On the cohomology of arrangements of subtori

Given an arrangement of subtori of arbitrary codimension in a torus, we compute the cohomology groups of the complement. Then, using the Leray spectral sequence, we describe the multiplicative structure on the graded cohomology. We also provide a differential model for the cohomology ring by considering a toric wonderful model and its Morgan algebra. Finally we focus on the divisorial case, proving a new presentation for the cohomology of toric arrangements.


Introduction
The cohomology ring of the complement of an arrangement of affine hyperplanes in a complex vector space admits a renowed combinatorial presentation in term of the poset of intersections of the arrangement, due to Orlik and Solomon [OS80]. For a toric arrangement, i.e. a collection of 1-codimensional subtori in a complex algebraic torus, a similar presentation was recently provided by [CDD + 19].
A different way of genealizing arrangements of hyperplanes is considering a family of affine subspaces, not necessarily of codimension 1. The complement of such a subspace arrangement was studied by several authors (see [GM88,DCP95,Yuz02,Yuz99,DGM00,dLS01]; see also [Bjo92] and the bibliography therein). In particular, Goresky and MacPherson provided the following description of the cohomology groups: Theorem A ([GM88, III.1.5, Theorem A]). Let A be a subspace arrangement in R d , and let M A = R d \ ∪A be its complement. The cohomology of the complement is given byH where L >0 is the poset of flats L without the minimum0 = R d , and ∆(0, W ) is the order complex of the corresponding interval.
Later, De Concini and Procesi constructed in [DCP95] a wonderful model for subspace arrangements, i.e. a smooth projective variety Y that contains M A as an open subset whose complement is a simple normal crossing divisor. They also applied a result of Morgan [Mor78] to present the rational cohomology ring of M A as the cohomology of a differential graded algebra explicitly constructed from the combinatorial data only.
In 1996, Yuzvinsky simplified the differential graded algebra (see [Yuz02]) by using the order complex of the poset of intersection. He also showed a connection between the results of [GM88] and of [DCP95]. A further simplification was obtained in [Yuz99] by replacing the order complex with the atomic complex. Yuzvinsky also conjectured an integral version of this presentation. This conjecture was proven in [DGM00] and [dLS01]: Deligne, Goresky, and MacPherson proved their result using diagram of spaces, de Longueville and Schultz by showing that the isomorphism of Theorem A is canonical.
In this paper we consider arrangements of subtori of arbitrary codimension in a complex algebraic torus. Given the complement of such an arrangement, we determine its cohomology groups (see Theorem 2.8): Theorem B. Let A be an arrangements of subtori of a torus T and L be it poset of layers. Then the cohomology groups of the complement M A are Our proof is based on a suitable embedding of a d-dimensional complex algebraic torus T in the Alexandroff compactification of C d , that is, the sphere S 2d . The embedding is chosen so that the complement of T in S 2d does not intersect the toric subspaces; hence the arrangement decomposes in a wedge of two simpler ones, given respectively by the compactifications of the coordinate hyperplanes and of the subtori in the original arrangement (Proposition 1.4). Then we apply some results on homotopy colimits [WZŽ99], following a strategy outlined by Deshpande in [Des18]. In that paper the same result is announced, but the proof therein does not seem to be correct, as several steps fail if the compactification is not chosen carefully.
Moreover we describe the multiplicative structure on the graded of the cohomology, by using the Leray spectral sequence for the inclusion map j : M A → T . We show, by using results of the previous section, that the second page of the spectral sequence is a finitely generated Z-module isomorphic to the cohomology as a module. It follows that the spectral sequence degenerates at the second page and this gives the isomorphism E p,q 2 ∼ = gr L p+q H p+q (M A ; Z) (Theorem 3.2). Furthermore, we provide a model for the cohomology of M A : we use the wonderful model for toric arrangements introduced by De Concini and Gaiffi (see [DCG18,DCG19]) to construct a differential graded algebra whose cohomology is isomorphic to the rational cohomology ring of the complement (Theorem 4.9). Since our method are based on the aforementioned Morgan algebra, this d.g.a. codifies also the rational homotopy type of the complement.
Finally we focus on the case of an arrangement of subtori of codimension 1. Given such a toric arrangement, and chosen its maximal building set, we find a subalgebra of the Morgan model isomorphic to the cohomology ring. This subalgebra, explicitly presented by generators and relations in Theorem 5.11, yields an analogue of the Orlik-Solomon algebra for toric arrangements. This new presentation depends on the oriented arithmetic matroid only (see [Pag20]) and, compared to the previous result of [CDD + 19], exhibits more clearly the dependence from the orientation. Furthermore, it seems more suitable to be generalized to arrangement of subtori of arbitrary codimensions. We also conjecture that a similar presentation holds for cohomology with integer coefficients (Conjecture 5.16).

Contents
Introduction 1 1. Positive systems and embedding of subtori 3 2. Cohomology groups of arrangements of subtori 4 3. Graded of the cohomology ring 7 4. A model for the complement 8 5. Divisorial case 12 References 24 1. Positive systems and embedding of subtori A d-dimensional complex torus T is an algebraic group isomorphic to (C * ) d . A character is a morphism of algebraic groups T → C * . The group Λ of all characters is a lattice of rank d, i.e., it is isomorphic to Z d . A subtorus of T is a translate of a subgroup isomorphic to (C * ) k , 0 ≤ k < d.
We denote by M A the complement of this arrangement, that is, The set of characters that are constant on a subtorus S i is a subgroup of Λ, that we denote by Λ i . Let B be a basis (over Z) of Λ and, for every i = 1, . . . , n, let B i be a basis (over Z) of Λ i . Proof. For each S i ∈ A choose a basis B i of Λ i and a basis B of Λ. Consider the matrix A that represent the elements b i,j ∈ B i , for i = 1, . . . , n and j = 1, . . . , |B i | in the basis B. By changing b i,j with −b i,j we suppose that the last non-zero entry of the j−th column of A is a positive integer: we call this entry the pivot of the column. We perform a sequence of elementary row operation in order to make A a non-negative matrix. The columns with pivot in the first row are already nonnegative. We proceed by induction, suppose that all columns with pivot in the first k − 1 rows are non-negative. By adding a suitable multiple of the kth row to the previous rows, we can make all the columns with pivot in the kth row non-negative. This operation does not change the columns with pivot in the first k − 1 rows. By repeating the procedure for every k = 2, . . . , d, we obtain a non-negative matrix. The elementary row operations correspond to a change of basis from B to a new basis B ′ that form a positive system (B ′ , B 1 , . . . , B n ).
We denote by S d the d-dimensional real sphere, and by B d the Boolean arrangement, i.e. the set of the coordinate hyperplanes in C d .
Given a topological space X, its Alexandroff compactification X is defined as the topological space on the set X ∪ {∞} whose basis of open sets is given by the open sets of X and the sets (X \ C) ∪ {∞}, where C ranges over all the closed and compact sets of X. For instance, the Alexandroff compactification of C d is isomorphic to the sphere S 2d .
Proposition 1.4. Let A be a arrangements of subtori in a d-dimensional torus T . Then there exists an embedding M A ֒→ S 2d such that Proof. We choose a positive system (B, B 1 , . . . , B n ). The basis B defines an isomorphism T ∼ = (C * ) d , and consider the composition Notice that C d \ M A is the disjoint union of ∪A and ∪B 2d because the system z n1 1 z n2 2 · · · z n d d = c z j = 0 for n i ∈ N, c ∈ C * , and j ≤ d, has no solutions. The condition of positive system ensure that each subtorus S i ∈ A is contained in a hypertorus of the form

Cohomology groups of arrangements of subtori
Let P be a poset. We recall that the order complex ∆(P) is the simplicial complex whose simplexes are the the totally ordered subsets of P. For any W, L ∈ P with W > L we denote ∆(L, W ) the order complex of the sub-poset { X ∈ P | W > X > L } .
Definition 2.1. Given two pointed CW-complexes (X, x) and (Y, y), we define: LetL >0 be the poset obtained from the poset of layers L by removing the minimum0 = T and adding a maximum1. We think ofL >0 as a category, having one morphism p → q for every p, q ∈L >0 such that p ≤ q.
Given a poset P, a P-diagram is a functor from the category P to the category Top * of pointed topological spaces. The colimit of a P-diagram F is the union of all topological spaces F (p) for all p ∈ P with the identification given by the maps between them (i.e. x = F (p → q)(x) for all maps p → q in P and all x ∈ F (p)).
The homotopy colimit of F can be constructed by replacing all the maps F (p → q) with homotopy equivalent cofibrations and then taking the colimit of the resulting diagram.
We recall the following results of Welker, Ziegler andŽivaljević: i.e. a basis B of Λ W and a basis B i of Λ Si for each atom S i ∈ A W . The basis B gives an isomorphism between W and (C * ) dim W . Let ǫ ∈ R + be the minimum of the distance between 0 ∈ C dim W and S i for all atoms S i ∈ A W . Each layer L ∈ L >W of the restricted arrangement A W is contained in a hypertorus {z ∈ C dim W | z n1 1 . . . z n dim W dim W = c} for some n i ∈ N and some c ∈ C * . Hence ǫ is positive and each layer L is disjoint to the ball D ǫ ⊂ C dim W ⊂ S 2 dim W of center 0 and radius ǫ.
Choose a continuous, increasing and surjective function f : [0, ǫ) → [0, ∞) and defineh W : where |x| is the distance of x from 0 and ∞ is the unique point in S 2 dim W \ C 2 dim W . It easy to see thath W induces a homotopy equivalence h W : W → W . The commutativity of the diagram above follows from L ∩ D ǫ = ∅.
The previous results now allow us to describe the Alexandroff compactification of the union of the subtori of the arrangement: Lemma 2.7. There exists a homotopy equivalence Proof. Consider the maps h W given by Lemma 2.6 and let h1 : {∞} → {∞} be the only map. This data define a morphism h : D → E ofL >0 -diagrams. We have where the first isomorphism follow by the definition of colim, the others by the projection Lemma 2.3, the homotopy Lemma 2.4 applied to h, and the wedge Lemma 2.5 respectively.
Theorem 2.8. Let A be an arrangements of subtori of a torus T and L be it poset of layers. Then the cohomology groups of the complement M A are Again Alexander duality for the embedding B d ⊂ S 2d gives the isomorphism where the last isomorphism is the Kunneth isomorphism for reduced cohomology applied to the smash product. We conclude the proof by the Poincarè duality on W between Borel-Moore homology and cohomology (see [Bre97,Theorem 9.2]):

Graded of the cohomology ring
In this section we study the Leray spectral sequence for the inclusion map j : M A → T to give a description of the graded cohomology ring gr . We refer to [Bre97, Sect. 6] as a general reference on this spectral sequence.
Let R q j * be the higher direct image functor of j. In our case the Leray spectral sequence converges to H p+q (M A ; Z); the second page of this spectral sequence is The following lemma generalizes [Bib16, Lemma 3.1].
Lemma 3.1. Let A be an arrangement of subtori. Then Proof. First we prove that ǫ q ∼ = R q j * Z MA : for each point x ∈ T there exists an open set U x isomorphic to an open subset V x of the tangent space T x T (containing the origin). We also take a neighborhood basis U given for every x ∈ T by the inverse image of all open balls in V x centered in 0 ∈ T x T . Notice that the arrangement of subtori A defines a central arrangement of subspaces We define a morphism of sheaves f : where the first isomorphism is given by Theorem A and the second one is given by the composition Since f (U ) is an isomorphism for all U ∈ U then f is an isomorphism of sheaves. Now, the isomorphism completes the proof.
The minimum of the poset L (and of L ≤W for all : this module depends only on the cohomology of W and on the poset L ≤W .
The multiplication in E 2 is induced by the maps Theorem 3.2. The Leray spectral sequence E p,q r for the inclusion M A ֒→ T degenerates at the second page, i.e.
∞ is a subquotient of E p,q 2 and that the last page is E p,q ∞ ∼ = gr L p+2q H p+q (M A ; Z). By theorem 2.8 and lemma 3.1, E p,q ∞ and E p,q 2 are isomorphic and finitely generated; hence E p,q 2 = E p,q ∞ .

A model for the complement
As in the previous sections, we denote by A an arrangement of subtori in T and by Λ the character group of T . Let Λ * be the dual lattice of Λ. We refer to [CLS11] for a general introduction to fans and toric varieties. Let ∆ be a smooth and complete fan in Λ * . Every ray of ∆ is generated by a (uniquely determined) primitive vector in Λ * : we denote by R ∆ ⊂ Λ * the set of primitive vectors corresponding to the rays of ∆. Let P(R ∆ ) be its set of parts; we denote by C ∆ ⊆ P(R ∆ ) the collection of the sets of primitive vectors that span a cone in ∆. Thus from now on we identify a cone in ∆ with the set of its extremal primitive vectors.
Definition 4.1. A fan ∆ in Λ * describes a good toric variety X ∆ (with respect to A) if ∆ is complete and smooth and each maximal cone C ∈ C ∆ can be completed to a positive system (C, C 1 , . . . , C n ) (where C is the dual basis of C).
The second condition in the above definition can be reformulated as follow: for each W ∈ A, there exists a basis β 1 , . . . , β cd W of Λ W such that for each maximal cone C ∈ C ∆ and each i = 1, . . . , cd W the natural pairing β i , c is nonnegative (or nonpositive) for all c ∈ C. In this case, we say that the basis β 1 , . . . , β cd W of Λ W has the equal sign property with respect to ∆ (see [DCG18, Definition 3.2]).
Let G ⊆ L >0 be a well connected building set in the sense of [DCG19, Definition 4.1] and ∆ a good toric variety. These data define a smooth projective variety Y (∆, G) obtained from X ∆ by subsequently blowing up (the strict transform of) W for all W ∈ G in any total order refining the partial order given by inclusion (so that smaller layers are blown up first). The variety Y (∆, G) is the wonderful model for M A described in [DCG19], i.e. a smooth projective variety containing M A and such that the complement Y (∆, G) \ M A is a simple normal crossing divisor.
We want to describe the Morgan algebra (cf [Mor78]) for the pair (Y (∆, G), M A ). For the convenience ot the reader, we will briefly recall here the definition of this algebra. Consider a smooth complete algebraic variety Y and a simple normal cross- . The multiplication is given by i.e. the composition of the restriction maps and the cap product. The differential is induced by the Gysin morphisms Let E be the exterior algebra over Q on generators s W , t W , b j , c j for W ∈ G and j ∈ R ∆ , where the bi-degree of s W and b j is (0, 1) and the bi-degree of t W and c j is (2, 0).
In order to understand what relations should put on E, we start by recalling the definition of some polynomials P W L (t), which were introduced in Section 8 of [DCG19] as good lifting of the Chern polynomials 1 .
For each pair W ≤ L in L, and for each t-uple β 1 , . . . , β cd W ∈ Λ W with the equal sign property with respect to ∆ such that β cd L−cd W +1 , . . . , β cd W form a integral basis of Λ L , define The polynomial P W L (t) depends on the choice of β 1 , . . . , β cd L−cd W . Definition 4.2. A set A ⊂ G is nested if the irreducible components of the normal crossing divisor of Y (∆, G) that correspond to the elements of A have non-empty intersection. When we want to emphasize the dependence on G we will say that A is G-nested.
The property of being nested does not depend on the choice of ∆, and can be expressed in a purely combinatorial way (see [DCG19, Definition 2.7]).
We recall the following result of De Concini and Gaiffi. 1 The authors of [DCG19] forgot to specify that, in order to define a good lifting, the basis of Λ W must have the equal sign property with respect to ∆.
(T1) j∈C c j if C ∈ C ∆ , (T2) j∈R∆ β, j c j for every β ∈ Λ (or equivalently for β in a fixed basis of Λ), for all W ∈ G and all C ⊆ G <W , the relations Although the polynomials P V W (t) in (W3a) depend on the choice of a basis, the ideal generated by all the relation is independent from this choice, as shown in [DCG19, Proposition 6.3].
Remark 4.4. Another possible choice of P V W (t) consist of taking the polynomials: Let A be a nested set, W be any element in G, and B ⊆ G be such that each L ∈ B is smaller than W (L W in L). We define the element F (A, W, B) in E by where V is the connected component of L∈A<W ∪B L containing W (so V ≤ W ).
Definition 4.5. Let (D, d) be the differential graded algebra given by E with relations: ( j∈R∆ β, j c j for every χ ∈ Λ (or equivalently for χ in a fixed basis of Λ), (5) F (A, W, B) for A G-nested set, W ∈ G, and B ⊆ G be such that each L ∈ B is smaller than W (ie B ⊆ G <W ), and differential d defined on generators by d( Lemma 4.6. The ideal generated by (1)-(5) is stable with respect to d, so (D, d) is a differential graded algebra.
Proof. It is obvious that the ideal generated by (1)-(4) is d-stable. The relation d (F (A, W, B) show that the ideal generated by (5) is d-stable.
2 In [DCG19] these relations are stated only for |C| = 1; however they hold, before performing blow-ups, for any set C, by the well-known theory of toric varieties. Thus, by adding the relations with |C| > 1 to the presentation given in [DCG19], we get a correct presentation.
Let M be the Morgan algebra associated to the pair (Y (∆, G), M A ). The complement Y (∆, G) \ M A is a simple normal crossing divisor W ∈G D W ∪ j∈R∆ D j , whose irreducible component are indexed by G ⊔ R ∆ .
For each A ⊆ G⊔R ∆ we denote with Y A the intersections of all divisors associated to A. The graded differential algebra M is the direct sum of vector spaces on which: • the total degree of the elements in H p (Y A ) is |A| + p; • the multiplication is induced by the restriction maps and the cup product We define a morphismf : E → M on generators by Lemma 4.7. The mapf is a surjective morphisms of differential graded algebras.
Proof. As shown in [DCG19, Theorem 9.1], the restriction maps H for A ⊂ B are surjective. Since Imf contains H • (Y (∆, G)) and the elements 1 ∈ H 0 (D) for all divisors D, the morphismsf is surjective. By construction of the cohomology algebra, the elements t W and c j of H 2 (Y (∆, G)) are t W = (i W ) * (1) and c j = (i j ) * (1), where i * is the Gysin morphism for the regular embedding i : D ֒→ Y (∆, G). Therefore,f is a morphism of differential graded algebras.
The mapf factors trough f : D → M, indeed we have the following theorem.
Lemma 4.8. The map f is well defined and is an isomorphism.
Proof. We first check that (1)-(5) belong to kerf : (1) there are four cases to check: We have proven that f is well defined and surjective, sincef is. Let I be the ideal in E generated by (1)-(5) and notice that I is a monomial ideal in the variable s W and b j for W ∈ G and j ∈ R ∆ . It is enough to prove that in D for all subsets A ⊆ G, B ⊆ R ∆ and all polynomials z in the variables {t W } W ∈G and {c j } j∈R∆ .
The monomials W ∈A s W j∈B b j with A a non-nested belong to I by (2), the ones with B not a cone belong to I by (3), and the ones with B ⊂ W ∈A Ann Λ W are in I by (1). Now, let A be a G-nested set and B ∈ C ∆ be a cone contained in W ∈A Ann Λ W . We define a map H • (Y A⊔B ) → D by using the presentation of Theorem 4.3: the morphism is defined by z → W ∈A s W j∈B b j z for all z in the exterior algebra on generators {t W } W ∈G and {c j } j∈R∆ . It is well defined:

Divisorial case
In this section we consider arrangements of subtori of codimension 1, usually known in the literature as toric arrangements. Given such an arrangement A = {S 1 , . . . , S n } we consider the toric wonderful model Y (∆, G) where G = L >0 is the maximal building set. In this case, the G-nested subsets coincide with the chains in L >0 .
Inspired by Yuzvinsky [Yuz02,Yuz99], we introduce a different set of generators σ W , τ W for the d.g.a. D and we determine the relations between them (Lemma 5.2). By using this generators we define some elements Ξ W,A of D (Definition 5.3) that belongs to the kernel of d (Lemma 5.4). We study the multiplication between them in Lemma 5.5 and their relation with the cohomology of the ambient torus (Lemma 5.6). The linear relation between Ξ W,A are rather complicated to prove (Lemmas 5.7, 5.9 and 5.10 and corollary 5.8). In the main result of this section, Theorem 5.11, we introduce a Orlik-Solomon type algebra R and we prove that the composition is an isomorphism. The map R → H(D, d) is well defined by all the Lemmas preceding the main Theorem, is injective by Lemmas 5.1, 5.12 and 5.13, and surjective by dimensional argument (Lemma 5.12). Although the d.g.a. D depends on the choice of a good fan ∆, the algebra R and its isomorphic image in D are independent from the choice of the fan.
In this section we will use basic notions of matroid theory such as those of independent set,circuit, no broken circuit, that can be found for instance in the paper [OS80].
Define the elements σ W = L≥W s L and τ W = L≥W t L in D for all W ∈ L ≥0 . Moreover, for every χ ∈ Λ define As in the previous Section, we consider the bi-gradation of D given by deg(s W ) = deg(b j ) = (0, 1) and deg(t W ) = deg(c j ) = (2, 0), so that the differential d has bidegree (2, −1).
where A runs over all the G-nested sets and C ∈ C ∆ over all the cones contained in ∩ W ∈A Ann Λ W , is a linear basis of D 0,• .
Moreover, the set { W ∈A σ W j∈C b j } A,C (where A and C runs over the range described above) is a linear basis of D 0,• .
Proof. Notice that D 0,• is the exterior algebra on generators s W and b j with relations: ( is not a cone in ∆. These relations generate a monomial ideal, so can be easily seen that these element divide a monomial if and only if it is not in the first basis.
For the second basis, we choose a total order on the set G that refines the partial order on it. This total order induces a lexicographical order on the set of G-nested sets. The matrix that represents the elements { W ∈A σ W j∈C b j } A,C in the basis { W ∈A s W j∈C b j } A,C is upper triangular with ones on the diagonal entries. This proves the claim.
Lemma 5.2. In D we have the following relations: ( if W ⋗ V and χ ∈ Λ W is an element that generates Λ W /Λ V , then: (1) Let The claimed equality can be rewritten as (x 1 + x 3 )(x 2 + x 3 ) = (x 1 − x 2 )x 3 . Since x 3 has degree one we have x 2 3 = 0 and we need to prove that x 1 x 2 = 0. This follows from x 1 x 2 = s V s U where the sum runs over all V ≥ W , V ≥ L and U ≥ W , U ≥ L: we have s V s U = 0 because V and U do not form a chain.
(2) Notice that for χ ∈ Λ W we have min(0, χ, j )a j r W = 0 for a = b or a = c and r = s or r = t, by Relation (1) of Definition 4.5. Since W ≥ V implies Λ W ⊇ Λ V , we have Let η ∈ Λ U be an element that generates Λ U /Λ L and notice that χ = aη+η ′ with η ′ ∈ Λ L and a = |L ∨ W |. Observe that by Relation (5) of Definition 4.5. The proof of τ V (τ W + γ − χ ) = τ 2 W is analogous.
Let A ⊆ {1, . . . , n} be an independent set and W a connected component of ∩ a∈A S a . Following [Moc12], we denote by m(A) the number of connected components in such intersection. A flag adapted to A and W is a list F = (a 1 , a 2 , . . . , a k ) of distinct elements of A; m(F ) is defined accordingly. The list F determines a flag in the usual sense by setting F 0 = T and F i being the unique connected component of S ai ∩ F i−1 containing W . Then F i is a flat ∀i = 0, . . . , k and F i ⋖ F i+1 . Viceversa, every maximal flag (F 0 = T ⋖ F 1 · · · ⋖ F k ) between T and F k with F k ≤ W determines a unique flag F = (a 1 , a 2 , . . . , a k ) adapted to A and W .
where the sum is taken over all the flags adapted to A and W .
In order to simplify the notations, in the definition above we denoted by a∈A the exterior product taken in the order of A ⊆ {1, . . . , n}. The same notation will be used from now on. Proof. We have that . Moreover for i = 1, k, F = (a 1 , a 2 , . . . , a k ), we consider the flag F ′ = (a 1 , . . . , a i−2 , a i , a i−1 , a i+1 , . . . , a k ) and notice that If k > 0, then: 1 , a 2 , . . . , a k ) we consider the flag F ′ = (a 1 , . . . , a k−2 , a k , a k−1 ) and we have: Finally, we have: where all the products are taken in the order of A with τ Fi in position of a i , σ Fi in position a i−1 and σ Fj in position a j . This completes the proof.
We recall that, given two positive integers k, h, a (k, h)-shuffle is an element p of the symmetric group on the elements {1, . . . , k + h} such that p(i) < p(j) for every couple i < j such that either i, j ∈ {1, . . . , k}, or i, j ∈ {k + 1, . . . , k + h}. Proof. Let F = (a 1 , . . . , a k ) and G = (a k+1 , . . . , a k+h ) be two flags and let A = {a 1 , . . . , a k+h }. If A is independent of cardinality k + h, then for each (k, h)-shuffle p and each element V ∈ F k ∨ G h we have a flag F * p G := (a p(1) , . . . , a p(k+h) ) adapted to A and V . By using only equation (1) of Lemma 5.2, we have  Lemma 5.6. If χ ∈ Λ W , then Ξ W,A β χ = 0.
Proof. Let F be a flag adapted to A and W , we show that β χ a∈A x(F , a) = 0. Notice that k i=0 β χi = 0 if the elements χ i , i = 0, . . . , k are linearly dependent because k + 1 distinct rays in a k-dimensional vector space cannot span a simplicial cone.
Let L be a layer contained in F k , if χ ∈ Λ L then β χ s L = 0 and hence β χ σ F k = 0. If χ ∈ Λ F k , then χ and χ a for a ∈ A \ F are nonzero and dependent in Λ W /Λ F k therefore s L β χ a∈A\F β χa = 0.
Finally we have β χ σ F k a∈A\F β χa = 0 and so the claim holds.
Lemma 5.7. Let A be an independent set and s a ∈ {+, −} for a ∈ A. Let Z be the set {v ∈ Λ * | v, s a χ a > 0 for all a ∈ A}. Consider the projection π : Λ * → Λ * / Ann Λ A . We have that where the last product is taken in any order such that the two bases (s a χ a ) a∈A and (π(c)) c∈K are both positive or both negative.
Proof. Let K ∈ C |A| ∆ be a |A|-dimensional cone not contained in Z: then there exists c ∈ K such that c ∈ Z. So, for some a ∈ A, we have min(0, s a χ a , c ′ ) = 0 for all c ′ ∈ K by using the equal sign property. It easy to see that the monomial c∈K b c does not appear in a∈A β sa χa . Now suppose that K = (k 1 , . . . , k l ) ∈ C l ∆ is contained in Z, the coefficient of Now notice that s i χ i , k σ(i) = s i χ i , π(k σ(i) ) for all i and σ. The equality follows from the multilinearity in the entries s i χ i and π(k i ). Since the two bases (s a χ a ) a∈A and (π(k)) k∈K are both positive (resp. negative) then det(s i χ i ) det(π(k i )) is positive and equals to m(A) Vol(π(K)).
The proof of this corollary follows from the proof of Lemma 5.7 by omitting some steps.
Corollary 5.8. Let A be an independent set, then: where the last product is taken in any order such that the two bases (χ a ) a∈A and (π(c)) c∈K are both positive or both negative.
Lemma 5.9. Let X be a subset such that |X| = rk(X) + 1, C ⊆ X be the unique circuit, A ⊂ X be a independent set, F be a connected component of ∩ a∈A S a , and j be an element of C \ A. There exists a minimal relation i∈C c i m(C \ {i})χ i = 0 for some c i ∈ {+, −}. Suppose that C ′ := C \ A has cardinality at least 2, then Proof. For the sake of simplifying the notation, let us suppose C ′ = {0, 1, . . . , l}. The first step of the proof is to reduce to the case c i = − for i < k and c i = + for i ≥ k for some k ∈ C ′ . Let µ ∈ S |C ′ | be the unique shuffle that reorders C ′ in such a way that c i = − for i < k and c i = + for i ≥ k. We have where we use sgn(µ) = (−1) j−µ(j) sgn(µ |C ′ \{j} ). Moreover notice that δ(i, j) = δ(µ(i), µ(j)) since (i, j) is an inversion of µ only if c i c j = −. Thus from now on we assume c i = − for i < k and c i = + for i ≥ k. Define The following properties follows easily from the definition: where in the last equality we used the property (P) of arithmetic matroids (see [BM14]). For j = k the bases (δ(i, j)χ i ) i =j and (δ(i, j − 1)χ i ) i =j−1 have the same orientation. The bases (−χ i ) i>0 and (δ(i, k)χ i ) i =k have the same orientation if and only if (−1) k−1 = 1. Since it is enough to consider the following: so the claim follows.
Let C ⊆ {1, . . . , n} be a circuit oriented by the signs (c i ) i∈C . We recall the following definition, which was introduced by Postnikov in [Pos06]. For each A ⊆ {1, . . . , n}, we say that C/A is a positroid if c i = c j for all i, j ∈ C \ A. Proof. We may assume that X = {0, 1, . . . , rk(X)} and C = {0, 1, . . . , rk(C)}. Let R = X \ C, we can rewrite the left hand side as follow: we need the following equality: B⊆C<j\F B∪{j} pos.
We also need, for |C ′ | > 1 the following: by Lemma 5.9. Finally, we have: where k(F ) is the last element of F that belongs to C, j(F ) the unique element in C \ F . LetF be the flag obtained from F substituting k(F ) with j(F ). Notice that F ∪ C =F ∪ C and that the addendum associated to F and toF differ by the sign (−1) k(F )−j(F )−1 . This prove the claimed equality.
Let ω be the generator of H 1 (C * ; Z).
Theorem 5.11. Let A be a toric arrangement. The rational cohomology algebra H * (M (A); Q) is isomorphic to the algebra where A ranges over all the idependent subsets of {1, . . . , n} and W ranges over all connected components of ∩ a∈A S a . The degree of the generator e W,A is |A|. The ideal I is generated by the following elements: • for any two generators e W,A , e W ′ ,A ′ , • For any ψ ∈ H • (T ) such that ψ |W = 0, • For every X ⊆ {1, . . . , n} such that rk(X) = |X| − 1 write X = C ⊔ F with C the unique circuit in X. Consider the associated linear dependency i∈C n i χ i = 0 with n i ∈ Z, and for every connected component L of ∩ i∈X H i a relation for all X such that rk(X) = |X| − 1 and all L connected components of ∩ a∈X S a . Notice that gr F R is L-graded and isomorphic to We want to construct a bijection for any geometric lattice L ≤W between no broken circuit sets and certain maximal flags. For any maximal flag of layers F = (T = F 0 ⋖F 1 ⋖· · ·⋖F k = W ) we define the edge labelling ǫ(F ) as the list (b 1 , . . . , b k ) where b k = max{i ∈ {1, . . . , n} | F k ∈ F k−1 ∨ S i }. We say that F is increasing if b i < b j for all i < j (where ǫ(F ) = (b 1 , . . . , b k )).
Notice that if F is a maximal flag adapted to A and W , ǫ(F ) may not be a subset of A.
Lemma 5.13. We fix a layer W of rank k and consider the geometric lattice L ≤W . If A = {a 1 < a 2 < · · · < a k } is a no broken circuit set, then a maximal flag F adapted to A and W is increasing in L ≤W if and only if F = (a 1 , a 2 , . . . , a k ).  The coefficient of z in ( L∈F σ L )g(ψ W ,A ψ) must be zero, but it is (up to a sign) equal to α W ,A m(B(A)) Vol(π(C(A))) c.f. Corollary 5.8. The volume Vol(π(C(A))) is different from zero because Λ A ⊗ Q ⊕ Λ B(A) ⊗ Q = Λ ⊗ Q. We have α W ,A = 0 contradicting the assumption, hence g is injective.
Notice that the range of g is contained in ker d by Lemma 5.4 and in the subalgebra D 0,• . The map g induces an injective map Remark 5.14. Theorem 5.11 is a generalization of [DCP05, Theorem 5.2] and analogous to [CDD + 19, Theorem 6.13]. Indeed, if A is totally unimodular and the circuit C = {0, 1, . . . , n} is oriented with c 0 = −, c i = + for i > 0, we obtain the Equation (20) of [DCP05].
We have chosen the generator associated with an hypertorus S a as Ξ Sa,{a} = σ Sa + β − χa that depends on the choice of one between χ a and −χ a . Another possible choice of generators were Ξ Sa,{a} = 2σ Sa + β − χa + β + χa , this would be lead to the same presentation of [CDD + 19, Theorem 6.13].
Conjecture 5.16. Substituting in eq. (3) m(A) m(X\{j}) ψ B with |B| i=1 ψ χi , where (χ i ) i form a basis of Λ C /Λ A with the same orientation of (χ b ) b∈B , the cohomology ring with integer coefficients have a presentation analogous to the one in Theorem 5.11. This approach to the computation of cohomology ring for toric arrangements may be generalised to the non divisorial case.