QUANTUM COMPLETE RINGS AND BLOCKS WIH ONE SIMPLE MODULE

MILES HOLLOWAY AND RADHA KESSAR

Abstract. We generalize a construction of Benson and Green to realize a large class of quantum complete intersections as basic algebras of non-principal blocks of certain finite groups. The realization arises from an isomorphism of a quantum complete ring to a skew group ring. We also show that blocks of finite groups with normal abelian defect groups, abelian inertial quotients and, up to isomorphism, only one simple module have basic algebras amongst this class of quantum complete intersections. We also study the Ext rings and finite p′-coverings of these quantum complete intersections.

1. Introduction

The inspiration for this paper has been the work of Benson and Green in [2], where they showed that a class of quantum complete intersections occur as basic algebras of blocks of finite groups. In this paper we extend these results to a larger (related) class of quantum complete intersections. We also prove a converse result showing that if a finite group has a block with one simple module, abelian defect groups and abelian inertial quotient, then the basic algebra of the block is a quantum complete intersection very similar to that considered in [2]. Our main result is

Theorem 1.1. Let p be a prime and k an algebraically closed field of characteristic p. Let q = (q_{ij}) ∈ Mat_{r}(k) be such that q_{ij}q_{ji} = 1, q_{ii} = 1 for 1 ≤ i, j ≤ r.

Let A be the k-algebra given by generators X_{i} (1 ≤ i ≤ r), and relations X_{i}X_{j} = q_{ij}X_{j}X_{i} and X_{i}^{p^{n_{i}}} = 0, 1 ≤ i, j ≤ r, where n_{i} ≥ 1 are given integers.

(i) Suppose that q_{ij}^{p−1} = 1 for all i, j, 1 ≤ i, j ≤ r. Then there exists a finite group G, a centrally primitive idempotent e ∈ kG and an integer d > 1 such that the block kGe is isomorphic to Mat_{d}(A). The block kGe has normal defect group isomorphic to ℤ/p^{n_{1}}ℤ × ⋯ × ℤ/p^{n_{r}}ℤ, abelian inertial quotient, and up to isomorphism, a unique simple module.

(ii) Suppose that G is a finite group, kGe a block of kG with normal abelian defect group, abelian inertial quotient and up to isomorphism a unique simple module. Let N ≅ ℤ/p^{n_{1}}ℤ × ⋯ × ℤ/p^{n_{r}}ℤ be the defect group of kGe. Then there exists q = (q_{ij}) ∈ Mat_{r}(k) with q_{ij}q_{ji} = 1, q_{ii} = 1 for 1 ≤ i, j ≤ r such that any basic algebra of kGe is isomorphic to the algebra A defined above.

(iii) Suppose p^{n_{i}} > 2 for all 1 ≤ i ≤ r. Then the algebra Ext_{A}^{∗}(k, k) is the graded commutative algebra given by generators x_{i} and y_{i} for 1 ≤ i ≤ r, with deg(x_{i}) = 2, deg(y_{i}) = 1, and relations

x_{i}x_{j} = q_{ji}^{p^{n_{i}+n_{j}}} x_{j}x_{i}, y_{i}x_{j} = q_{ji}^{p^{n_{j}}} x_{j}y_{i}, y_{i}y_{j} = −q_{ji}y_{j}y_{i}, y_{i}^{2} = 0.

(iv) A has a finite p′-cover Ã which supports a Hopf algebra structure, which in general is neither commutative nor cocommutative. The category of (finite-dimensional) graded A-modules is equivalent to the category of (finite-dimensional) Ã-modules.

Statements (i), (iii) and (iv) with the additional hypothesis that all n_{i} = 1 are proved in [2].

The paper is organized as follows. In §2 we recall the construction of the blocks in [2]. In §3 we prove (i) by extending this construction of blocks, but with arbitrary abelian defect group, and produce an explicit isomorphism to a matrix ring over a quantum complete intersection algebra A. This isomorphism comes from an isomorphism of a quantum complete ring to a skew group ring which uses properties of the Artin-Hasse function. In §4 we employ Külshammer’s structure theorem on blocks and prove (ii). Finally in §5 we observe that the results from [2] concerning Ext-rings and p′-covers extend easily to the more general quantum complete intersections considered here and thus we prove parts (ii) and (iii) of Theorem 1.1. In Appendix A we list some basic facts about multiplicative representatives and the Artin-Hasse function needed in §3.

Acknowledgement. Part of this work was carried out while the first author was visiting Ohio State University with the aid of an EPSRC postgraduate fellowship grant. The first author would like to thank both EPSRC for its financial support and Ohio State University for their hospitality.

2. The construction of Benson and Green

For the benefit of the reader and to establish some of our notation we briefly recall the relevant results from [2]. Throughout this section, k will denote a field of characteristic p with p an odd prime which has enough roots of unity to be a splitting field for the group H_{1} constructed below. Here we differ slightly from [2] where k is an arbitrary field of characteristic p. We do this in order to cover the case where H_{1} has exponent greater than (p − 1). In §§2.1 we recall the quantum complete intersection algebra in [2], in §§2.2 we briefly describe the construction of the Benson-Green blocks and in §§2.3 we state the main isomorphism from [2].

2.1. The quantum complete intersection algebra A. In [9] Manin defined the concept of a quantum symmetric algebra. With R a commutative ring, a square matrix q = (q_{ij}) (for 1 ≤ i, j ≤ r) of elements of R is said to be a commutation matrix if

q_{ij}q_{ji} = q_{ii} = 1 for all 1 ≤ i, j ≤ r.

The quantum symmetric algebra, R_{q}[X_{1}, … X_{r}], is then defined to be

R_{q}[X_{1}, … X_{r}] := R⟨X_{1}, …, X_{r}⟩/(X_{i}X_{j} − q_{ij}X_{j}X_{i}).

There is a notion of regular sequences for R_{q}[X_{1}, … X_{r}] analogous to those for ordinary commutative rings, see [1].

The case of interest in [2] is for R = k and q_{ij} ∈ 𝔽_{p} (viewed as elements in k, in particular they also satisfy q_{ij}^{p−1} = 1). The k_{q}[X_{1}, … X_{r}]-regular sequence X_{1}^{p}, …, X_{r}^{p} is taken and the following quotient is defined.

A := k_{q}[X_{1}, … X_{r}]/(X_{1}^{p}, …, X_{r}^{p}).

This is the quantum complete intersection algebra referred to in the title of [2].

2.2. Construction of non-principal blocks.

2.2.1. Construction of the group. A central extension H, dependent upon certain parameters q_{ij} ∈ 𝔽_{p}, is defined,

(2.1)  1 → Z → H → \overbrace{ℤ/(p − 1)ℤ × ⋯ × ℤ/(p − 1)ℤ}^{r} → 1

where Z = ℤ/(p − 1)ℤ.

An isomorphism z : 𝔽_{p}^{×} ⟶^{≅} Z is fixed. This is equivalent to choosing a faithful character χ ∈ Irr(Z) such that χ(z(q)) = q, ∀q ∈ 𝔽_{p} (i.e. an inverse to z).

The central extension group H is defined by generators and relations as follows. Let h_{i} ∈ H be the preimage of a generator of the i^{th} factor on the right hand side of the extension (2.1). Identify the elements z(q) (q ∈ 𝔽_{p}^{×}) with their embedding in H. Thus {z(q), h_{i}}, for q ∈ 𝔽_{p}^{×}, i ∈ {1, ⋯, r} are a set of generators of H. The relations that define H are:

(1) h_{i}^{p−1} = 1 for 1 ≤ i ≤ r.

(2) h_{i}h_{j} = z(q_{ji})h_{j}h_{i} for 1 ≤ i, j ≤ r. (Equivalently χ([h_{i}, h_{j}]) = q_{ji}.)

(3) h_{i}z(q) = z(q)h_{i} for q ∈ 𝔽_{p}^{×}, i ∈ {1, …, r}.

The group H is now defined to act on an elementary abelian p-group, N ≅ (ℤ/(p−1)ℤ)^{r}, of rank r. With g_{i} a generator of the i^{th} cyclic factor of N, the action of H on N is defined by:

(4) ^{h_{i}}g_{j} = g_{j}^{q_{ji}} for 1 ≤ i, j ≤ r. (Identifying q_{ji} with the integer in {1, …, p − 1} that represents it.)

(5) ^{z(q)}g_{j} = g_{j} for 1 ≤ j ≤ r and q ∈ 𝔽_{p}^{×}.

The group G is defined to be the semidirect product, G = N ⋊ H.

2.2.2. Construction of the block. Let H_{0} be defined as the kernel of the map H → Aut(N) defining the action for G. This H_{0} can be interpreted as Z(H) (so in particular Z ≤ H_{0}). We have a non-degenerate symplectic form, f, on H/H_{0} (viewed as a vector space over 𝔽_{p}, since Z ≅ 𝔽_{p}^{×}) defined by commutation. That is we have

f : H/H_{0} × H/H_{0} → 𝔽_{p}^{×}, where f(h, h′) = χ([hH_{0}, h′H_{0}]).

Because f is non-degenerate, the dimension of the vector space H/H_{0} must be a square. That is |H : H_{0}| = d^{2}. The group H_{1} is defined to be the preimage in H of a maximal isotropic subspace of H/H_{0}. In particular we have |H : H_{1}| = |H_{1} : H_{0}| = d. The group H_{1} can be interpreted as a maximal abelian subgroup of H.

An irreducible character χ_{1} ∈ Irr(H_{1}) is taken such that its restriction to Z is χ (it is here that we need k to have sufficient roots of unity, to obtain such a χ_{1}). That is χ_{1}|_{Z} = χ. Define χ_{0} ∈ Irr(H_{0}) by χ_{0} = χ_{1}|_{H_{0}}.

Taking ρ : kH → Mat_{d}(k) to be the induced representation, with character ψ = χ ↑^{H}, it’s easy to see that ψ(h) = 0 if h ∉ H_{1} and also that ψ ↓_{H_{1}}= ∑_{t∈[H:H_{1}]} ^{t}χ (with {^{t}χ|t ∈ [H : H_{1}]} distinct linear characters). It follows by Frobenius reciprocity that ρ is an irreducible representation and the corresponding idempotent, e, can easily be shown to be

e = \frac{1}{|H_{0}|} ∑_{h_{0}∈H_{0}} χ_{0}(h_{0})h_{0}^{−1}.

By construction ρ induces an isomorphism kHe ⟶^{≅} Mat_{d}(k). But e ∈ Z(kG) and hence is a primitive central idempotent determining a block, kGe, that has only one simple module, viz the one represented by ρ. This is the non-principal block in the title of [2].

2.3. The isomorphism. In §4 of [2] an explicit isomorphism is constructed

ϕ : kGe ⟶^{≅} Mat_{d}(A)
ϕ(h_{i}) = ρ(h_{i})
ϕ(g_{i}) = exp(ρ(h_{i})X_{i})

Here, for X ∈ A such that X^{p} = 0, then exp(X) := ∑_{i=0}^{p−1} X^{i}/i! makes sense and has all the usual properties of an exponential such as exp(X)^{a} = exp(aX) (a ∈ ℤ). With these properties it’s easy to check that the algebra structure in kGe (in particular (4) of §§§2.2.1) is respected by ϕ.

It is also clear that this isomorphism won’t extend to N having a cyclic factor of order p^{2} (or greater) so that the corresponding X satisfies X^{p^{2}} = 0, because then exp(X) would involve division by p and char(k) = p.

3. Benson-Green blocks with arbitrary abelian defect

In this section we find a suitable isomorphism that generalizes that in §§2.3. In §§3.1 we examine the construction in §§2.2 for arbitrary abelian group N. In §§3.2 we define a skew group ring and prove the main result for this section, Theorem 3.1; an explicit isomorphism from the skew group ring to a quantum complete ring. A consequence of this, Corollary 3.3, will be an isomorphism that extends that in §§2.3 for the blocks kGe defined in §§3.1, thus proving Theorem (i) of 1.1. We continue to assume p an odd prime and k a field having sufficient roots of unity.

3.1. Definition of the blocks considered. We keep the same construction for H in §2 and consider a general finite abelian p-group,

N := ℤ/p^{n_{1}}ℤ × ⋯ × ℤ/p^{n_{r}}ℤ,

with integers n_{i} ≥ 1 (1 ≤ i ≤ r).

We need to define an action of H on N with Z in the kernel. This really amounts to defining an action of H on ℤ/p^{n}ℤ for n ≥ 1 any integer. If g is a generator for ℤ/p^{n}ℤ and h_{i} is one of the generators for H considered in §§2.2.1, then we need to specify an integer a ∈ {1, …, p^{n} − 1} such that ^{h_{i}}g = g^{a}. Clearly a ∈ U(ℤ/p^{n}) and in order to be an action a must have order p − 1 in U(ℤ/p^{n}ℤ).

The group N has a unique subgroup, N′, that is elementary abelian of rank r. The action of H on N induces an action on N′. It clearly makes sense to demand that this action on N′ is as given in §§2.2.1. In other words we assume that the induced action on ⟨\overline{g} := g^{p^{n−1}}⟩ ≅ ℤ/pℤ is determined. This action is given by^{h_{i}} \overline{g} = \overline{g}^{b} where b ∈ \{1, …, p−1\} is fixed. But as integers, b = a (mod p). This together with a having order p − 1 in U(ℤ/p^{n}ℤ) uniquely determines a ∈ {1, …, p^{n} − 1}, in fact a = b^{p^{n−1}} (mod p^{n}). This is nothing other than the image of the multiplicative representative, r_{0}(b) ∈ ℚ_{p}, of b ∈ 𝔽_{p}^{×} under the map ℚ_{p} → ℚ_{p}/(p^{n}) (see Appendix A). We’ll therefore write r_{0}(b)_{n} := r_{0}(b) (mod p^{n}).

We can now define the action of H on N. We keep the commutation matrix q = (q_{ij}) from §2, in particular q_{ij} ∈ 𝔽_{p}, and define the action by

(1) ^{h_{i}}g_{j} = g_{j}^{r_{0}(q_{ji})_{n_{j}}} for 1 ≤ i, j ≤ r.

(2) ^{z(q)}g_{j} = g_{j} for 1 ≤ j ≤ r and q ∈ 𝔽_{p}^{×}.

Let G := N ⋊ H, of order |G| = p^{n}(p − 1)^{r+1} (where n = ∑ n_{i}). Then kG has a block idempotent, e, as constructed in §2 and kGe is a block with defect group N having up to isomorphism only one simple module.

The field k can be taken to be any extension of 𝔽_{p} that has (p−1)^{t}th roots of unity (see §§§2.2.2). For the sake of definiteness we’ll take k to be a finite extension of 𝔽_{p} having such roots of unity and we will take as a p-modular system (K, O, k) where K is a local field that is a finite (unramified) extension of ℚ_{p}. If a larger residue field k is needed we could extend K as required.

We first observe that with our p-modular system, the character χ_{1} used in §§2.2.2 lifts to a character χ̃_{1} ∈ Hom(H_{1}, O^{×}). Moreover the lift of χ ∈ Hom(Z, k^{×}) (which takes values in the prime field 𝔽_{p}) is χ̃ ∈ Hom(Z, O^{×}), where χ̃(z(q)) = r_{0}(χ(z(q))) (for q ∈ 𝔽_{p}).

The construction of the irreducible representation ρ therefore lifts to a representation ρ̃ (over O) and we have an O -isomorphism ρ̃: OGẽ ⟶^{≅} Mat_{d}(O), where ẽ ∈ Z(OH) is the lift of e.

3.2. Quantum complete rings and skew group rings. With commutation matrix r_{0}(q) := (r_{0}(q_{ij})) ∈ Mat_{r}(O) we can consider completing the quantum symmetric ring O_{r_{0}(q)}[X_{1}, …, X_{r}] with respect to the ideal n = (X_{1}, …, X_{r}), obtaining the quantum complete ring O_{r_{0}(q)}[[X_{1}, …, X_{r}]].

Our aim is to generalize the isomorphism in §§2.3 by replacing A with the quantum complete ring above. If we do this replacement then we also need a suitable replacement for kGe. Regarding the group algebra kG as a skew group ring (kN)H then it is clear that we’ll need to find an appropriate replacement for kN in this skew group ring. To this end we take the complete local ring T = O[[Z_{1}, …, Z_{r}]] and define an O-linear action of H on T by

h_{j}Z_{i} := (1 + Z_{i})^{r_{0}(q_{ji})} − 1.

This does induce an automorphism of T because^{h_{j}}(Z_{i}) ≡ r_{0}(q_{ji})Z_{i} (mod Z_{i}^{2}) and r_{0}(q_{ji}) ∈ O^{×}. Moreover this action is well defined because the order of h_{j} is p − 1 and r_{0}(a^{p−1}) = 1 for all a ∈ 𝔽_{p}. So we can form the skew group ring T H. The idempotent ẽ is central in T H, and hence we have the summand T Hẽ. Our main result in this section is

Theorem 3.1. There is an isomorphism of O-algebras

ϕ : O[[Z_{1}, …, Z_{r}]]Hẽ ⟶^{≅} Mat_{d}(O_{r_{0}(q)}[[X_{1}, …, X_{r}]]).

Proof. We define ϕ by

ϕ(h_{i}) = ρ̃(h_{i}) for h_{i} ∈ H, 1 ≤ i ≤ r,
ϕ(Z_{i}) = E(ρ̃(h_{i})X_{i}) − 1 for 1 ≤ i ≤ r,

where E(X) ∈ O[[X]] is the Artin-Hasse function (see Appendix A) and extend ϕ by O linearity and algebra multiplication to obtain a map between the two rings. This makes sense because E(X) − 1 ≡ X (mod X^{2}) (see Appendix A). To have a well defined map we must check that the relations are preserved. That is we have to check

ϕ(h_{j}(1 + Z_{i})h_{j}^{−1}) = ϕ(h_{j})(1 + ϕ(Z_{i}))ϕ(h_{j})^{−1}

By definition the left hand side is ϕ((1 + Z_{i})^{z_{0}(q_{ji})}) = (ϕ(1 + Z_{i}))^{z_{0}(q_{ji})} which is

E(ρ(h_{i})X_{i})^{r_{0}(q_{ji})},

while the right hand side is

E(r_{0}(q_{ji})ρ(h_{i})X_{i}).

But these are known to be equal by properties of the Artin-Hasse function (see Appendix A). So ϕ does indeed define an O-algebra homomorphism.

To check that ϕ induces an isomorphism it suffices (just as in the commutative case -see Theorem 7.16 of [4]) to check that the induced map grϕ on the associated graded rings is an isomorphism.

Set R = O[[Z_{1}, …, Z_{r}]]Hẽ, S = Mat_{d}(O_{r_{0}(q)}[[X_{1}, …, X_{r}]]) and let m = (Z_{1}, …, Z_{r}), n = (X_{1}, …, X_{r}) be corresponding ideals and recall that

grR = R/m ⊕ m/m^{2} ⊕ m^{2}/m^{3} ⊕ · · · .

Using the facts that R/m ≅ OHẽ and ^{h_{j}}(Z_{i}) ≡ r_{0}(q_{ji})Z_{i} (mod m^{2}), it’s easy to see that

grR ≅ O[Y_{1}, …, Y_{r}]Hẽ and grS ≅ Mat_{d}(O_{r_{0}(q)}[X_{1}, …, X_{r}]),

where the action of H on O[Y_{1}, …, Y_{r}] is given by ^{h_{j}}Y_{i} = r_{0}(q_{ji})Y_{i}. Clearly the induced map

grϕ : O[Y_{1}, …, Y_{r}]Hẽ → Mat_{d}(O_{r_{0}(q)}[X_{1}, …, X_{r}])

is given by grϕ(h_{i}) = ρ̃(h_{i}), grϕ(Y_{i}) = ρ̃(h_{i})X_{i}. This is clearly an isomorphism with inverse X_{i} → h_{i}^{−1}Y_{i}. Hence ϕ is an isomorphism. □

Corollary 3.2. For positive integers t_{1}, …, t_{r}, there is an isomorphism

ϕ : (O[Z_{1}, …, Z_{r}]/(Z_{1}^{t_{1}}, …, Z_{r}^{t_{r}}))Hẽ ⟶^{≅} Mat_{d}(O_{r_{0}(q)}[X_{1}, …, X_{r}]/(X_{1}^{t_{1}}, …, X_{r}^{t_{r}}))

Proof. The map ϕ from Theorem 3.1 is such that ϕ(Z_{i}^{t}) = (ρ(H_{i})X_{i})^{t} (mod X_{i}^{t+1}). The result follows immediately. □

Corollary 3.3. There is an isomorphism

ϕ : kGe ⟶^{≅} Mat_{d}(k_{q}[X_{1}, …, X_{r}]/(X_{1}^{p^{n_{1}}} , …, X_{r}^{p^{nr}}))

Proof. The result follows on applying k ⊗_{O} − to the isomorphism obtained from Corollary 3.2, taking t_{i} = p^{n_{i}} for 1 ≤ i ≤ r, and then composing the result with the isomorphism

kGe ≅ (k[Z_{1}, …, Z_{r}]/(Z_{1}^{p^{n_{1}}} , …, Z_{r}^{p^{nr}}))He

given by sending g_{j} − 1 to Z_{j}. This latter isomorphism makes sense because, for q ∈ 𝔽_{p}, we have (1 + Z_{j})^{r_{0}(q)} ≡ (1 + Z_{j})^{r_{0}(q)_{n_{j}}} (mod Z_{j}^{p^{n_{j}}}) so that the action of H on N agrees with the action defined by (1) and (2) of §3.1 under the isomorphism sending g_{j} − 1 to Z_{j}, for 1 ≤ j ≤ r. □

4. Blocks with normal abelian defect groups and abelian inertial quotient

In this section we consider the blocks described in Theorem 1.1. That is blocks of finite groups with abelian defect and inertial quotient groups which have, up to isomorphism, only one simple module. We prove part (ii) of Theorem 1.1 in Corollary 4.3. This corollary is a consequence of combining Külshammer’s theory on the structure of blocks with normal defect groups with our main result in this section, Theorem 4.2. Throughout this section p will be a prime and k a field of characteristic p having sufficient roots of unity to define necessary characters. In the sequel, whenever L is a finite group of order prime to p, by a character of L we will mean a character of an L representation over k.

We begin with the following elementary lemma.

Lemma 4.1. Let

1 → Z → H → L → 1

be a central extension of finite groups such that L is abelian, Z cyclic and Z = Z(H). For l ∈ L, define the map ϕ_{l} : L → Z by ϕ_{l}(k) = [l̃, k̃] = l̃^{−1}k̃^{−1}l̃k̃ where l̃and k̃ are lifts of l and k respectively in H. Then ϕ_{l} is a group homomorphism and the map ϕ : L → Hom(L, Z) given by ϕ(l) = ϕ_{l} is an isomorphism of groups.

Proof. Since Z is central in H, the maps ϕ_{l} and ϕ are well defined. Let a, b, c be elements of H. Then [a, bc] = [a, c][a, b]^{c} = [a, b][a, c] since [a, c] ∈ Z(L). Similarly [ab, c] = [a, c][b, c]. Thus the maps ϕ_{l} and ϕ are group homomorphisms. The injectivity of ϕ is immediate from the fact that Z(H) = Z. The surjectivity of ϕ follows from the fact that Hom(L, Z) is a subgroup of the character group L̂ := Hom(L, ℂ^{×}) and the fact that L ≅ L̂. □

Theorem 4.2. Let N = ℤ/p^{n_{1}}ℤ × ⋯ × ℤ/p^{n_{r}}ℤ be an abelian p-group and L an abelian p′-subgroup of Aut(N). Let

1 → Z → H → L → 1

be a central extension of finite groups such that Z is cyclic.

Let G := N ⋊ H be the semidirect product (with Z acting trivially on N) and let χ be a faithful irreducible character of Z with e = \frac{1}{|Z|} ∑_{z∈Z} χ(z)z^{−1} the corresponding idempotent, which is central in kG.

Suppose the algebra kGe has only one isomorphism class of simple modules, which has dimension d. Then there is a commutation matrix q ∈ Mat_{r}(k) and an isomorphism of algebras

kGe ≅ Mat_{d}(k_{q}[X_{1}, …, X_{r}]/(X_{1}^{p^{n_{1}}} , …, X_{r}^{p^{nr}})).

Before proving Theorem 4.2, it’s worth recalling Külshammer’s structure theory for blocks with normal defect groups (see [8]). If e is a block idempotent of a finite group G which has normal defect group N and inertial quotient L, then there is an injection of L into Aut(N) and a central extension of finite groups

1 → Z → H → L → 1

with Z cyclic and a faithful character χ ∈ Irr(Z), with corresponding idempotent e, such that the block kGe is Morita equivalent to k(N ⋊ H)e (in the semi-direct product the action of H is through the action of L, with Z acting trivially). It follows that Theorem 4.2 has the following consequence.

Corollary 4.3. Let G be a finite group and kGe a block with normal defect group N. Suppose N is abelian, with N ≅ ℤ/p^{n_{1}}ℤ × ⋯ × ℤ/p^{n_{r}}ℤ, and kGe has abelian inertial quotient.

If kGe has one isomorphism class of simple modules, then there is a commutation matrix q ∈ Mat_{r}(k) such that kGe is Morita equivalent to the quantum symmetric algebra k_{q}[X_{1}, …, X_{r}]/(X_{1}^{p^{n_{1}}} , …, X_{r}^{p^{nr}}).

Proof of Theorem 4.2. The hypothesis implies that the algebra kHe has one isomorphism class of simple modules. Since H is a p′-group, this means that kHe ≅ Mat_{d}(k) where d^{2} = |L|. In particular, Z = Z(H).

Thus, Lemma 4.1 applies. With ϕ_{l} and ϕ as in Lemma 4.1, let χ_{∗} : Hom(L, Z) → Hom(L, k^{×}) be given by χ_{∗}(f) = χ ∘ f. The faithfulness of χ implies that χ_{∗} is injective and so the map χ_{∗} ∘ ϕ : L → Hom(L, k^{×}) is an injective group homomorphism. On the other hand, since L is an abelian group of order prime to p, L ≅ Hom(L, k^{×}). So we have an isomorphism of groups

(4.1)  χ_{∗} ∘ ϕ : L ⟶^{≅} Hom(L, k^{×}).

Because kHe is a central simple subalgebra of kGe we have kGe ≅ C_{kGe}(kHe) ⊗_{k} kHe. Thus, in order to prove the theorem, we need to show that C_{kGe}(kHe) ≅ A where A := k_{q}[X_{1}, …, X_{r}]/(X_{1}^{p^{n_{1}}} , …, X_{r}^{p^{nr}}) is the quantum complete intersection algebra in the statement of the theorem (we still have to define q). We’ll do this by finding suitable elements in C_{kGe}(kHe) that can act as the images of the X_{i} ∈ A. To this aim we need to examine the action of H on N.

Since Z acts trivially on N and L = H/Z is abelian, we may write N as a direct product N ≅ Q_{1} ×⋯×Q_{s} where for each u, 1 ≤ u ≤ s, Q_{u} is a non-trivial H-invariant homocyclic subgroup of N (see [6], Chapter 3, Theorem 2.2.). Moreover if Q_{u} ≅ (ℤ/p^{m_{u}}ℤ)^{r_{u}} (1 ≤ u ≤ s) then we have

\{\overbrace{m_{1}, …, m_{1}}^{r_{1}}, …, \overbrace{m_{s}, …, m_{s}}^{r_{s}}\} = \{n_{1}, …, n_{r}\}.

For 1 ≤ u ≤ s we can consider kQ_{u} as a kH-module (via the conjugation action of H on Q_{u}). The subspaces J(kQ_{u}) and J^{2}(kQ_{u}) are kH-submodules of kQ_{u}. As Z acts trivially on Q_{u}, they are kL-modules and since L is a p′-abelian group we have

J(kQ_{u}) ≅ J^{2}(kQ_{u}) ⊕ W_{u}

where W_{u} is a direct sum of r_{u} 1-dimensional irreducible kL (and kH)-modules. If kw is one of these 1-dimensional modules with basis w ∈ J(kQ_{u}) \ J^{2}(kQ_{u}) then w^{m_{u}} = 0. It follows that, by considering all W_{u}, 1 ≤ u ≤ s, we can find basis elements, say w_{1}, …, w_{r}, for the r 1-dimensional kH-modules indexed so that w_{i}^{p^{n_{i}}} = 0.

For 1 ≤ i ≤ r, let η_{i} ∈ Hom(H, k^{×}) be the character associated to the kH-module kw_{i}. As Z acts trivially on w_{i}, η_{i} factors through the character η̄_{i} ∈ Hom(L, k^{×}). Through the isomorphism (4.1) this corresponds to some element in L. Applying this argument to the character η_{i}^{−1} ∈ Hom(H, k^{×}), it corresponds to l_{i} ∈ L say. Let h_{i} ∈ H be a lift of l_{i} then we have, for h ∈ H,

(4.2)  hw_{i}h^{−1} = η_{i}(h)w_{i} = χ([h_{i}, h])^{−1}w_{i}

Defining v_{i} := h_{i}^{−1}w_{i} (=w_{i}h_{i}^{−1}) then v_{i}^{p^{n_{i}}} = 0 and for h ∈ H,

\begin{aligned}hv_{i} & = [h^{−1}, h_{i}]h_{i}^{−1}hw_{i} \\ & = [h^{−1}, h_{i}]h_{i}^{−1}η_{i}(h)w_{i}h \\ & = [h_{i}, h]η_{i}(h)v_{i}h \\ & = [h_{i}, h]χ([h_{i}, h])^{−1}v_{i}h. \end{aligned} ; (4.3)

Let ρ : kG → kGe; x → xe be the natural projection, so that in particular, ρ(z) = χ(z)e for all z ∈ Z. By (4.3), for 1 ≤ i ≤ r and h ∈ H,

ρ(h)ρ(v_{i}) = ρ([h_{i}, h](χ([h_{i}, h]))^{−1}v_{i}h)
= χ([h_{i}, h])χ([h_{i}, h])^{−1}ρ(v_{i}h)
= ρ(v_{i})ρ(h).

Hence, for 1 ≤ i ≤ r, ρ(v_{i}) ∈ C_{kGe}(kHe).

Let B be the subalgebra of kGe generated by the ρ(v_{i})’s, 1 ≤ i ≤ r and E the subalgebra generated by B and kHe. Then

(4.4)  dim_{k}(B) ≤ dim_{k}(C_{kGe}(kHe)) = |N|, and dim_{k}(E) ≤ dim_{k}(B) dim_{k}(kHe)

Letting D be the unitary subalgebra of kN generated by {w_{i} : 1 ≤ i ≤ r}, we have, by construction of the w_{i}, D + J^{2}(kN) = kN. Hence, by Proposition 1.2.8 of [3], D = kN. Since w_{i} = h_{i}v_{i}(1 ≤ i ≤ r) it’s clear that E contains ρ(D) = ρ(kN) and hence E = kGe. It follows from (4.4) that dim_{k}(B) = |N| and hence that B = C_{kGe}(kHe).

We’ll finish by showing A ≅ B. For 1 ≤ i, j ≤ r set q_{ij} = χ([l_{j}, l_{i}]); clearly q_{ij} = q_{ji}^{−1}, and take q = (q_{ij}) ∈ Mat_{r}(k) as our commutation matrix.

By equation 4.2

ρ(v_{i})ρ(v_{j}) = q_{ij}ρ(v_{j})ρ(v_{i}), for 1 ≤ i, j ≤ r.

This, together with ρ(v_{i})^{p^{n_{i}}} = 0, shows that the map A → B given by X_{i} → ρ(v_{i}) (1 ≤ i ≤ r) extends to a surjective algebra homomorphism. But dim_{k}(A) = dim_{k}(B) = |N| and hence this homomorphism must be an isomorphism. □

Remark: There is an interesting similarity in our approach in this section and that of [10] in the context of deformation theory.

5. The Ext-ring and p′-cover of A

In this section we recall the results from §5 and §6 in [2], and state the corresponding results for the more general A. For notational convenience, throughout this section A will stand for the quantum complete intersection algebra

A := k_{q}[X_{1}, …, X_{r}]/(X_{i}^{p^{m_{i}}})

where m_{1}, … m_{r} are integers ≥ 1, and we assume p^{m_{i}} > 2 for all 1 ≤ i ≤ r, and q ∈ Mat_{r}(k) is some commutation matrix.

5.1. The ring Ext_{A}^{∗}(k, k). The calculations and results are almost identical to those in §5 of [2] and so we’ll just record those parts of the construction requiring minor modification and describe the structure of Ext_{A}^{∗}(k, k). We’ll try to use as much of the notation in [2] as possible.

A complex K_{∗} of free A-modules is constructed. For each \underline{n} = (n_{1}, …, n_{r}) ∈ (ℤ_{≥0})^{r}, Θ(\underline{n}) is the basis element for a free A-module of rank 1 which lies in degree n_{1} + ⋯ + n_{r} of K_{∗}. Define functions σ_{i}, τ_{i} : ℕ → ℕ, for 1 ≤ i ≤ r, by

σ_{i}(n) = \begin{cases}1 & if n is odd, \\ p^{m_{i}} − 1 & if n is even. \end{cases} and τ_{i}(n) = ∑_{j=1}^{n} σ_{i}(j)

and extend τ_{i} by defining τ_{i}(0) = 0.

For 1 ≤ i ≤ r, set Q_{i}(\underline{n}) := ∏_{ℓ<i}(−1)^{n_{ℓ}}q_{ℓi}^{σ_{i}(n_{i})τ_{ℓ}(n_{ℓ})} and define d_{i} : K_{∗} → K_{∗−1} by

d_{i}Θ(\underline{n}) = \begin{cases}Q_{i}(\underline{n})X_{i}^{σ_{i}(n_{i})}Θ(⋯, n_{i} − 1, ⋯) & if n_{i} > 0, \\ 0 & if n_{i} = 0. \end{cases}

Here we’ll use the convention that Θ(⋯, n_{i} − 1, ⋯) in the right hand side of equations just records the changed entries to Θ(\underline{n}).

It can be checked that d_{i}d_{j} + d_{j}d_{i} = 0 (for i ≠ j) and, as σ_{i}(n_{i}) + σ_{i}(n_{i} − 1) = p^{m_{i}}, d_{i}^{2} = 0. So d = d_{1} + ⋯ + d_{r} satisfies d^{2} = 0 and (K_{∗}, d) is a complex of free A-modules.

A contracting homotopy is constructed as follows. For α ∈ A, writing α = ∑_{j=0}^{p^{m_{i}}−1} α_{j}X_{i}^{j}, with α_{j} in the subalgebra generated by X_{ℓ} (ℓ ≠ i). Let

s_{i}(α_{j}X_{i}^{j}Θ(\underline{n})) = \begin{cases}0 & j = 0 \\ Q_{i}(⋯, n_{i} + 1, ⋯)^{−1}α_{j}X_{i}^{j−1}Θ(⋯, n_{i} + 1, ⋯) & j > 0 \end{cases}  if n_{i} is even,

s_{i}(α_{j}X_{i}^{j}Θ(\underline{n})) = \begin{cases}0 & j < p^{m_{i}} − 1 \\ Q_{i}(⋯, n_{i} + 1, ⋯)^{−1}α_{j}X_{i}Θ(⋯, n_{i} + 1, ⋯) & j = p^{m_{i}} − 1 \end{cases} if n_{i} is odd,

and set s = s_{1} +⋯+s_{r}, then sd+ds = 0. Augmenting the complex K_{∗} by K_{−1} = k and d : K_{0} → K_{−1}, the radical quotient map, and s : K_{−1} → K_{0}, the unit, then sd + ds = 1. Hence K_{∗} is a free minimal resolution of k.

Calculations show that we have chain maps ξ_{i} : K_{∗} → K[2]_{∗}, η_{i} : K_{∗} → K[1]_{∗} where

ξ_{i}(Θ(\underline{n})) = (∏_{ℓ>i} q_{iℓ}^{p^{m_{i}}τ_{ℓ}(n_{ℓ})}) Θ(⋯, n_{i} − 2, ⋯)

η_{i}(Θ(\underline{n})) = (∏_{ℓ<i} q_{ℓi}^{(σ_{i}(n_{i})−1)τ_{ℓ}(n_{ℓ})}) (∏_{ℓ>i}(−1)^{n_{ℓ}}q_{iℓ}^{τ_{ℓ}(n_{ℓ})}) X_{i}^{σ_{i}(n_{i})−1}Θ(⋯, n_{i} − 1, ⋯).

Setting x_{i} = [ξ_{i}] and y_{i} = [η_{i}] in Ext_{A}^{∗}(k, k) we have

Theorem 5.1. For q ∈ Mat_{r}(k) a commutation matrix, (m_{1}, …, m_{r}) ∈ ℕ^{r} such that p^{m_{i}} > 2 for all 1 ≤ i ≤ r and

A := k_{q}[X_{1}, …, X_{r}]/(X_{i}^{p^{m_{i}}})

then the ring Ext_{A}^{∗}(k, k) is generated by elements x_{i} in degree two and y_{i} in degree one, for 1 ≤ i ≤ r subject to the relations

x_{i}x_{j} = q_{ji}^{p^{m_{i}+m_{j}}} x_{j}x_{i}, y_{i}x_{j} = q_{ji}^{p^{m_{j}}} x_{j}y_{i}, y_{i}y_{j} = −q_{ji}y_{j}y_{i}, y_{i}^{2} = 0.

Remark: If p^{m_{i}} = 2 for some (or all) i, then as noted in [2], y_{i} (in degree one) becomes the only corresponding generator with x_{i} = y_{i}^{2}.

5.2. A finite p′ cover for A. Here we just want to record the fact that the results of §6 in [2] extend effortlessly to our algebra A considered in this section.

We’ll use the notation from §6 in [2] and refer the reader to [2] for an explanation of the notation and description of the various constructions (see also [7] for more background).

The result is (see part (iii) of Theorem 1.1 in [2]):

Theorem 5.2. The algebra A has a finite p′-cover Ã which supports a Hopf algebra structure (neither commutative nor cocommuative in general). The category of (finitedimensional) graded A-modules is equivalent to the category of (finite-dimensional) Ãmodules.

The construction of Ã is almost identical to that in [2] except that relations (12) in §6 of [2] generating the ideal I are obviously replaced by

(5.1)  X_{i,(p^{m_{i}−1})w_{i}+s} ⋯ X_{i,w_{i}+s}X_{i,s}  (p_{i}^{m} terms)

The only issue is whether Δ defines a comultiplication modulo these (new) relations (5.1) or equivalently whether I is a Hopf ideal.

As in [2] applying Δ to (5.1) gives

Δ(X_{i,(p^{m_{i}−1})w_{i}+s} ⋯ X_{i,w_{i}+s}X_{i,s}) = ∑_{s′+s′′=s} {v_{s′} ⊗ X_{i,(p^{m_{i}}−1)w_{i}+s′′} ⋯ X_{i,s′′}

+ ∑_{1≤j≤p^{m_{i}}−1} \binom{p^{m_{i}}}{j}f_{i}(s′′)^{j}X_{i,(j−1)w_{i}+s′} ⋯ X_{i,s′} ⊗ X_{i,(p^{m_{i}}−j−1)w_{i}+s′′} ⋯ X_{i,s′′}

+f_{i}(s′′)^{p^{m_{i}}} X_{i,(p^{m_{i}}−1)w_{i}+s′} ⋯ X_{i,s′} ⊗ v_{s′′}} .

Just as in [2] the middle terms in the right hand side summation all vanish as char(k) = p and p divides \binom{p^{m_{i}}}{j} (1 ≤ j ≤ p^{m_{i}} − 1) (this observation is already in §5 of [7]). So I does become a Hopf ideal.

Appendix A. Multiplicative representatives and the Artin-Hasse function

We briefly recall the definition and basic facts about multiplicative representatives and the Artin-Hasse function. Reference for this can be found in Chapter 1 of [5].

A.1. Multiplicative representatives. Denote the image of the map ℤ_{p} → 𝔽_{p} by bars α → \overline{α}. Given a ∈ 𝔽_{p} an element α ∈ ℤ_{p} is said to be a multiplicative representative (or Teichmüller representative) of a if \overline{α} = a and α ∈ ∩_{m≥0}ℚ_{p}^{p^{m}}. It’s easy to construct such a representative, r_{0}(a), for a ∈ 𝔽_{p}. Simply take r_{0}(a) = lim α^{p^{m}} for α ∈ ℤ_{p} any lift of a (such as a, viewed as an element in ℤ_{p}). The following holds

Lemma A.1. For a ∈ 𝔽_{p}, r_{0}(a) ∈ ℤ_{p} is unique and r_{0}(ab) = r_{0}(a)r_{0}(b). In particular r_{0}(1) = 1 and r_{0}(a)^{p} = r_{0}(a) for all a ∈ 𝔽_{p}.

A.2. The Artin-Hasse function. Let ℤ_{(p)} denote the subring of the complete ring ℤ_{p} defined by

ℤ_{(p)} = ℚ ∩ ℤ_{p} = {m/n|m, n ∈ ℤ, (n, p) = 1}.

The formal power series E(X) ∈ ℚ[[X]], given by E(X) = exp(∑_{i≥0} \frac{X^{p^{i}}}{p^{i}}) is called the Artin-Hasse function and some of its properties are summarised in the following Lemma.

Lemma A.2. E(x) ∈ ℤ_{(p)}[[X]] and in fact we have

(1) E(X) = ∏_{(i,p)=1}(1 − X^{i})^{−μ(i)/i},

(2) E(X) = 1 + X (mod X^{2}),

where μ is the Möbius function. Moreover, for a ∈ 𝔽_{p}, E(X)^{r_{0}(a)} = E(r_{0}(a)X).

References

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[2] D.J. Benson and E.L. Green. Non principal blocks with one simple module. The Quarterly journal of Mathematics 55(1) (2004), 1-11.

[3] D.J. Benson. Representations and Cohomology, vol.1, Cambridge University Press, Cambridge, (1991).

[4] David Eisenbud. Commutative algebra with a view towards algebraic geometry. Springer GTM 150 (1994).

[5] I.B.Fesenko and S.V.Vostokov. Local fields and their extensions. AMS Trans. Mathematical Monographs (121).

[6] D. Gorenstein. Finite groups, Chelsea New York, (1980).

[7] E.L. Green. Constructing quantum groups and Hopf algebras from coverings. J.Algebra 176 (1995) 12-33.

[8] B. Külshammer. Crossed products and blocks with normal defect group, Comm. Algebra 13, (1985), pp. 147-168.

[9] Yu.I.Manin. Some remarks on Koszul algebras and quantum groups, Ann. Inst. Fourier (Grenoble) 37 (1987), 191-205.

[10] C. Mejer and M. Schaps. Separable deformations of blocks with abelian normal defect group and of derived equivalent global blocks. Representation theory of algebras (Cocoyoc, 1994), CMS Conf. Proc., 18, 505-517, (1996), Amer. Math. Soc., Rhode Island.

Miles Holloway, Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LB, England

   E-mail address: holloway@maths.ox.ac.uk

Radha Kessar, Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH43210 USA

   E-mail address: kessar@ohio-state.edu