{∗}}(q^{i}) = Ψ(\underline{s}, q^{i}) · Ψ(\underline{t}, q^{i}) .

To compute those two factors, we take a closer look at the q^{i}’s. Let

q^{i} = e^{R^{∗}} − e^{r(i)} + ∑_{v} λ_{v}^{i} e^{[c^{v};η^{v}]} ,

and the sum is taken over those v’s meeting the condition ⟨a^{i}, −c^{v}⟩ = η^{v}. Then, by definition of Ψ in (7.2),

Ψ(\underline{s}, q^{i}) = ∑_{j=1}^{(i+1)−1} s_{j} ⟨d^{j}, r(i)⟩ − ∑_{v} λ_{v}^{i} · (∑_{j=1}^{i−1} s_{j} ⟨d^{j}, c^{v}⟩) .

On the other hand, we know that q^{i} is a relation, i.e. the equation

R^{∗} − r(i) + ∑_{v} λ_{v}^{i}[c^{v}; η^{v}] = 0

is true in M. Hence,

Ψ(\underline{s}, q^{i}) = ∑_{j=1}^{i} s_{j} ⟨d^{j}, r(i)⟩ − ∑_{j=1}^{i−1} s_{j}⟨d^{j}, r(i)⟩
= s_{i} · ⟨d^{i}, r(i)⟩
= −s_{i} . □

T_{Y}^{1} ⊆ I C^{N} is the tangent space of the versal base space S of our singularity Y. It is given by the linear equation ∑_{i} t_{i} · d^{i} = 0. On the other hand, the cup product T_{Y}^{1} × T_{Y}^{1} → T_{Y}^{2} shows the quadratic part of the equations defining S ⊆ I C^{N}. By the previous proposition, it equals ∑_{i} t_{i}^{2} · d^{i}.

These facts suggest the form of all equations of S ⊆ I C^{N}. In [Al 3] we have, indeed, shown that S is given by the equations

∑_{i=1}^{N} t_{i}^{k} · d^{i} = 0 (k ≥ 1) .

References

[Al 1] Altmann, K.: Computation of the vector space T^{1} for affine toric varieties. J. Pure Appl. Algebra 95 (1994), 239-259.

[Al 2] Altmann, K.: Minkowski sums and homogeneous deformations of toric varieties. Tôhoku Math. J. 47 (1995), 151-184.

[Al 3] Altmann, K.: The versal Deformation of an isolated, toric Gorenstein Singularity. Preprint M.I.T., February 1994; preprint 59 “Europäisches Singularitätenprojekt”; e-print alg-geom/9403004.

[Ke] Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings I. Lecture Notes in Mathematics 339, Springer-Verlag, Berlin-Heidelberg-New York, 1973.

[KPR] Kurke, H., Pfister, G. Roczen, M.: Henselsche Ringe und algebraische Geometrie. Berlin: Deutscher Verlag der Wissenschaften 1975.

[Ld] Laudal, O.A.: Formal Moduli of Algebraic Structures.

Lecture Notes in Mathematics 754 (1979), Springer-Verlag.

[LdSl] Laudal, O.A., Sletsjœ, A.: Cohomology of groups, monoids and their algebras. Preprint, University of Oslo.

[LiS] Lichtenbaum, S., Schlessinger, M.: The cotangent complex of a morphism. Trans. AMS 128 (1967), 41-70.

[Od] Oda, T.: Convex bodies and algebraic geometry.

Ergebnisse der Mathematik und ihrer Grenzgebiete (3/15), Springer-Verlag, 1988.

[Pa] Palamodov, V.P.: Deformations of complex spaces.

Russian Math. Surveys 31 (1976), 129-197.

[Sm] Smilansky, Z.: Decomposability of Polytopes and Polyhedra.

Geometriae Dedicata. 24 (1987), 29-49.