QUANTUM COMPLETE INTERSECTIONS, AND NONPRINCIPAL BLOCKS OF FINITE GROUPS

D. J. BENSON AND E. L. GREEN

Abstract. We give a general construction which shows that a large class of quantum complete intersections can be realized as the basic algebras of nonprincipal blocks of finite groups. We investigate the Ext rings of these algebras. We describe how to construct a finite p′-covering for one of these quantum complete intersections, which supports a Hopf algebra structure which is neither commutative nor cocommutative in general. We use this to investigate a problem concerning the definition of nucleus for a nonprincipal block.

1. Introduction

In this paper, we show that a large class of quantum complete intersections occur as basic algebras of blocks of finite groups, and we calculate the Ext ring of the simple module in this context. Each of the blocks described has just one simple module, and is usually not nilpotent in the sense of Broué and Puig [8, 17]. Our main theorem is as follows.

Theorem 1.1. Let k be a field of characteristic p, and let q = (q_{ij}) ( 1 ≤ i, j ≤ r) be a square matrix of elements of k^{×} satisfying q_{ij}q_{ji} = 1, q_{jj} = 1 and q_{ij}^{p−1} = 1 for 1 ≤ i, j ≤ r.

(i) Let A be the k-algebra of dimension p^{r} given by generators X_{i} ( 1 ≤ i ≤ r), and relations X_{i}X_{j} = q_{ij}X_{j}X_{i} and X_{i}^{p} = 0 for 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 B = e.kG is isomorphic to Mat_{d}(A). The block B is nilpotent if and only if q_{ij} = 1 for all 1 ≤ i, j ≤ r.

(ii) 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}x_{j}x_{i}, y_{i}x_{j} = q_{ji}x_{j}y_{i}, y_{i}y_{j} = −q_{ji}y_{j}y_{i}, y_{i}^{2} = 0.

The subalgebra generated by the elements x_{i}^{p−1} ( 1 ≤ i ≤ r) is a central polynomial subalgebra, and Ext_{A}^{∗}(k, k) is finitely generated as a k[x_{1}^{p−1}, …, x_{r}^{p−1}]- module.

Both authors were partly supported by grants from the NSF.

(iii) The algebra A has a finite p′-cover Ã, described explicitly in Section 6, which supports a structure of Hopf algebra which is in general neither commutative nor cocommutative. The category of (finite dimensional) graded A-modules is equivalent to the category of (finite dimensional) Ã-modules.

In Section 2 we define the algebra A, and in Section 3 we give the construction of the group G appearing in part (i) of the theorem. In Section 4 we describe the idempotent e giving the block B of kG, and we give the isomorphism between B and Mat_{d}(A). In Section 5 we calculate the Ext ring of A, and in Section 6 we construct the finite p′-covering Ã of A and describe its Hopf algebra structure.

In Section 7 we investigate some of the implications of the definition of Ã in understanding the nucleus for the nonprincipal blocks B appearing in the theorem. We formulate a theory of rank varieties for A-modules, and we ask whether the tensor product theorem holds for Ã-modules.

2. Definition of the algebra A

Manin [16] introduced the concept of a quantum symmetric algebra, as a noncommutative version of a polynomial ring. Let k be a commutative coefficient ring. A square matrix q = (q_{ij}) (1 ≤ i, j ≤ r) of elements of k is said to be a commutation matrix if it satisfies q_{ij}q_{ji} = 1 and q_{jj} = 1 for 1 ≤ i, j ≤ r. The quantum symmetric algebra k_{q}[X] is defined to be the algebra generated over k by elements X_{1}, …, X_{r}, subject only to the quantum commutation relations

X_{i}X_{j} = q_{ij}X_{j}X_{i}  (1)

for 1 ≤ i, j ≤ r. Let E be an k_{q}[X]-module, which is the same thing as an k-module together with r endomorphisms corresponding to the X_{i}, satisfying the relations (1).

If ϕ_{1}, …, ϕ_{c} is a sequence of monomials in X_{1}, …, X_{r}, then they also satisfy quantum commutation relations, with coefficients given by multiplying suitable entries from q. The sequence ϕ = ϕ_{1}, …, ϕ_{c} is said to be E-regular (Avramov [1]) if E ≠ (ϕ)E, and ϕ_{j} is injective on E/(ϕ_{1}, …, ϕ_{j−1})E for j = 1, …, c. This notion has properties analogous to those of an ordinary regular sequence in commutative algebra.

We are interested in the particular case where p is an odd prime, k is a field of characteristic p and the elements q_{ij} satisfy q_{ij}^{p−1} = 1 for 1 ≤ i, j ≤ r. The sequence ϕ is taken to be X_{1}^{p}, …, X_{r}^{p}, which is always k_{q}[X]-regular. The quotient

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

is a k-vector space of dimension p^{r}, with a basis consisting of the monomials X_{1}^{i_{1}} … X_{r}^{i_{r}} with 0 ≤ i_{j} ≤ p − 1 for 1 ≤ j ≤ r. An A-module is the same as a k-vector space with r endomorphisms corresponding to the X_{i}, satisfying relations (1) and

X_{i}^{p} = 0.  (2)

3. Definition of the groups H and G

The multiplicative subgroup 𝔽_{p}^{×} of k^{×} is cyclic of order p − 1. Now let H be the finite group defined as follows. The underlying set consists of ordered (r +1)-tuples (λ_{1}, …, λ_{r}; q) where the λ_{i} are elements of ℤ/_{p−1} and q is in 𝔽_{p}^{×} (which is isomorphic to ℤ/_{p−1}, but not canonically so). The multiplication is given by

(λ_{1}, …, λ_{r}; q)(λ′_{1}, …, λ′_{r}; q′) = (λ_{1} + λ′_{1}, …, λ_{r} + λ′_{r}; qq′q′′)

where q′′ is defined by

q′′ = ∏_{1≤j<i≤r} q_{ij}^{λ_{i}λ′_{j}}.

In other words, we have used the matrix q to define an element of the cohomology group H^{2}((ℤ/_{p−1})^{r}, 𝔽_{p}^{×}), and we have defined H to be the corresponding central extension

1 → 𝔽_{p}^{×} → H → (ℤ/_{p−1})^{r} → 1.

We define elements h_{i} and z(q) ∈ H by h_{i} = (0, …, 0, 1, 0, …, 0; 0) with 1 in the ith place, and z(q) = (0, …, 0; q) for q ∈ 𝔽_{p}^{×}. These elements satisfy the relations

h_{i}h_{j} = z(q_{ji})h_{j}h_{i},  (3)
h_{i}^{p−1} = 1,  (4)
z(q)z(q′) = z(qq′),  (5)
h_{i}z(q) = z(q)h_{i}  (6)

for 1 ≤ i, j ≤ r. In fact, these relations could have been used to define the group H. We define Z to be the subgroup consisting of the elements z(q), so that Z ≅ 𝔽_{p}^{×}.

Remark. As a variation on this presentation, it is possible to alter the (p−1) power relations on the h_{i}. Namely, if α_{1}, …, α_{r} are elements of 𝔽_{p}^{×}, then we could set h_{i}^{p−1} = z(α_{i}) instead of h_{i}^{p−1} = 1.

We build a group G as a semidirect product G = N⋊H, where N ≅ (ℤ/p)^{r} is generated by elements g_{1}, …, g_{r}, and the conjugation action of H on N is given by

h_{i}g_{j}h_{i}^{−1} = g_{j}^{q_{ji}}  (7)

for 1 ≤ i, j ≤ r. Thus |G| = p^{r}(p−1)^{r+1}. Note that relation (7) makes sense, because the qth power of g_{j} only depends on the residue class of q modulo p.

4. The isomorphism

In this section, we describe an isomorphism between a matrix ring over the algebra A described in Section 2 and a block of the group algebra kG of the finite group described in Section 3.

Let H_{0} be the kernel of the map H → Aut(N) given by the conjugation relations (7). Then H_{0} is abelian, and q defines a nonsingular symplectic form on H/H_{0} with values in 𝔽_{p}^{×}. This means that |H : H_{0}| is a perfect square, say |H : H_{0}| = d^{2}. If H_{1} is the preimage in H of a maximal isotropic subgroup of H/H_{0}, then |H : H_{1}| = |H_{1} : H_{0}| = d. Let χ be an irreducible character of the abelian group H_{1} whose restriction to Z is “the identity”; in other words, χ(z(q)) = q ∈ k^{×}. Let ρ: H → Mat_{d}(k) be the induced representation χ ↑^{H}. Then the restriction of ρ to H_{1} is a direct sum of the images of χ under conjugation in H. So ρ is irreducible, and elements of Z act as the corresponding scalar matrices. This means that the commutation relation (3) gives

ρ(h_{i})ρ(h_{j}) = q_{ji} ρ(h_{j})ρ(h_{i}).  (8)

Let e be the idempotent ∑_{g∈H_{0}} χ(g^{−1})g in kH_{0} corresponding to the restriction of χ to H_{0}. Then e.kH is isomorphic to Mat_{d}(k) via the representation ρ. Since e is also in the center of kG, the fact that N is a p-group and k has characteristic p implies that the algebra e.kG has only one simple module, namely the one determined by ρ. So setting B = e.kG is a block of kG of dimension p^{r}d^{2}. We claim that B is isomorphic to Mat_{d}(A). To prove this, we need first to introduce the exponential and logarithm maps.

If X is an element of Mat_{d}(A) satisfying X^{p} = 0, then it makes sense to define an exponential

exp(X) = ∑_{t=0}^{p−1} \frac{X^{t}}{t!} .

This satisfies the usual properties of exponentials. Namely, if X and Y commute then exp(X + Y) = exp(X) exp(Y). If a ∈ 𝔽_{p} then exp(aX) = exp(X)^{a}. In particular, exp(X) is a unit of order p in Mat_{d}(A).

Conversely, if U is a unit of order p in Mat_{d}(A), then (U − 1)^{p} = 0, and it makes sense to define a logarithm

log(U) = ∑_{t=0}^{p−1}(−1)^{t+1} \frac{(U − 1)^{t}}{t} .

This satisfies the usual properties of logarithms. Namely, if U and V commute then log(UV) = log(U) + log(V). If a ∈ 𝔽_{p} then log(U^{a}) = a log(U). Moreover, log and exp are mutually inverse: exp(log(U)) = U and log(exp(X)) = X.

Now define a homomorphism ϕ: kG → Mat_{d}(A) as follows. The generators h_{i} of H are taken via ρ to Mat_{d}(k) ⊆ Mat_{d}(A),

ϕ(h_{i}) = ρ(h_{i}).

These matrices commute with the elements X_{j} times the identity, and so the commutation relations (1) and (8) imply that the elements ρ(h_{i})X_{i} commute with each other. It follows that the matrices exp(ρ(h_{i})X_{i}) are commuting elements of order p in Mat_{d}(A), so that it makes sense to send g_{i} to exp(ρ(h_{i})X_{i}),

ϕ(g_{i}) = exp(ρ(h_{i})X_{i}).

Since

ρ(h_{j}) (ρ(h_{i})X_{i}) = q_{ij} (ρ(h_{i})X_{i}) ρ(h_{j})

it follows that

ρ(h_{j}) exp(ρ(h_{i})X_{i}) = exp(ρ(h_{i})X_{i})^{q_{ij}} ρ(h_{j}),

so that the conjugation action (7) of H on N is preserved, and ϕ does indeed define a group homomorphism from G to the units in Mat_{d}(A). We extend linearly to a ring homomorphism ϕ: kG → Mat_{d}(A). Since e maps to the identity under this homomorphism, 1 − e is in the kernel. So ϕ is really a homomorphism from e.kG to Mat_{d}(A). The image certainly contains Mat_{d}(k). Since

ϕ(h_{i}^{−1} log(g_{i})) = X_{i},

the image also contains the diagonal matrices X_{i}. So ϕ is surjective. Since both e.kG and Mat_{d}(A) have k-dimension p^{r}d^{2}, it follows that

ϕ: e.kG ⟶^{≅} Mat_{d}(A)

is an isomorphism. This completes the proof of part (i) of Theorem 1.1.

The statement about nilpotence of the block B follows immediately from the main theorem of Puig [17]. Namely, the basic algebra of a nilpotent block is isomorphic to the group algebra of the defect group. The defect group of B is N, which is abelian. The algebra A is commutative if and only if q_{ij} = 1 for 1 ≤ i, j ≤ r, in which case it is isomorphic to the group algebra kN.

5. The Ext ring of A

In this section, we describe the algebra Ext_{A}^{∗}(k, k). For this purpose, we do not assume that the relations q_{ij}^{p−1} = 1 hold, because the calculations work more generally. At the end, we make note of the minor simplifications introduced by these relations.

We define a complex K_{∗} of free A-modules as follows. It has one free generator Θ(n_{1}, …, n_{r}) for each r-tuple (n_{1}, …, n_{r}) with each n_{i} ≥ 0. The degree of the generator Θ(n_{1}, …, n_{r}) is n_{1} + ⋯ + n_{r}. We define a function σ on the natural numbers via σ(n) = 1 if n is odd and σ(n) = p − 1 if n is even. We define τ(n) = ∑_{j=1}^{n} σ(j). For 1 ≤ i ≤ r, set

d_{i}Θ(n_{1}, …, n_{r}) = ∏_{ℓ<i}(−1)^{n_{ℓ}}q_{iℓ}^{σ(n_{i})τ(n_{ℓ})}X_{i}^{σ(n_{i})}Θ(n_{1}, …, n_{i} − 1, …, n_{r})

if n_{i} > 0, and d_{i}Θ(n_{1}, …, n_{r}) = 0 if n_{i} = 0. This extends uniquely to make an A-module homomorphism. If i < j then

d_{i}d_{j}Θ(n_{1}, …, n_{r}) = (∏_{ℓ<i}(−1)^{n_{ℓ}}q_{iℓ}^{σ(n_{i})τ(n_{ℓ})})( ∏_{m<j}(−1)^{n_{m}}q_{jm}^{σ(n_{j})τ(n_{m})})

X_{j}^{σ(n_{j})}X_{i}^{σ(n_{i})}Θ(n_{1}, …, n_{i} − 1, …, n_{j} − 1, …, n_{r}).

Now d_{j}d_{i}Θ(n_{1}, …, n_{r}) is the same, except that in the second product, the term with m = i changes from (−1)^{n_{i}}q_{ji}^{σ(n_{j})τ(n_{i})} to (−1)^{n_{i}−1}q_{ji}^{σ(n_{j})τ(n_{i}−1)},

while X_{j}^{σ(n_{j})}X_{i}^{σ(n_{i})} is replaced by X_{i}^{σ(n_{i})}X_{j}^{σ(n_{j})} = q_{ji}^{σ(n_{j})σ(n_{i})}X_{j}^{σ(n_{j})}X_{i}^{σ(n_{i})}. Since τ(n_{i}) − τ(n_{i} − 1) = σ(n_{i}), we have

d_{i}d_{j} + d_{j}d_{i} = 0.

Since σ(n_{i}) + σ(n_{i} − 1) = p and X_{i}^{p} = 0, we have d_{i}^{2} = 0. So d = d_{1} + ⋯ + d_{n} satisfies d^{2} = 0 and makes K_{∗} into a complex. To check exactness, we construct a contracting homotopy as follows. If α ∈ A, we can write α = ∑_{j=0}^{p−1} α_{j}X_{i}^{j} where α_{j} is in the subalgebra generated by the X_{ℓ} with ℓ ≠ i. We set

s_{i}(α_{j}X_{i}^{j}Θ(n_{1}, …, n_{r})) = \begin{cases}0 & j = 0 \\ (∏_{ℓ<i}(−1)^{n_{ℓ}})α_{j}X_{i}^{j−1}Θ(…, n_{i} + 1, …) & j > 0 \end{cases}

if n_{i} is even, and

s_{i}(α_{j}X_{i}^{j}Θ(n_{1}, …, n_{r})) = \begin{cases}0 & j < p − 1 \\ (∏_{ℓ<i}(−1)^{n_{ℓ}})α_{j}Θ(…, n_{i} + 1, …) & j = p − 1 \end{cases}

if n_{i} is odd. Then in degree > 0 we have s_{i}d_{i} + d_{i}s_{i} = 1 and s_{i}d_{j} + d_{j}s_{i} = 0 for i ≠ j. So s = s_{1} + ⋯ + s_{r} satisfies sd + ds = 0 so that s is a contracting homotopy, in degree > 0. If we augment the complex with k in degree −1 with d: A → k the radical quotient map and s: k → A the unit, then sd + ds = 1 in all degrees including zero. It follows that K_{∗} is a free resolution of k as an A-module. Since the image of d lies in the radical, this resolution is minimal. So we deduce that dim_{k} Ext_{A}^{n}(k, k) is equal to the free rank of K_{n}, namely \binom{n+r−1}{r}.

We have chain maps ξ_{i}: K_{n} → K_{n−2} and η_{i}: K_{n} → K_{n−1} defined by

ξ_{i}(Θ(n_{1}, …, n_{r})) = ∏_{ℓ>i} q_{ℓi}^{pτ(n_{ℓ})}Θ(n_{1}, …, n_{i} − 2, …, n_{r})

and

η_{i}(Θ(n_{1}, …, n_{r})) = ∏_{ℓ>i}(−1)^{n_{ℓ}}q^{τ(n_{ℓ})σ(n_{i})}X_{i}^{σ(n_{i})−1}Θ(n_{1}, …, n_{i} − 1, …, n_{r}).

A calculation shows that for 1 ≤ i, j ≤ r we have d_{j}ξ_{i} = ξ_{i}d_{j} and d_{j}η_{i} = η_{i}d_{j}, so that ξ_{i} and η_{i} define chain maps on K_{∗}.

Setting x_{i} = [ξ_{i}] and y_{i} = [η_{i}] in Ext_{A}^{∗}(k, k), and examining the effect on low degree basis elements, we see that we have the following relations:

x_{i}x_{j} = q_{ji}^{p^{2}}x_{j}x_{i},  y_{i}x_{j} = q_{ji}^{p}x_{j}y_{i},  y_{i}y_{j} = −q_{ji}y_{j}y_{i},  y_{i}^{2} = 0. (9)

It is easy to see that these relations allow any element in the algebra generated by the x_{i} and y_{i} to be written as a linear combination of elements of the form

x_{1}^{n_{1}} … x_{r}^{n_{r}}y_{1}^{ε_{1}} … y_{r}^{ε_{r}}

with n_{i} ≥ 0 and ε_{i} = 0 or 1. Since this element takes

Θ(2n_{1} + ε_{1}, …, 2n_{r} + ε_{r}) to Θ(0, …, 0)

and the other free generators to elements not involving Θ(0, …, 0), these elements form a basis for Ext_{A}^{∗}(k, k). So we have proved the following theorem.

Theorem 5.1. Let A be the finite dimensional k-algebra defined by generators X_{i} for 1 ≤ i ≤ r subject to relations (1) and (2), but without necessarily satisfying the assumption that q_{ij}^{p−1} = 1. Then Ext_{A}^{∗}(k, k) is generated by elements x_{i} in degree two and y_{i} in degree 1, for 1 ≤ i ≤ r, subject only to relations (9).

Remarks: (i) A similar but easier calculation shows that without the relations (2), the Ext ring is generated by the elements y_{i} subject only to the relations y_{i}y_{j} = −q_{ji}y_{j}y_{i} and y_{i}^{2} = 0; a quantum exterior algebra.

(ii) If we assume that q_{ij}^{p−1} = 1, then relations (9) reduce to

x_{i}x_{j} = q_{ji}x_{j}x_{i},  y_{i}x_{j} = q_{ji}x_{j}y_{i},  y_{i}y_{j} = −q_{ji}y_{j}y_{i},  y_{i}^{2} = 0. (10)

This proves part (ii) of Theorem 1.1.

(iii) The assumption that p is odd is definitely used in the proof of the theorem. If p = 2, the answer is very similar except that y_{i}^{2} = x_{i}. In this case, the Ext ring reduces to a quantum symmetric algebra in the generators y_{i}.

6. A covering of the algebra A

In this section, we investigate a finite p′-cover Ã of the algebra A, with the property that Ã is a Hopf algebra which is neither commutative nor cocommutative in general. The construction is closely related to a special case of the constructions described in Green [13].

We define elements w_{i} = (0, …, 1, …, 0) ∈ (ℤ/_{p−1})^{r}, with 1 in the ith place and zero in the remaining places. We define f_{i}: (ℤ/_{p−1})^{r} → 𝔽_{p}^{×} for 1 ≤ i ≤ r to be the homomorphisms determined by setting f_{i}(w_{j}) = q_{ij} and then extending linearly, so that

f_{i}(λ_{1}, …, λ_{r}) = q_{i1}^{λ_{1}} … q_{ir}^{λ_{r}}.

The maps f_{i} will be used in the definition of the diagonal map Δ in equation (15) below. We remark that f_{i}(w_{i}) = 1 and f_{j}(w_{i}) = f_{i}(w_{j})^{−1} for 1 ≤ i, j ≤ r. Next, we define Γ to be the quotient of the group (ℤ/_{p−1})^{r} by the intersection of the kernels of the f_{i}, so that each f_{i} determines a well-defined homomorphism, still denoted f_{i}: Γ → 𝔽_{p}^{×}. We also denote by w_{i} the images of these elements in Γ.

The algebra structure on Ã is described as a quiver with relations. We define a quiver Q with one vertex v_{s} for each s ∈ Γ. We define arrows X_{i,s} : v_{s} → v_{w_{i}+s} for 1 ≤ i ≤ r and s ∈ Γ. This completes the description of Q. The ideal I of kQ is generated by the elements

X_{i,w_{j}+s}X_{j,s′} − q_{ij}X_{j,w_{i}+s′}X_{i,s},  (11)
X_{i,(p−1)w_{i}+s} ⋯ X_{i,w_{i}+s}X_{i,s} (p terms)  (12)

which correspond to lifts of the relations (1) and (2). We define Ã to be the algebra kQ/I. A basis for Ã is given by the elements

[X_{1,α_{1}w_{1}+···+α_{r}w_{r}+s} ⋯ X_{1,w_{1}+α_{2}w_{2}+···+α_{r}w_{r}+s}] ⋯ [X_{r−1,α_{r−1}w_{r−1}+α_{r}w_{r}+s} ⋯ ⋯ X_{r−1,w_{r−1}+α_{r}w_{r}+s}][X_{r,α_{r}w_{r}+s} ⋯ X_{r,w_{r}+s}X_{r,s}] (13)

where s ∈ Γ, there are r brackets, the ith bracket has α_{i} terms, and 0 ≤ α_{i} ≤ p − 1 for 1 ≤ i ≤ r. The socle of Ã has k-dimension equal to |Γ|, and is spanned by the above elements with α_{1} = ⋯ = α_{r} = p − 1.

There is an obvious action of Γ on Q given by s′(v_{s}) = v_{s+s′} and s′(X_{i,s}) = X_{i,s+s′}. This action extends uniquely to algebra automorphisms of kQ. The action on kQ preserves the ideal I, and hence defines an action of Γ on Ã.

Lemma 6.1. The fixed points of the action of Γ on Ã are isomorphic to A, via the map sending 1 to ∑_{s∈Γ} v_{s} and X_{i} to ∑_{s∈Γ} X_{i,s}. As an A-module, Ã is free on the generators v_{s}, s ∈ Γ.

Proof. It is easy to check that the given map preserves relations (1) and (2). Since A is a local algebra with simple socle, and the dimension of Ã is |Γ| times the dimension of A, it will suffice to prove that the images of the elements v_{s} under a generator for the socle of A are linearly independent. The socle of A is generated by X_{1}^{p−1} … X_{r}^{p−1}. It is not hard to check that multiplying the generators v_{s} by the image of this element in Ã gives rise to the set of monomials (13) which provide a basis for the socle of Ã.

We define a comultiplication Δ: kQ → kQ ⊗_{k} kQ by setting

Δ(v_{s}) = ∑_{s′+s′′=s} v_{s′} ⊗ v_{s′′}  (14)

and

Δ(X_{i,s}) = ∑_{s′+s′′=s} v_{s′} ⊗ X_{i,s′′} + f_{i}(s′′) X_{i,s′} ⊗ v_{s′′}.  (15)

The antipode S: kQ → kQ is defined to be the antiautomorphism determined by the equations

S(v_{s}) = v_{−s}

and

S(X_{i,s}) = −f_{i}(s)X_{i,−w_{i}−s}.

The fact that f_{i}(−w_{i} − s) = f_{i}(−s) = f_{i}(s)^{−1} implies that S^{2} is equal to the identity. The counit ε: kQ → k is defined by ε(v_{0}) = 1, ε(v_{s}) = 0 if s ≠ 0, and ε(X_{i,s}) = 0 for all i and s.

These definitions complete the description of the Hopf algebra structure on kQ, so we must now check that I is a Hopf ideal. It is proved in proposition 4.1 of [13] that Δ defines a comultiplication on kQ modulo relations (11). A computation (using the fact that f_{i}(w_{i} + s′′) = f_{i}(s′′)) shows that applying Δ to relation (12) gives

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

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

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

The exact form of the terms is not so important. The main point is that since we are in characteristic p, the binomial coefficients in the middle terms vanish. The outside terms are taken care of by relations (12). So we conclude that Δ passes down to a well defined comultiplication on kQ/I. Similar but easier computations show that S and ε pass down to operations on kQ/I, making the latter into a Hopf algebra.

Example. If r = 2 then there is only one parameter q_{ij} to worry about, namely q = q_{12}, and we have q_{11} = q_{22} = 1, q_{21} = q^{−1}. The map f_{1} sends (λ_{1}, λ_{2}) ∈ (ℤ/_{p−1})^{2} to q^{λ_{2}}, while f_{2} sends (λ_{1}, λ_{2}) to q^{−λ_{1}}.

The following theorem is a special case of theorem 3.2 of [12], and explains part (iii) of Theorem 1.1.

Theorem 6.2. The category of Ã-modules is equivalent to the category of Γ-graded A-modules, where each generator X_{i} is given weight w_{i}.

7. Varieties and the nucleus

The theory of rank varieties for finitely generated modules over the group algebra of a finite group was first introduced by Carlson [11], and has played a major role in the representation theory of finite groups. There is an obvious problem with trying to mimic this theory for the quantum complete intersections A described in part (i) of Theorem 1.1. Namely, since the elements X_{i} do not commute, it is not usually true that a linear combination of the X_{i} has its pth power equal to zero. So it does not make much sense to restrict a module to the subalgebra generated by such a linear combination, and ask about freeness of the restriction in order to define a rank variety.

However, the isomorphism described in §4 allows for a more sneaky definition of rank variety. Namely, since A and Mat_{d}(A) are Morita equivalent via the bimodule A^{d}, any A-module M corresponds to a uniquely determined Mat_{d}(A)-module M^{d}. Then we can use the map ϕ to make M^{d} into a kGmodule, and restrict to the normal elementary abelian p-subgroup N. At this point, we can take the variety of M^{d} as a kN-module. This serves not only to define the rank variety of a finitely generated A-module, but also the rank variety in the sense of Benson, Carlson and Rickard [4] for an infinitely generated A-module. We shall denote the varieties constructed this way by V_{A}^{r}(M) in the finitely generated case and 𝒱_{A}^{r}(M) in the infinitely generated case.

The nucleus was first introduced in a 1990 paper by Benson, Carlson and Robinson [6] in an attempt to understand modules in the principal block of a finite group, whose cohomology vanishes. Since then, it has played an important role in understanding the module theory for the principal block, both for finite groups and p-restricted Lie algebras [2, 5, 7, 9, 10]. Linckelmann [15] has made a tentative group theoretic definition of the nucleus for a nonprincipal block which generalizes this, but it is not at all clear that his definition has the properties one would desire in terms of representation theory. In Benson [3], a definition is given for the representation theoretic nucleus of a nonprincipal block. The smallest example where we do not know whether this agrees with Linckelmann’s nucleus is the nonprincipal block of a group of the form (ℤ/p × ℤ/p)⋊Q_{8} (p odd), which is a special case of the blocks described in this paper. This block is examined in detail in [3]. In fact, for any noncommutative block of the type described in part (i) of Theorem 1.1, the question remains open as to what the representation theoretic nucleus is.

The issue boils down to asking the following question. Let G be the semidirect product described in the previous paragraph, and let k be an algebraically closed field of characteristic p. Let g_{1} and g_{2} be generators for the elementary abelian normal p-subgroup, so that g_{1} is fixed under conjugation by i ∈ Q_{8} and g_{2} is fixed under conjugation by j ∈ Q_{8}. Let B be the nonprincipal block of kG. Do there exist finitely generated B-modules M and N, neither of which is free on restriction to the subgroup generated by g_{1}, and which satisfy Ext_{B}^{n}(M, N) = 0 for all n > 0? For this block, there exist such modules if and only if the nucleus defined in Linckelmann [15] is equal to the representation theoretic nucleus described in Benson [3].

Now, B ≅ Mat_{2}(A), where A = k⟨X, Y ⟩/(X^{p}, Y^{p}, XY + Y X), and the B-modules M and N correspond to A-modules M_{0} and N_{0}. So we wish to know whether there are finitely generated A-modules M_{0} and N_{0}, neither of which is free on restriction to the subalgebra generated by X, and such that Ext_{A}^{n}(M_{0}, N_{0}) = 0 for all n > 0.

If A itself were a Hopf algebra, we would be able to argue as follows. For finitely generated modules U and V over a finite dimensional Hopf algebra (not necessarily commutative or cocommutative), the map ϕ: V^{∗} ⊗_{k} U → Hom_{k}(V, U) given by ϕ(α⊗u)(v) = uα(v) is an isomorphism. See proposition 1.2 of [14]. Applying this to A and dimension shifting, we would have induced isomorphisms

Ext_{A}^{n}(k, N_{0}^{∗} ⊗_{k} M_{0}) ≅ Ext_{A}^{n}(N_{0}, M_{0}) = 0

for all n > 0. Since A is a finite dimensional local algebra, an examination of minimal resolutions shows that N_{0}^{∗} ⊗_{k} M_{0} would have to be projective as an A-module. If one could then show that the variety of a tensor product is the intersection of the varieties, as happens in the group case, this would contradict the existence of such M_{0} and N_{0}.

The point of constructing the Hopf algebra Ã as in Section 6 is that one can obtain the same sort of information, but just about Γ-gradable A-modules. In the notation of Section 2, we have X_{1} = X, X_{2} = Y, and q_{12} = q_{21} = −1. Then the group Γ is isomorphic to (ℤ/_{2})^{2}.

Suppose that M_{0} and N_{0} are A-modules with the properties discussed above. If, say, M_{0} is Γ-gradable, then there exists an Ã-module M_{1} such that M_{1}↓_{A} ≅ M_{0}. Let ε_{i} (i = 1, 2, 3) be representatives of the three isomorphism classes of nontrivial one dimensional Ã-modules. Then the modules ε_{i} ⊗_{k} M_{1} also have the property that their restriction to A is isomorphic to M_{0}. Furthermore,

M_{0}↑Ã ≅ M_{1}↓_{A}↑Ã ≅ k_{A}↑Ã ⊗_{k} M_{1} ≅ (k ⊕ ε_{1} ⊕ ε_{2} ⊕ ε_{3}) ⊗_{k} M_{1}

and so M_{0}↑Ã↓_{A} ≅ 4M_{0}. So for n > 0 we have

0 = Ext_{A}^{n}(N_{0}, 4M_{0}) ≅ Ext_{A}^{n}(N_{0}, M_{0}↑Ã↓_{A})

≅ Extn ; Ã(N_{0}↑Ã, M_{0}↑Ã)

≅ Extn ; Ã(N_{0}↑Ã, (k ⊕ ε_{1} ⊕ ε_{2} ⊕ ε_{3}) ⊗_{k} M_{1})

≅ Extn ; Ã(k ⊕ ε_{1} ⊕ ε_{2} ⊕ ε_{3}, (N_{0}↑Ã)^{∗} ⊗_{k} M_{1}).

It follows that for n > 0, the nth term in the minimal projective resolution of (N_{0}↑Ã)^{∗} ⊗_{k} M_{1} as an Ã-module has no copies of the projective cover of k, ε_{1}, ε_{2} or ε_{3} as summands. The only way this can happen is that (N_{0}↑Ã)^{∗} ⊗_{k} M_{1} is projective.

These observations prompt the following general question about the algebras Ã constructed in Section 6.

Question. Does the tensor product theorem hold for Ã-modules? In other words, if M and N are finitely generated Ã-modules, is it true that

V_{A}^{r}(N ⊗_{k} M) = V_{A}^{r}(M) ∩ V_{A}^{r}(N)?

Similarly, one would like to know that V_{A}^{r}(N^{∗}) = V_{A}^{r}(N) (duality theorem). The reason, as usual, why it is hard to attack this problem is that the comultiplication and antipode in the Hopf algebra structure on Ã do not behave well with respect to the restriction maps used to construct the variety.

Of course, at the same time, we would also like to know whether the tensor product theorem and the duality theorem hold for infinitely generated modules with respect to the varieties 𝒱_{A}^{r}.

References

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2. D. J. Benson, Cohomology of modules in the principal block of a finite group, New York Journal of Mathematics 1 (1995), 196–205.

3. , The nucleus, and extensions between modules for a finite group, To appear in the proceedings of ICRA9, Beijing, 2000.

4. D. J. Benson, J. F. Carlson, and J. Rickard, Complexity and varieties for infinitely generated modules, II, Math. Proc. Camb. Phil. Soc. 120 (1996), 597–615.

5. , Thick subcategories of the stable module category, Fundamenta Mathematicae 153 (1997), 59–80.

6. D. J. Benson, J. F. Carlson, and G. R. Robinson, On the vanishing of group cohomology, J. Algebra 131 (1990), 40–73.

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8. M. Broué and L. Puig, A Frobenius theorem for blocks, Invent. Math. 56 (1980), 117– 128.

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12. E. L. Green, Graphs with relations, coverings and group-graded algebras, Trans. Amer. Math. Soc. 279 (1983), 297–310.

13. , Constructing quantum groups and Hopf algebras from coverings, J. Algebra 176 (1995), 12–33.

14. E. L. Green, E. N. Marcos, and Ø. Solberg, Representations and almost split sequences for Hopf algebras, Representation Theory of Algebras, Seventh International Conference on Representations of Algebras, August 22–26, 1994 (Bautista, Martínez-Villa, and de la Peña, eds.), CMS Conference Proceedings, vol. 18, AMS, 1996, pp. 237–245.

15. M. Linckelmann, Varieties in block theory, J. Algebra 215 (1999), 460–480.

16. Yu. I. Manin, Some remarks on Koszul algebras and quantum groups, Ann. Inst. Fourier (Grenoble) 37 (1987), 191–205.

17. L. Puig, Nilpotent blocks and their source algebras, Invent. Math. 93 (1986), 77–129.

Department of Mathematics, University of Georgia, Athens GA 30602, USA

E-mail address: djb@byrd.math.uga.edu

Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg VA 24061, USA

E-mail address: green@calvin.math.vt.edu