As a consequence, we can provide a more profound structure result concerning the group gradings on the incidence algebras, and we can classify their isomorphism classes of group gradings

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ALGEBRAS

EDNEI A. SANTULO JR, JONATHAN P. SOUZA, AND FELIPE Y. YASUMURA

Abstract. In this work, we classify the group gradings on finite-dimensional incidence algebras over a field, where the field has characteristic zero, or the characteristic is greater than the dimension of the algebra, or the grading group is abelian.

Moreover, we investigate the structure of G-graded (D1, D2)-bimodules, whereGis an abelian group, andD1 andD2 are the group algebra of finite subgroups ofG. As a consequence, we can provide a more profound structure result concerning the group gradings on the incidence algebras, and we can classify their isomorphism classes of group gradings.

1. Introduction

When one deals with an algebraic structure, it is usually useful to consider an additional structure on it, when possible. Considering group gradings in algebras has proved to be useful to solve relevant mathematical problems more than once.

That is the case of the classification of finite-dimensional semisimple Lie algebras, which are graded by the root system (see e.g. [8], or [15]). That is also the case of the positive solution for the Specht problem given by Kemer in [12]. Furthermore, group gradings naturally appear in several branches of Mathematics and Physics.

We briefly recall the definition of graded algebras. Let A be any algebra (not necessarily associative, not necessarily with unit), and letGbe any group. We say that A is G-graded if there exists a vector space decomposition A = L

g∈GA_g, whereAgare possibly zero subspaces, such thatAgAh⊆Agh, for allg, h∈G. The elements of∪_g∈GAgare called homogeneous, and a non-zerox∈Ag is said to have degreeg, denoted by deg_Gx=g, or simply degx=g if there is no ambiguity. An extensive theory concerning graded algebras can be found in the monograph [6].

An interesting question is, given an algebra, determine all the possible group gradings on it up to graded isomorphisms. In this direction, the works [1, 3] gave the answer for the matrix algebras (they investigate a more general situation), and then, in [2], the authors show how to obtain group gradings on some simple Lie and Jordan algebras from the knowledge of the group gradings on matrix algebras.

These works started an extensive research in the subject. The monograph [6] is a complete state-of-art of the theory.

2010Mathematics Subject Classification. 16W50.

Key words and phrases. Group Gradings, Incidence algebras, Associative algebras.

The second author was financed by the Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior - Brasil (CAPES) - Finance Code 001.

The third author was supported by S˜ao Paulo Research Foundation (Fapesp), grant number 2013/22.802-1, and by the Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior - Brasil (CAPES) - Finance Code 001.

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In this paper, we are interested in the non-simple associative algebras called incidence algebras. These are a generalization for the upper triangular matrices U Tn. In short, an incidence algebra is a subalgebra of U Tn, generated by matrix units and containing all diagonal matrices (see a precise definition below). Thus, the classification of group gradings on upper triangular matrices is a particular result of the classification of group gradings on the incidence algebras. The classification of group gradings on the upper triangular matrices was obtained in two works [17, 5].

To state the classification, we give some definitions. We call a grading onU T_n good if all matrix unitseij are homogeneous in the grading; and we call aG-grading on U Tnelementaryif there exists a sequence (g1, g2, . . . , gn)∈Gⁿsuch that everyeijis homogeneous of degreegig_j⁻¹. Clearly every elementary grading is a good grading, and, in the context of upper triangular matrices, the converse holds. Hence, good and elementary gradings are equivalent notions for the algebra of upper triangular matrices. In [17], it is proved that every group grading onU Tn is elementary, up to an isomorphism. In [5], the isomorphism classes of elementary gradings onU Tnare classified, thus obtaining a complete classification of group gradings on this algebra.

Note that the notions of elementary gradings and good gradings can be defined for subalgebras of matrix algebras generated by matrix units. Hence, in particular, one can define these notions for the incidence algebras. It is worth mentioning that these notions were generalized in [4] for a larger class of algebras. It is natural to conjecture that these two notions are equivalent, and that every group grading on an incidence algebra is elementary up to an isomorphism. However, these two notions are not always equivalent: not every group grading on an incidence algebra is elementary or good (see below). Thus, incidence algebra turns out to be a more intricate combinatorial object in the point of view of gradings by a group.

In this paper, we classify group gradings on the incidence algebraI(X) over a field F, where X is finite and one of the following conditions hold: the base field has characteristic zero, charF>dimI(X), orGis abelian.

Some works were dedicated to study the group gradings on the incidence algebras [13, 10, 14], but none of them provide a complete understading of its group gradings.

Some different phenomenon happens in this algebra. The first surprising fact is that not every good grading onI(X) is necessarily elementary, even if X is finite (see, for instance, [10, Example 12]). Moreover, there exist group gradings onI(X) that are not good (see also [13, Example 1]):

Example 1. Let X ={1,2,3} be partially ordered by 1 ≤2 and 1≤3. Then we obtain the algebra

I(X) =











∗ ∗ ∗

∗ 0

∗









 .

LetG=Z2×Z2, and define

A_(0,0)= Span{e11, e₂₂+e₃₃}, A(0,1)= Span{e22−e33}, A_(1,0)= Span{e12−e13}, A_(1,1)= Span{e₁₂+e₁₃}.

This is a well-defined grading on I(X). Moreover, it is not isomorphic to a good grading, since there are not three orthogonal homogeneous idempotents.

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Thus, the study of group gradings on incidence algebras is more complicated than the case of upper triangular matrices. Recall that a grading by a group is equivalent to the action of the group of characters on the algebra, given that the group is abelian and the base field is algebraically closed of characteristic zero (see, for instance, [6, Section 1.4]). The automorphism group of the incidence algebra is known [16, Theorem 7.3.6], and it is more intricate than the automorphism group of the upper triangular matrices (see, for instance, [9]). Hence, the automorphism group ofI(X) suggests the existence of new gradings, other than the gradings on upper triangular matrices.

Another surprising fact is the existence of group gradings on incidence algebras without any homogeneous multiplicative basis (see example in Section 7). We recall that a basisB is multiplicative if for allu, v∈B, there exists a scalarλsuch that λw = uv, for some w ∈B. In contrast, it is clear that any good grading admits a multiplicative homogeneous basis. Thus, every group grading onU Tn admits a multiplicative homogeneous basis.

In the next section, we are going to prove our main result:

Theorem 1. Let F be a field, X a finite poset, and let I(X) be endowed with a G-grading. Assume at least one of the following conditions: charF= 0, charF>

dimI(X), or G is abelian. Then, up to a graded isomorphism, there exist finite abelian subgroupsH₁, . . . , H_t⊆G, such that: for eachi= 1, . . . , t,charFdoes not divide |Hi|, andF contains a primitiveexpH_i-root of1; and

I(X)∼=







FH1 M1,2 . . . M1,t

FH2 . .. ... . .. M_t−1,t

FHt





 ,

where each Mi,j is a graded (FHi,FHj)-bimodule.

The setE ={1,2, . . . , t}has a natural structure of poset, and it plays an essential role in establishing the isomorphism classes of group gradings on the incidence algebras. The partial order ofE is given byiEj ifM_i,j 6= 0 (see Remark 19).

Further, if we assume the grading group abelian, then we can prove a stronger statement concerning the bimodulesM_ij:

Theorem 2. Assume that G is abelian, and, using the notation of Theorem 1, fix i < j and let H_ij = H_i∩H_j. Then, there exist pairwise distinct characters χ^(ij)₁ , . . . , χ^(ij)sij ∈Hˆij, andm1, . . . , ms_ij ∈Mij homogeneous such that

M_ij =FH_im₁FH_j⊕ · · · ⊕FH_im_s_ijFH_j,

where hm_` = χ^(ij)_` (h)m_`h, for all h∈ Hij, for ` = 1,2, . . . , s_ij. Moreover, as a G-graded vector spaces, FHim`FHj ∼= (F(HiHj))^[deg^m^`^], for each `= 1,2, . . . , sij. Also,sij ≤ |Hij|.

Finally, given that the grading group is abelian, we solve the problem of classify- ing the isomorphism classes of group gradings on the incidence algebras (Theorem 34).

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Example 2. In Example 1, the graded algebra I(X) is isomorphic to F M

0 FZ2

,

whereM is isomorphic toFZ2 as a rightFZ2-module.

2. Preliminaries

2.1. Incidence algebras. We provide the definition of an incidence algebra over a field F. Let (X,≤) be any partially ordered set (poset, for short). Assume that (X,≤) is locally finite, i.e., for allx, y∈X, there exists a finite number ofz∈X such thatx≤z≤y. Define I(X) ={f :X×X →F|f(x, y) = 0,∀x6≤y}. Then I(X) has a natural sum (pointwise sum), and natural scalar multiplication, which givesI(X) a structure ofF-vector space. Forf, g∈I(X), we defineh=f·gas the functionh, such thath(x, y) =P

z∈Xf(x, z)g(z, y). Note that the unique possibly non-zero elements in the previous sum are thez ∈X such that x≤z ≤y; hence, since X is locally finite, the sum is well-defined. So h∈ I(X). It is standard to prove thatI(X), with the defined operations, is an associative algebra. The algebra I(X) is called anincidence algebra.

Note that the defined multiplication onI(X) is similar to the product of matrices. Moreover, ifX is totally ordered and containsnelements, then I(X)∼=U T_n. In connection, if (X,≤X) is arbitrary (not necessarilly totally ordered) and finite withnelements, then we can rename the elements ofX ={1,2, . . . , n}in such way thati≤Xj impliesi≤jin the usual ordering of the integers. With this identifica- tion, we see thatI(X)⊂U Tn is a subalgebra. Thus, finite-dimensional incidence algebras are subalgebras of U Tn, generated by matrix units, and containing the diagonal matrices. As we are interested in finite-dimensional incidence algebras, we will assume from now on thatI(X)⊂U Tn.

The incidence algebras are very interesting by its own and, moreover, they are related with other branches of Mathematics. They give rise to interesting and chalenging combinatorial problems. For an extensive theory on incidence algebras see, for instance, the book [16].

2.2. Graded Jacobson radical. It is fundamental in our arguments to guarantee that the Jacobson radical of the incidence algebras are graded ideals.

For this, we have the following result, due to Gordienko:

Lemma 3 (Corollary 3.3 of [7]). Let A be a finite-dimensional algebra, graded by any group G. Suppose charF = 0or charF >dimA. Then the Jacobson radical

J(A)is a graded ideal.

Now, if I(X) has a G-grading, and G is abelian, then the commutator is a homogeneous operation. Moreover, [I(X), I(X)] =J(I(X)) is the Jacobson Radical of the incidence algebraI(X). Hence,J(I(X)) is a graded ideal. Thus, we proved Lemma 4. Let X be finite, and letI(X)be G-graded. Assume at least one of the following conditions: charF = 0, charF > dimI(X), or G is abelian. Then the

Jacobson radical ofI(X)is graded.

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2.3. Group algebras over fields. We include in this subsection a few facts concerning the group algebras, which will be essential for our purposes. We do not include the proofs, since the results are classical, or an adaptation of classical proofs when the base field isC, the field of complex numbers.

LetFbe a field,Ga group, and we denote ˆG={χ:G→F group homomorphisms}.

Lemma 5. Assume thatGis abelian finite. The following assertions are equivalent:

(i) G∼= ˆG,

(ii) FG∼=F^G^ˆ (as F-algebras),

(iii) for everyg∈Ghaving orderq,charFdoes not divide q, andFcontains a primitiveq-root of unit.

For instance, we will have the following situation. AG-graded algebraAadmits a subalgebraD⊆Awhich is, simultaneously isomorphic toF^mas ordinary algebras, and to FH as graded algebras, for some abelian subgroupH ⊆G. Then, in this case, the fieldFmust satisfy the properties (iii) of the lemma above.

Lemma 6. Let G be a finite abelian group, assume that Gˆ ∼= G, and let Gˆ = {χ1, . . . , χm}. Then the mapψ:FG→F^m, given by

(1) ψ(g) = (χ1(g), . . . , χm(g))

is an isomorphism of algebras. Moreover, every isomorphism of algebrasψ⁰ :FG→ F^m is given by (1), up to a permutation of the indexes.

3. Group gradings on incidence algebras

Let (X,≤X) be a finite poset, and let n = |X|. We fix a field F and G any group with multiplicative notation and neutral element 1, and we assume one of the conditions of Lemma 4. As mentioned above, we can rename the elements ofX asX ={1,2, . . . , n}, andi≤X jimpliesi≤jin the usual ordering of the integers.

Hence, we can identifyI(X)⊂U T_n. If 1≤i≤j≤nandx∈I(X),x(i, j) denotes the entry (i, j) of the elementx.

By direct computation, it is easy to see that for anyM ∈I(X), which is invertible inU T_n, we haveM⁻¹∈I(X).

Lemma 7. Lete∈I(X)be an idempotent. Then there existsM ∈I(X)such that M⁻¹eM is diagonal.

Proof. The proof is similar to the proof of Lemma 1 of [17].

We assume that I(X) acts on the left on the n×1 matrices. For each i = 1,2, . . . , n, let

ei =





 0 0 ... 1 ... 0





 ,

having non-zero entry only in the position (i,1). One, and only one, betweene·e_i or e_i −e·e_i will have non-zero entry at (i,1), where e·e_i is the usual matrix

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multiplication. Let fi be either e·ei, or ei−e·ei, the unique matrix with the non-zero entry in (i,1).

LetM be the matrix such that the columns aref1, f2, . . . , fn. By construction, M ∈I(X). Moreover, since the diagonal entries ofM are non-zero, it is invertible.

Also,f₁, f₂, . . . , f_n are eigenvectors ofe. Thus,M⁻¹eM is diagonal.

A classical result says that a set of diagonalizable matrices, which pairwise com- mutes, is simultaneously diagonalizable. We have a similar result forI(X):

Lemma 8. Let e1, . . . , es ∈ I(X) be orthogonal idempotents. Then there exists M ∈I(X)such thatM⁻¹eiM is diagonal, for eachi= 1,2, . . . , s.

Proof. The proof will be by induction ons. Ifs= 1, then the result follows from the previous lemma. So, assumes >1, andf₁, . . . , f_n is a basis such thate₁, . . . , e_s−1 are diagonal, and the matrixM having f₁, . . . , f_n as its columns lies inI(X).

Fixj ∈ {1, . . . , n}. Ife_sf_j = 0, then f_j is an eigenvector of e_s with eigenvalue 0, and there is nothing to do. If there is i < s such that e_if_j =f_j, then e_sf_j = eseifj = 0, hence we are in the previous case. So, assume that esfj 6= 0 and eifj = 0, for all i < s. As in the previous lemma, we can change fj to esfj or fj−esfj, in such a way that this new fj still form a basis{f1, . . . , fn}of Fⁿ. By construction,eifj = 0, for this new fj, andesfj is either 0 orfj. Continuing the process, we obtainf1, . . . , fs such that the matrixM, having these elements as its columns, is inI(X), and everyei is diagonal in this basis.

Lemma 9. Let x∈ U Tn be such that x=e+b, where e² =e6= 0 is a diagonal matrix, and b is a strict upper triangular matrix. Then there exists a non-zero idempotent inSpan{x, x², . . .}. In other words, there exists a non-zero polynomial p(λ)∈F[λ], such that p(0) = 0 andp(x)is a non-zero idempotent.

Proof. Let ¯Fan algebraic closure of F. By Jordan canonical form, there exists an invertible matrixmsuch that

mxm⁻¹=







1 ∗ 0 · · · 0 . .. ... ... 1 0 · · · 0

0 ∗

. .. 0





 .

With an adequates >1, we obtain mx^sm⁻¹=

J 0 0 0

,

whereJ is a triangular matrix, with 1’s in the diagonal. Now, there exists a poly- nomialp(λ) such thatp(J) = 1, the identity matrix. Indeed, let q(λ) =P

i≥0a_iλⁱ be the characteristic polynomial ofJ. By the Cayley-Hamilton Theorem,q(J) = 0.

Moreover, sinceq(0) =a₀= detJ = 1, we obtain thatp(λ) =P

i>0−a_iλⁱ satisfies p(J) = 1.

Lete⁰=p(x^s), thenme⁰m⁻¹=

1 0 0 0

, hencee⁰²=e⁰, as desired.

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Fix anyG-grading onI(X). Define

E(X) ={e∈I(X) non-zero homogeneous idempotent}.

Of course it is non-empty since 1∈E(X). Fore1, e2∈E(X), lete1e2denote (the non-necessarily partial order)e1I(X)⊆e2I(X) (usually,is called a preordering).

Lete∈E(X) be a minimal element according to(sinceI(X) has finite dimension, we can always find such minimale). Note that dege= 1, sincee²=e. By Lemma 7, we can assumeediagonal, up to a graded isomorphism. The algebraB=eI(X)e has a naturalG-grading, induced from the grading of I(X). We will prove thatB is a (commutative) graded division algebra.

Lemma 10. For every homogeneous x∈B,x /∈J(B),degxhas finite order.

Proof. Note that, ifx /∈J(B), thenxhas at least one non-zero entry in the diagonal.

So,x^m∈/ J(B) for allm∈N. In particular, x, x², x³, . . . are non-zero elements. If degxhas infinite order, then the elementsx, x², . . .have all different degree, thus they are linearly independent. However, B has finite dimension, so we obtain a contradiction. As a conclusion, degxhas finite order.

Recall, from Lemma 4, that J(B) is graded. Thus B/J(B) has a natural G- grading, induced from the grading onB.

Lemma 11. (i) The algebra B/J(B) is a commutative graded division algebra.

(ii) WriteB/J(B) =L

g∈GB¯g. Thendim ¯Bg≤1, for allg∈G.

Proof. If dim ¯B1 is not 1, then we claim that there would exists an idempotent e⁰ ∈B¯1 such that e⁰ e. Indeed, let ¯x∈B¯1 be a non-zero homogeneous element, not a scalar multiple ofe, moduloJ(B). Then there exists a homogeneousx∈Bof degree 1, having non-zero entries in the diagonal. We can writex=α1e1+α2e2+

· · ·+α_se_s+b, whereb∈J(B),e₁, e₂, . . . , e_sare diagonals orthogonal idempotents (sum of diagonal matrix units), andα₁, . . . , α_s∈Fare non-zero elements such that α_i 6= α_j for all i 6= j. Since x has non-zero entry in the diagonal, s > 0; and since xand e are linearly independent modulo J(B), we have necessarillys > 1.

The powers x, x², . . . , x^sare homogeneous elements of degree 1. By Vandermonde argument, we conclude thaty=e₁+b⁰ is a linear combination ofx, x², . . . , x^s, for some b⁰ ∈J(B), hence y is homogeneous of degree 1. Using Lemma 9, we obtain a homogeneous idempotent e⁰, which is a linear combination of y, y², . . .. Thus e⁰ ∈ B. Moreover, we have e⁰I(X) ( eI(X). But e is minimal, so we obtain a contradiction. As a conclusion, dim ¯B1= 1.

Now, let ¯x∈B/J(B) be non-zero and homogeneous. Thus, ¯xhas non-zero entry in the diagonal. Hence, ¯x^m6= 0, for allm∈N. By Lemma 10, ¯x^m∈B¯1, for some m∈N. This means ¯x^m=λ¯e, for someλ6= 0. As a consequence, ¯xis invertible.

Thus, B/J(B) is a graded division algebra. Moreover, since B/J(B) is a sum of copies ofF, as an ordinary algebra, then it is necessarily commutative, proving (i).

Since dim ¯B₁= 1 andB/J(B) is a graded division, then dim ¯B_g≤1, for allg∈G,

thus proving (ii).

Corollary 12. The algebraBis semisimple as an ordinary algebra, that isJ(B) = 0. In particular,B is a commutative graded division algebra with unite.

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Proof. Let J = J(B) and ¯B = B/J. Note that J/J² is a graded ¯B-module.

The right annihilator Ann^rB¯(J/J²) is a homogeneous subspace of ¯B. However, no nonzero homogeneous element of ¯B annihilates a nonzero homogeneous element of J/J², since ¯B is a graded division algebra, by previous lemma. Hence,J(B) = 0.

Since, by previous lemma,B/J(B) is a commutative graded division algebra, then B is a commutative graded division algebra as well.

From now on,ewill always denote a minimal idempotent as before. Denote by D(e) ={i∈X |e(i, i)6= 0}.

Corollary 13. The elements of D(e) are not pairwise comparable.

Proof. If not, supposei < j, withi, j∈D(e). Recall thatB=eI(X)e∼=I(Y), for some posetY (whereY =D(e)). Theneij ∈B. Moreover, one haseij ∈J(B), so

J(B)6= 0, contrary to Corollary 12.

Now, ife6= 1, consider the set

E1(X) ={e⁰∈I(X) idempotent|e⁰I(X)⊂(1−e)I(X)}.

We remark that E1(X) 6=∅, since 1−e ∈ E1(X). We claim that we can choose the idempotents belonging to E1(X) so that they commute with e. Indeed, if e⁰ ∈ E1(X), then ee⁰ = ee⁰² ∈ e(1−e)I(X) = 0. Also, let f1 = e⁰ −e⁰e. Then f₁² =f1, f1e =ef1 = 0 and f1I(X) =e⁰I(X). Thus,f1 ∈E1(X) is the required element. Let e₁ = e, and chose a homogeneous idempotent e₂ ∈ E1(X) which is minimal with respect to, and orthogonal toe₁.

Proceeding by induction, we assume that we have a set e₁, . . . , e_r of minimal homogeneous orthogonal idempontents. Then we define

Er(X) ={e⁰∈I(X) idempotent |e⁰I(X)⊆(1−e1· · · −er)I(X)}.

As before, we can find a minimal homogeneous idempotenter+1∈Er(X) which is orthogonal toe1, . . . , er. We can continue the process, and, since dimI(X) is finite, it ends within finitely many steps. Lete1, e2, . . . , et be the homogeneous idempotents, which are minimal with respect to, obtained by the previous construction.

By Lemma 8, we can assumee1, e2, . . . , etdiagonal.

Definition 14. The diagonal homogeneous idempotents e1, . . . , et will be called theminimal idempotents. We denoteE ={e1, . . . , et}.

We denote D_i =e_iI(X)e_i which is, according to Corollary 12, a commutative graded division algebra. Moreover,M_ij=e_iI(X)e_j is a graded (D_i, D_j)-bimodule.

The triple (D_i, D_j, M_ij) constitutes a triangular algebra.

If e_i and e_j are two minimal idempotents, we denote e_i E e_j if there exist x∈D(ei) andy∈D(ej) such thatx≤y.

Lemma 15. The relation Eis a partial order inE.

Before we prove Lemma 15, we need to investigate the relation between each Mij and the poset.

Recall that the structure of graded modules over graded division algebras is similar to the structure of vector spaces over division rings (see, for instance, [6, page 29]). More precisely, let D be a G-graded division algebra and let M be a

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graded right D-module. Then M ∼= L

αMα, where each Mα = ^[h^α^]D, for some hα∈G. Here, ifD=L

g∈GDg, then

[h]D=M

g∈G

[h]Dg, where^[h]Dg=Dh⁻¹g.

In particular, every graded rightD-module is free. A similar discussion is valid for graded leftD-modules.

Let ei and ej be minimal idempotents. For each x ∈ D(ei), we define the link of x to ej as the non-negative integer `(x, ej) = |{y ∈ D(ej) | x ≤ y}|.

Similarly, we define the link of ei to y, denoted `(ei, y). The link of ei to ej is

`(ei, ej) =|{(x, y)|x≤y, x∈D(ei), y∈D(ej)}|.

Lemma 16. Let e_i, e_j be minimal idempotents. Then, for any x ∈ D(e_i), y ∈ D(ej),

`(ei, ej) =|D(ei)|`(x, ej) =`(ei, y)|D(ej)|.

Proof. Let M_ij = e_iI(X)e_j, which is a free G-graded left D_i-module. We shall prove that`(x, e_j) = dim_D_iM_ij, for anyx∈D(e_i).

On one hand, asF-linear space, we have

Mij = M

(x,y)∈D(e_i)×D(e_j);x≤y

Fexy.

Thus,`(ei, ej) = dim_FMij = dim_FDidimD_iMij. On the other hand, letv1, . . . , v`

be aDi-basis ofMij, where` >0 (the case`= 0 is trivial).

Claim. Fork∈ {1, . . . , `}, letw∈D_iv_k, and asumew(x, y) = 0, butv_k(x, y)6= 0.

Thenw(x, y⁰) = 0, for ally⁰∈D(e_j).

Indeed, in this case,B={e_xxv_k |x∈D(e_i)}is a set of |D(e_i)| F-linearly independent elements. Since dim_FD_iv_k= dim_FD_i=|D(e_i)|,Bis a (non-homogeneous) F-vector space basis ofD_iv_k. So, writew=P

x∈D(ei)λ_xe_xxv_k. Ifv_k(x, y)6= 0, but w(x, y) = 0, then necessarilyλx= 0. This impliesw(x, y⁰) = 0, for ally⁰∈D(ej).

As a conclusion, sincev1, . . . , v`generateMijas aDi-linear space, every element ofMij is a linear combination of elements inDiv1, . . . , Div`. By the claim, in order to be possible to fill all the entries of the coordinates of type (x,·) in Mij, it is necessary that`≥`(x, ej). Hence,`≥max{`(x, ej)|x∈D(ei)}. And, therefore,

dim_FD_idim_D_iM_ij=`(e_i, e_j) = X

x∈D(e_i)

`(x, e_j)≤ |D(e_i)|`= dim_FD_idim_D_iM_ij which implies`(x, ej) = dimD_iMij, for any x∈D(ei) and we are done.

Now, we can prove the lemma:

Proof of Lemma 15. It is clear thatE is reflexive and transitive. So, let ei 6= ej

(thus, by construction,D(ei)∩D(ej) =∅), and assume thatei Eej, andej Eei. Then, by Lemma 16, every x ∈ D(e_i) is ≤ to at least one y ∈ D(e_j). In the same way, every y ∈ D(e_j) is ≤ to at least one x ∈ D(e_i). Hence, we can find x₁, x₂∈D(e_i), andy∈D(e_j) such thatx₁< y < x₂, so x₁< x₂. This contradicts

Corollary 13. Hence,Eis antisymmetric.

As a consequence of Lemma 15, we can rename the elements of X, in such a way that D(e₁) = {1,2, . . . , i₁}, D(e₂) = {i1+ 1, i₁+ 2, . . . , i₂}, ..., D(e_t) = {it−1+ 1, . . . , n}. Since e_iI(X)e_j = Span{ek` | k ∈ D(e_i), ` ∈ D(e_j), k < `}, we

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have dimeiI(X)ej ≤ dimDidimDj. Moreover, as graded vector space, I(X) = P

i,jeiI(X)ej. In this way, we obtain the proof of the Theorem 1.

Corollary 17. Up to a graded isomorphism, there exists commutative graded division algebras D1, D2, . . . , Dt, all of them having all homogeneous subspaces with dimension at most 1, such that

I(X)∼=







D1 M1,2 . . . M1,t

D2 . .. ... . .. M_t−1,t

Dt





 ,

where eachMi,jis a graded(Di, Dj)-bimodule of dimension at mostdim_FDidim_FDj. IfD is a commutative graded division algebra, where each homogeneous component has dimension at most 1 andD∼=F^dim^D, thenD∼=FH, whereH = SuppD.

Indeed, a set of nonzero homogeneous elements{Xu|u∈H}, where degXu=uis a vector space basis ofD. Also,XuXv =σ(u, v)Xuv, for some (nonzero)σ(u, v)∈F. Since D is associative, we obtain that σ is a 2-cocycle. Thus, σ ∈ Z²(H,F^×), and D ∼=F^σH (the twisted group algebra). To complete the proof, we need the following result, communicated by M. Kochetov:

Proposition 18. Let H be a finite abelian group and σ∈ Z²(H,F^×). IfF^σH ∼= F^|H|, thenFH ∼=F^|H|.

Proof. Given a finite-dimensional unital commutative algebraA overF, let N(A) denote the number of algebra homomorphisms fromAto F.

Claim. Ais a direct sum of copies ofFif and only ifN(A) = dimA.

Indeed, (⇒) is obvious, and (⇐) follows from linear independence of distinct homomorphisms.

Note that bothN and dim are multiplicative over tensor products of algebras.

Now, writeH =hh1i × · · · × hhri, direct product of cyclic groups, where eachhi

has orderm_i. Then, we notice that

F^σH ∼=F[x1]/(x^m₁¹−a1)⊗ · · · ⊗F[xr]/(x^m_r^r−ar), whereai=Qm_i−1

j=1 σ(hi, h^j_i)∈F. In the same way,FH ∼=Nr

i=1F[xi]/(x^m_i ⁱ−1).

From the remarks above,F^σH is a direct sum of copies ofFif and only if each F[xi]/(x^m_i ⁱ−ai) is a direct sum of copies ofF. The former condition is equivalent to Fcontainsmi distinct mi-roots ofai; and this implies thatF contains a primitive mi-th root of 1. Thus, each F[xi]/(x^m_i ⁱ−1) is a direct sum of F’s. Hence, by the remarks above,FH is a direct sum of copies ofF. As a consequence, F^σH ∼=FH as ungraded algebras. From [11, Corollary 1.2], we obtain thatαis a coboundary map. Thus,F^σH∼=FH as graded algebras.

Remark 19. Consider the notation of Corollary 17. Letei ∈Di be the unity. Set E ={1,2, . . . , t}. We define the partial orderEonE by the following (equivalent) conditions (see Lemma 15):

(1) iEj, (2) e_iEe_j,

(3) there existsx∈D(e_i) andy∈D(e_j) such thati≤j,

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(4) Mij 6= 0.

Note that we can identify the present definition ofE with that given in Definition 14. We callE theassociated poset of the graded algebraI(X).

Note that we have the following equality:

|H_i|= dimD_i=|D(e_i)|.

4. On the structure of graded bi-vector spaces

In this section, we shall investigate graded bimodules. We are interested in the special case whereGis an abelian group,H₁, H₂⊆Gare finite subgroups, andV is aG-graded (FH1,FH2)-bimodule. We will conclude that, in this situation,V is a FH-vector space, whereH =H1H2. Hence, the bimodules have a nice description in this case.

The notion of tensor product will play a fundamental role in this section. Recall that, ifR is anyF-algebra, M, N areF-vector spaces,M is a rightR-module, and N is a leftR-module, thenM⊗RN is a uniquely definedF-vector space, satisfying a universal property. It is spanned bym⊗n, wherem∈M, andn∈N, and they satisfymr⊗n=m⊗rn,r∈R. For the special case whereM andN are algebras, thenM⊗RN is an algebra as well.

We are interested in the following special case: letR, A1, A2 beF-algebras, and assume ϕ_i : R →A_i homomorphism of algebras, for i = 1,2. Then A₁ becomes a rightR-module by a·r =aϕ₁(r), and, similarly, A₂ becomes a left R-module.

So we can construct the tensor product A₁⊗_RA₂. Since ϕ₁ and ϕ₂ will play a somewhat significant role, we use the notation (A₁⊗_RA₂, ϕ₁, ϕ₂). In this case, A₁⊗_RA₂ is spanned bya₁⊗a₂, for a_i ∈β_i, whereβ_i is any basis of A_i; also, we haveb1ϕ1(r)⊗b2=b1⊗ϕ2(r)b2, for anyr∈R,b1∈A1, b2∈A2.

Now, assume further that R, A1, A2 are G-graded algebras, and ϕ1, ϕ2 are G- graded homomorphisms of algebras. SoA1 andA2 areG-graded R-modules. The algebraA1⊗RA2 admits a natural structure ofG-grading, where the homogeneous component of degreeg∈Gis spanned bya1⊗a2,ai∈Ai homogeneous satisfying dega1dega2 = g. Moreover, A1⊗RA2 becomes a G-graded algebra. In what follows, we will see that, in the case of group algebras, the just defined grading has a nicer way to be constructed.

Lemma 20. Let G be an abelian group, and H1, H2⊆Gbe finite subgroups. Let H =H1H2, andH=H1∩H2. Let ιj :FH →FHj be G-graded monomorphisms, forj= 1,2. Then, as graded algebras,

(FH1⊗_F_HFH2, ι1, ι2)∼=FH.

Proof. First we note thatFH1⊗_FHFH2has a natural well-definedG-grading, given by degh₁⊗h₂=h₁h₂, where h₁∈H₁, h₂∈H₂.

Now, since every homogeneous component ofFH_j has dimension at most 1, we see that an inclusionι_j(h) =χ_j(h)h, for eachh∈ H, for someχ_j(h)∈F. Moreover, h 7→ χ_j(h) ∈ F is a character of H. Consider the character given by χ⁻¹_j . It is known that every character ofHcan be extended to a character of H_j. So, let ¯χ_j be an extension ofχ⁻¹_j .

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Letϕ:FH1⊗_FHFH2→FH be defined by ϕ(h1⊗h2) = ¯χ1(h1) ¯χ2(h2)h1h2. If h∈ H, then

ϕ(h1ι1(h)⊗h2) =χ1(h) ¯χ1(h1h) ¯χ2(h2)h1hh2=χ1(h) ¯χ1(h) ¯χ1(h1) ¯χ2(h2)h1hh2, ϕ(h₁⊗ι₂(h)h₂) =χ₂(h) ¯χ₁(h₁) ¯χ₂(hh₂)h₁hh₂=χ₂(h) ¯χ₂(h) ¯χ₁(h₁) ¯χ₂(h₂)h₁hh₂. Sinceχ₁(h) ¯χ₁(h) =χ₂(h) ¯χ₂(h) = 1, we obtain that ϕis well-defined.

SinceGis abelian,ϕis an onto homomorphism of graded algebras.

LetPm

i=1λ_ig_i⊗h_i∈Kerϕ. As Kerϕis a graded subspace, we can suppose that g₁h₁ = . . . = g_mh_m. Moreover, we can assume that h₁, . . . , h_m are FH-linearly independent. Thus, g_ih_i = g_jh_j implies h_ih⁻¹_j ∈ H, which is a contradiction if m >1. Hence,m≤1, and this is sufficient to conclude that Kerϕ= 0.

Lemma 21. LetGbe an abelian group,H1, H2⊆Gfinite subgroups,H=H1∩H2. Let V be a finite-dimensional G-graded (FH1,FH2)-bimodule. Then, there exist m1, . . . , ms∈V homogeneous, and χ1, . . . , χs∈H, such thatˆ

V =FH₁m₁FH₂⊕ · · · ⊕FH₁m_sFH₂, andhmi=χi(h)mih, for alli= 1, . . . , s,h∈ H.

Proof. Let W_g be the homogeneous component of degree g of V. Given h ∈ H, letL_h, R_h :V →V denote the maps L_h(m) = hm, R_h(m) =mh. Then L_h is a linear isomorphism fromWg to Whg. Also,R_h−1 : Wgh →Wg is an isomorphism as well. Thus,LhR_h−1 ∈GL(Wg). It is elementary to see that this defines a linear representationh∈ H →LhR_h−1 ∈GL(Wg).

Since H is abelian, the representation is completely reducible, and each irre- ducible representation has degree 1. So, we can find a vector space basisv1, . . . , vr∈ Wg, and characters χ1, . . . , χr ∈ H such that LhR_h⁻¹(vi) = χi(h)vi, for all i = 1, . . . , r. In particular, this giveshvi=χi(h)vih, for allh∈ H.

It is clear that every componentWg⁰, whereg⁰∈H1gH2belongs toFH1v1FH2⊕

· · · ⊕FH1vrFH2. Thus, we can takeg2 ∈/ H1gH2 and repeat the process. Since

dimV <∞, the process ends.

In the notation of the previous lemma, giveng1∈H1 andg2∈H2, we have g₁hm_ig₂=χ_i(h)g₁m_ihg₂,∀h∈ H.

Lemma 22. Assuming the same hypotheses of the previous lemma, let χ ∈ H,ˆ and V = FH₁mFH₂, where hm = χ(h)mh, for all h ∈ H. Then V is a graded FH₁⊗_F_HFH₂-module.

Proof. Consider the inclusions

ι₁:h∈FH →h∈FH₁, ι2:h∈FH →χ(h)h∈FH2.

Define the left action of FH1⊗_FHFH2 on V by g1⊗g2m =g1mg2. It suffices to prove that this action is well-defined. For, assumeg1∈H1,g2∈H2,h∈ H. Then g1h⊗g2=χ(h)g1⊗hg2, and

(g1h⊗g2)m=g1hmg2=χ(h)g1mhg2= (g1⊗χ(h)hg2)m.

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Recall that, ifR andS are any rings,ϕ:R→S is a ring homomorphism, and M is an S-module; then M becomes an R-module if we set m·r = mϕ(r), for m∈M andr∈R. We summarize the results of this section.

Corollary 23. Let G be an abelian group, H1, H2 ⊆ G finite subgroups, H = H1∩H2,H =H1H2. LetV be a finite-dimensionalG-graded(FH1,FH2)-bimodule.

Then, there exist homogeneousm₁, . . . , m_s∈V and charactersχ₁, . . . , χ_s∈Hˆ such that

(2) V =FH1m1FH2⊕ · · · ⊕FH1msFH2,

wherehm`=χ`(h)m`h, for eachh∈ H. Moreover, for eachi, one hasFH1miFH2∼= (FH)^[deg^mⁱ^], as graded vector spaces.

Proof. Lemma 21 says that we can find homogeneousm1, . . . , ms∈V such that V =FH1m1FH2⊕ · · · ⊕FH1msFH2.

By Lemma 22, everyFH1miFH2has a structure of graded (FH1⊗_FHFH2, ιi1, ιi2)- module. By Lemma 20, there exists a graded isomorphismϕi :FH →(FH1⊗_FH

FH2, ιi1, ιi2). Hence, FH1miFH2 has a structure of graded FH-module, and it is 1-dimensional, since it isFH-spanned bymi. This concludes the proof.

Remark 24. It should be noted that Corollary 23 is no longer valid when the grading group is not abelian. Indeed, letH be a finite abelian group,C₂={1, w}the cyclic group of order 2, and letG=H∗C₂ be the free product ofH andC₂. Consider V =FH⊗_FFH, and define theG-grading onV byV =L

g∈GV_g, where Vh1wh2 = Span{h1⊗h2}, h1, h2∈H.

ThenV is aG-graded vector space, andV has a structure ofG-graded (FH,FH)- bimodule. For every nonzero homogeneousv∈V, we haveFHvFH =V; and it is not isomorphic to a shift of FH. Moreover,V is not a G-graded left FH⊗_FFH- module.

Now, we are ready to prove Theorem 2.

Proof of Theorem 2. Fixi, j. We consider the triangular algebra A=

F^mⁱ Vij

F^m^j

,

where m`=|H`|, which is graded isomorphic to

FHi Mij

FHj

. Lete`∈H` be the unit. ThenA = (ei+ej)I(X)(ei+ej), and it is the incidence algebra of the subposetY =D(ei)∪D(ej)⊆X.

LetH=Hi∩Hj, ˆH={χ1, . . . , χm}, and assume ˆH= ˆHi∩Hˆj. Write Hˆ_`=χ^(`)₁ Hˆ∪˙. . .∪χ˙ ^(`)_n

`

H,ˆ wherem`=mn`, for`∈ {i, j}(we can assumeχ^(`)₁ = 1).

By Lemma 6, up to renaming the entries, the graded isomorphism φ : FH_i⊕ M_ij⊕FH_j→A is such that, forh₁∈H_i,h₂∈H_j,

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φ(h1+h2) = (χ⁽ⁱ⁾₁ χ1(h1), . . . , χ⁽ⁱ⁾n_iχm(h1)) 0

(χ^(j)₁ χ1(h2), . . . , χ^(j)n_jχm(h2))

! .

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LetMij=FHim1FHj⊕ · · · ⊕FHimsFHj, with charactersµ1, . . . , µs, as in (2). We only need to prove that µ1, . . . , µs are pairwise distinct characters. By Corollary 23, dimFHim`FHj =|HiHj|, so φ(m`)∈Vij has at least|HiHj|non-zero entries.

To make notation easier, we will identify the elements ofD(e_i)∪D(e_j) with the respective characterχ^(`)r χs, as in (3). So, for instance, to denote the entry (1,1) of φ(h1+h2) in (3), we will writeφ(h1+h2)(χ⁽ⁱ⁾₁ χ1, χ⁽ⁱ⁾₁ χ1).

Claim. For each χ⁰ ∈ Hˆ_i, we can find unique χ⁰_k

1, . . . , χ⁰_k

nj ∈ Hˆ such that φ(m`)(χ⁰, χ^(j)0r χ⁰_k

r)6= 0, r= 1,2, . . . , nj.

Indeed, let µ ∈ Hˆ_i and let µ⁰ ∈ Hˆ_j be such that φ(m_`)(µ, µ⁰) 6= 0. Now, if µ⁰⁰ ∈ Hˆ_j is such that φ(m_`)(µ, µ⁰⁰) 6= 0, then µ⁰ and µ⁰⁰ cannot be in the same left coset, unless they coincide, as we will prove now. For eachh∈ H,hm_`h⁻¹= µ_`(h)m_`. On the other hand, hφ(m_`)h⁻¹(µ, µ⁰) =µ(h)φ(m_`)(µ, µ⁰)µ⁰(h⁻¹), and a similar relation holds forµ⁰⁰. This impliesµ⁰|_H= (µ`µ⁻¹)|_H =µ⁰⁰|_H. Hence, ifµ⁰ andµ⁰⁰ are in the same coset of ˆH, thenµ⁰ =µ⁰⁰. So, for eachµ∈Hˆ_i, there exist at most|Hj|/|H|elementsµ⁰ such thatφ(m_`)(µ, µ⁰)6= 0. Thus,φ(m`) has at most

|Hi||Hj|/|H|non-zero entries.

This implies that φ(m_`) has exactly |H_iH_j| non-zero entries. Moreover, the non-zero entries of theφ(m₁), . . . , φ(m_s) are disjoint.

Now, let`1, `2 withµ`₁ =µ`₂, and fixχ⁰∈H ⊆ˆ Hˆi, and letχ⁰₁, χ⁰₂∈H ⊆ˆ Hˆj be such thatφ(m`_k)(χ⁰, χ⁰_k)6= 0,k= 1,2. For eachh∈ H, again we havehm`_kh⁻¹= µ`_k(h)m`_k. On the other hand, (recall thatχ⁽ⁱ⁾₁ = 1,χ^(j)₁ = 1)

(hφ(m`_k)h⁻¹)(χ⁰, χ⁰_k) =χ⁰(h)χ⁰⁻¹_k (h)φ(m`_k)(χ⁰, χ⁰_k).

Hence,χ⁰₁=χ⁰µ⁻¹_`

k =χ⁰₂. Thus, this impliesm`₁ =m`₂. We also have the following immediate consequence of Corollary 23:

Corollary 25. Let Gbe abelian, and leti < j. Then:

(i) ifH_i∩H_j={1}, thenM_ij is either 0 orFH_ij, whereH_ij ∼=H_i×H_j. (ii) if H_i ⊆ H_j (in particular, H_i = H_j), then M_ij is a graded FH_j-vector

space, and 0≤dim_FH_jMij ≤ |Hi|.

Given a graded (FH₁,FH₂)-bimoduleMwith decomposition given by (2), denote [M] = [(χ₁, h₁), . . . ,(χ_s, h_s)], whereh_` = degm_`. We finish this section showing that the knowledge of those characters determines the isomorphism classes of the bimodules, or certain triangular algebras.

Lemma 26. Let V and V⁰ be G-graded (FH₁,FH₂)-bimodules, and denote[V] = [(χ₁, h₁), . . . ,(χ_s, h_s)], [V⁰] = [(χ⁰₁, h⁰₁), . . . ,(χ⁰_s0, h⁰_s0)]. Then, as G-graded bimodules, V ∼=V⁰ if, and only ifs=s⁰ and there existsσ∈S_ssuch that

h`≡h⁰_σ(`) (modH1H2), χ`=χ⁰_σ(`), ∀`= 1,2, . . . , s.

Proof. Write V = FH₁m₁FH₂ ⊕ · · · ⊕ FH₁m_sFH₂, V⁰ = FH₁m⁰₁FH₂ ⊕ · · · ⊕ FH₁m⁰_s0FH₂, as in (2). Let ψ : V → V⁰ be a graded isomorphism of bimodules. Then ψ is an isomorphism of H₁∩H₂ representations, where the action is given by L_h◦R_h−1. Let V_χ =P

χi=χFH₁m_iFH₂. Note that V_χ is the sum of all subrepresentations of V isomorphic toχ. So, ψ(Vχ) = P

χ⁰_i=χFH1m⁰_iFH2 =:V_χ⁰.

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Moreover,ψ:Vχ →V_χ⁰ is a graded isomorphism ofF(H1H2)-modules. This proves one implication.

The converse is immediate.

Lemma 27. Let V and V⁰ be G-graded (FH1,FH2)-bimodules, and denote[V] = [(χ1, h1), . . . ,(χs, hs)],[V⁰] = [(χ⁰₁, h⁰₁), . . . ,(χ⁰_s0, h⁰_s0)]. Then, asG-graded algebras,

A:=

FH1 V FH2

∼=

FH1 V⁰ FH2

=:A⁰ if and only if s=s⁰ and there existχ∈H\₁∩H₂ andσ∈S_s such that

h_`≡h⁰_σ(`) (mod H₁H₂), χ_`=χχ⁰_σ(`), ∀`= 1,2, . . . , s.

Proof. The converse is an elementary computation.

Letψ:A→A⁰ be a graded isomorphism of algebras. Then we have an induced graded isomorphism ¯ψ : A/V → A⁰/V⁰, moreover, ¯ψ(FHi) = FHi. So, there exists χi ∈ Hˆi such that ¯ψ(hi) = χi(hi)hi, for each hi ∈ Hi, for i = 1,2. Write V⁰=FH1m⁰₁FH2⊕ · · · ⊕FH1m⁰_s0FH2, and consider the mapϕ:A⁰ →A⁰ given by

ϕ

h1 h⁰₁m⁰_ih⁰₂ h2

=

χ⁻¹₁ (h1)h1 χ⁻¹₁ (h⁰₁)χ⁻¹₂ (h⁰₂)h⁰₁m⁰_ih⁰₂ χ⁻¹₂ (h₂)h₂

.

It is elementary to prove that ϕis a graded isomorphism of algebras. Moreover, givenh_i∈H_i andm∈V, we have

ϕψ(h₁mh₂) =h₁ϕψ(m)h₂.

Thus, ϕψ : V → ϕ(V⁰) is a graded isomorphism of bimodules. Consider the product of the restriction of the characters, χ =χ₁|_H₁_∩H₂ ·χ₂|_H₁_∩H₂. Note that, if [V⁰] = [(χ⁰₁, h⁰₁), . . . ,(χ⁰_s0, h⁰_s0)], then the decomposition of ϕ(V⁰) is [ϕ(V⁰)] = [(χχ⁰₁, h⁰₁), . . . ,(χχ⁰_s0, h⁰_s0)]. Thus, the result follows from Lemma 26.

5. The isomorphism problem

Assume that the conditions of Theorem 1 hold. Consider aG-grading onI(X) and let E be the associated poset. Denote bye /ce⁰ ife, e⁰ ∈ E, and e⁰ is a cover fore; that is,e6=e⁰,eEe⁰, and ifeEf Ee⁰, then eitherf =eorf =e⁰.

Note that the Jacobson RadicalJ =J(I(X)) coincides withP

eEe⁰,e6=e⁰eI(X)e⁰. Also,J² is the sum of alleI(X)e⁰⁰ where there exists a chain of different elements e/e⁰/e⁰⁰;J³is the sum of alleI(X)e⁰⁰⁰, wheree/e⁰/e⁰⁰/e⁰⁰⁰, and so on. Givene, e⁰ ∈ E, e E e⁰, we can always construct a chain of elements containing only subsequent covers, starting withe, and ending withe⁰; however, we can have two different such chains having different number of elements. Nonetheless, ife /ce⁰, thene / e⁰ is the unique chain of subsequent covers linking eande⁰. LetJ1=L

e/_ce⁰eI(X)e⁰. The previous discussion is the proof of the following:

Lemma 28. Let I(X)be G-graded, and letJ1 be as above.

(i) The restriction of the natural projection π: J1 →J/J² is a graded linear isomorphism.

(ii) J =J₁⊕P

m>1J₁^m.