RANK VARIETIES FOR A CLASS OF FINITE-DIMENSIONAL LOCAL ALGEBRAS

DAVID J. BENSON, KARIN ERDMANN, AND MILES HOLLOWAY

Abstract. We develop a rank variety for finite-dimensional modules over a certain class of finite-dimensional local k-algebras, A_{q,m}^{n}. Included in this class are the truncated polynomial algebras k[X_{1}, …, X_{m}]/(X_{i}^{n}), with k an algebraically closed field and char(k) arbitrary. We prove that these varieties characterise projectivity of modules (Dade’s lemma) and examine the implications for the tree class of the stable Auslander-Reiten quiver. We also extend our rank varieties to infinitely generated modules and verify Dade’s lemma in this context.

1. Introduction

The theory of the rank variety associated to a (finite-dimensional) module over a group algebra kE, where E is an elementary abelian p-group of finite rank and k an algebraically closed field of characteristic p > 0, was introduced by Carlson in [10]. Since then the theory has proven to be extremely useful and in addition to the well known connections between it and the support (or cohomological) variety (see [5]) that were anticipated by Carlson, the theory has also been extended to cover infinite-dimensional modules over the group algebra kE (see [7] and [8]).

Because of this success for group algebras, analogous theories have been developed to cover algebras arising in different settings, most notably p-restricted Lie algebras, where a support and rank variety were introduced in [14] (although the case p = 2 was already in [18]). These further results have been subsumed in the work of Friedlander and Suslin in [16] establishing the finite generation of cohomology for finite group schemes, and subsequent work in [15] generalizing the rank variety.

Here we return to the group algebra kE and seek to develop a rank variety for a wider class of (not necessarily commutative) finite-dimensional k-algebras (with k algebraically closed but not necessarily of prime characteristic), which can be viewed as quantum complete intersections. Included in this class are the truncated polynomial algebras, Λ_{m}^{n} := k[X_{1}, …, X_{m}]/(X_{i}^{n}), and it is worth noting that the naive approach to extending Carlson’s rank variety for such algebras will not work because in general, for λ_{i} ∈ k, (∑ λ_{i}X_{i})^{n} is not necessarily zero.

Before describing our results further we should reflect on what properties can reasonably be expected to hold for a putative rank variety. For group algebras there are three properties that hold for their rank variety that make it a particularly useful construction. They are:

2000 Mathematics Subject Classification. Primary: 16G10, 16D40, Secondary: 16S35, 16S38.

Key words and phrases. rank variety, projectivity, skew group algebras.

(1) The rank variety is directly related to the structure of the module and is computationally explicit.

(2) The rank variety characterises projectivity. This property, referred to from now on as Dade’s lemma, was first proved by Dade (Lemma 11.8 in [11]), although it was not expressed in the language of rank varieties as they had yet to be defined. In terms of rank varieties, Dade’s lemma says that a module is projective if and only if the rank variety is trivial.

(3) The tensor product property holds. That is, the rank variety of a tensor product, over k, of two modules is the intersection of the rank varieties of the two modules.

Both properties (1) and (2) would be desirable for a putative rank variety for a finitedimensional k-algebra. However property (3) relies upon the fact that group algebras are Hopf algebras and therefore tensoring two modules over k still yields a module.

The rank variety we develop for the algebras A_{m}^{n} (m ≥ 1, n > 1) considered in this paper will have property (1) by definition and our main results will be to establish Dade’s lemma (property (2)) for finite-dimensional modules (Theorem 2.6) and later for infinitedimensional modules (Theorem 5.4). We extend, in §3 and §4, our rank variety theory to apply to algebras that are related (in a nice way) to A_{m}^{n}. One such class of related algebras, of particular interest, are the truncated polynomial algebras Λ_{m}^{n} (see Remark 4.7 (2)) mentioned above. Unfortunately neither the algebra A_{m}^{n} nor the related algebras considered, will obviously be Hopf algebras and so, as noted above, property (3) will not feature. By way of an application of our theory we examine the implications for the tree class structure of the stable Auslander-Reiten quivers of the algebras considered in this paper.

The paper is organised as follows. In §2 we define the algebras A_{m}^{n} and describe some of their properties. Then, in §2.1, we define the rank variety for a (finite-dimensional) A_{m}^{n}-module and prove Dade’s lemma for these varieties. In §2.2 we recall some results that restrict the tree class of the stable Auslander-Reiten quiver associated to an algebra provided the algebra has certain properties and we then show that our rank variety for A_{m}^{n} implies that A_{m}^{n} has these properties. In §3 we consider under what circumstances we can use an existing rank variety for some given algebra A, to define a rank variety for an algebra B that is suitably related to A. In §4 we apply the results from §2 and §3 to produce a rank variety for a wide class of algebras (which includes Λ_{m}^{n}) that can be viewed as quantum complete intersections (see Remarks 4.2 (2)). Finally in §5 we follow in the footsteps of [8] and develop a rank variety theory for infinite-dimensional A_{m}^{n}-modules, and establish the corresponding infinite-dimensional version of Dade’s lemma.

Throughout the paper k will denote an algebraically closed field of characteristic p ≥ 0. All algebras considered will be finite-dimensional k-algebras and given such an algebra Λ, J will denote its Jacobson radical (if the context does not make it clear which algebra we are considering then we will explicitly write J(Λ) for J). The category of arbitrary (resp. finite-dimensional) left Λ-modules will be denoted by Λ -Mod (resp. Λ -mod) and the full subcategory of projective (resp. finite-dimensional projective) Λ-modules will be denoted by Λ -Proj (resp. Λ -proj).

Remarks 1.1.

(1) The results in this paper significantly extend the results obtained in [12], which can be interpreted as having established a rank variety (with Dade’s lemma) for the case A_{m}^{2} only. It should also be pointed out that one of the main arguments used to establish Dade’s lemma in [12] has a serious error in it and so, while the statements of the results in [12] are correct, the approach and appropriate proofs (for Dade’s lemma) in [12] should be replaced by those used here.

(2) We understand that there is some related, independent, work by Pevtsova and Witherspoon.

2. Rank varieties for finite-dimensional A_{m}^{n}-modules.

Let m ≥ 1 and n > 1 be given integers and define

n′ := \begin{cases}n/ HCF(n, p) & if p > 0, \\ n & if p = 0, \end{cases}

to be the p′ part of n. Let ζ ∈ k be a fixed primitive n′th root of unity.

The concept of a quantum symmetric algebra, k_{q}[X], was introduced by Manin in [19]. If q = (q_{ij}) ∈ Mat_{m}(k) is a commutation matrix (that is to say q_{ii} = 1 and q_{ij}q_{ji} = 1, ∀i ≠ j) then k_{q}[X] = k_{q}[X_{1}, …, X_{m}] is defined by

k_{q}[X] := k⟨X_{1}, …, X_{m}⟩/⟨X_{i}X_{j} − q_{ij}X_{j}X_{i}⟩.

If q is an indeterminate over k then, with a slight abuse of notation, we are particularly concerned with the special case q_{ij} := q for i < j (so q ∈ Mat_{d}(k[q])). Then the q analogue of the multinomial theorem is

(X_{1} + ⋯ + X_{m})^{r} = ∑_{\begin{matrix}r_{1}+···+r_{m}=r \\ r_{i}≥0 \end{matrix}}(\begin{matrix}& r \\ r_{1} & … & r_{m} \end{matrix} )_{q}X_{1}^{r_{1}} ⋯ X_{m}^{r_{m}},

where, for integers r_{i} ≥ 0 (1 ≤ i ≤ m) such that r_{1} + ⋯ + r_{m} = r, the q analogue of the multinomial coefficient is

(\begin{matrix}& r \\ r_{1} & … & r_{m} \end{matrix} )_{q} := \frac{[r]_{q}!}{[r_{1}]_{q}! ⋯ [r_{m}]_{q}!}, with [m]_{q}! := \begin{cases}[m]_{q}[m − 1]_{q} ⋯ [1]_{q} & if m ≥ 1, \\ 1 & if m = 0. \end{cases}

Here

[m]_{q} := (1 − q^{m})/(1 − q)

is a polynomial in q.

We are interested in specializing to the case q = ζ, and we will simply write k_{ζ}[X] for k_{q}[X] (with q = (q_{ij}), q_{ij} = ζ for i < j). We can now define the truncations of these algebras which are the subject of this section.

Definition 2.1. Let m and n positive integers and ζ ∈ k a fixed primitive n′th root of unity. Define the k-algebra A_{m}^{n} by

A_{m}^{n} := k_{ζ}[X_{1}, …, X_{m}]/(X_{i}^{n}).

Remarks 2.2.

(1) The algebra A_{m}^{n} is a noncommutative, finite-dimensional, symmetric k-algebra and dim A_{m}^{n} = n^{m}.

(2) The algebra A_{m}^{n} is a quantum complete intersection (see Remarks 4.2 (3)).

(3) If p > 0 and n = p^{a} (for some positive integer a) then clearly A_{m}^{n} is isomorphic to the group algebra of a homocyclic group. In particular if n = p then

(2.1)  A_{m}^{p} = kE = k[X_{1}, …, X_{m}]/(X_{i}^{p})

is the group algebra of an elementary abelian p-group, E, of rank m.

2.1. One of the ideas behind the original rank variety for group algebras, kE, was to exploit the fact that, for such an algebra, a linear combination of the generators {X_{i} | 1 ≤ i ≤ m} behaved like a generator itself. The main reason for considering the algebra A_{m}^{n} is that we have the same phenomenon.

Lemma 2.3. Given λ := (λ_{1}, …, λ_{m}) ∈ k^{m} let u_{λ} ∈ A_{m}^{n}, be defined by

u_{λ} := λ_{1}X_{1} + ⋯ + λ_{m}X_{m}.

Then u_{λ}^{n} = 0.

Proof. Write n = p^{a}n′ ((n′, p) = 1). From the q analogue of the multinomial formula and the fact that

(\begin{matrix}& n′ \\ r_{1} & …. & r_{m} \end{matrix} )_{ζ} = \begin{cases}1 & if r_{i} = n′ for some 1 ≤ i ≤ m, \\ 0 & otherwise, \end{cases}

we have

u_{λ}^{n′} = (λ_{1}^{n′}X_{1}^{n′} + ⋯ + λ_{m}^{n′}X_{m}^{n′}).

Now note that X_{i}^{n′}X_{j}^{n′} = X_{j}^{n′}X_{i}^{n′} (for all 1 ≤ i, j ≤ m) and, since char(k) = p, we then have u_{λ}^{n} = (λ_{1}^{n′}X_{1}^{n′} + ⋯ + λ_{m}^{n′}X_{m}^{n′})^{p^{a}} = (λ_{1}^{n}X_{1}^{n} + ⋯ + λ_{m}^{n}X_{m}^{n}) = 0. □

Imitating Carlson’s original definition, we now define a rank variety for a finite-dimensional A_{m}^{n}-module as follows.

Definition 2.4. Let M ∈ A_{m}^{n} -mod and, for λ = (λ_{i}) ∈ k^{m}, let u_{λ} := ∑ λ_{i}X_{i} ∈ A_{m}^{n} and k[u_{λ}] be the subalgebra of A_{m}^{n} generated by u_{λ}.

The rank variety of M is defined by

V_{A_{m}^{n}}^{r}(M) := {0} ∪ {0 ≠ λ ∈ k^{m} | M ↓_{k[u_{λ}]} is not a projective k[u_{λ}]-module}.

Remarks 2.5.

(1) This is a homogeneous affine variety.

(2) If n = p > 0, then by Remark 2.2 (3), A_{m}^{n} = kE, and we recover Carlson’s original definition of the rank variety for elementary abelian p-group algebras.

(3) The subalgebra k[u_{λ}] is isomorphic to the Nakayama algebra k[Y]/(Y^{n}). Hence we can replace the requirement

M ↓_{k[u_{λ}]} is not a projective k[u_{λ}]-module

by the equivalent condition

rank(u_{λ}) < ((n − 1)/n) dim M.

Here we view u_{λ} as the k linear map M → M that is multiplication by u_{λ}. This makes it clear that V_{A_{m}^{n}}^{r}(M) is a homogeneous affine variety and explains the reason for it being called the rank variety.

(4) In the light of Remarks 2.5 (3), it’s worth noting that given N ∈ k[Y]/(Y^{n}) -mod then (for n > 1) N is not projective if and only if the complex

N ⟶^{Y^{(n−i)}} N ⟶^{Y^{i}} N

is not exact for some 1 ≤ i < n, which is itself equivalent to the complex not being exact for all 1 ≤ i < n.

(5) For a Nakayama algebra k[Y]/(Y^{n}), any Y′ ∈ Y + J^{2} (J = J(k[Y]/(Y^{n}))) is also a generator of the algebra. Given M ∈ k[Y]/(Y^{n}) -mod, note that rank(Y′) = rank(Y) (Y and Y′ are again viewed as linear maps M → M induced by multiplication by Y and Y′).

To be a useful definition of a rank variety we need to establish the following.

Theorem 2.6 (Dade’s Lemma for finite-dimensional A_{m}^{n}-modules). Let M ∈ A_{m}^{n} -mod. Then V_{A_{m}^{n}}^{r}(M) = {0} if and only if M is projective.

The proof of this uses the same inductive argument that was used in the original Dade’s Lemma [11]. We shall follow the presentation of that argument as given in [8] since it is more adapted to the language of rank varieties. However in [8] modifications of the argument are given to take account of infinite-dimensional modules. We don’t need these yet (but see Theorem 5.4) but we do need to take account of the fact that our algebra is not commutative. As in [8] we first need to establish Corollary 2.10 (Lemma 5.1 in [8]) in order to apply an inductive argument that is needed to prove Theorem 2.6. Rather than prove Corollary 2.10 directly we make some general observations about selfinjective k-algebras and derive Corollary 2.10 as a consequence.

Recall the following basic facts.

Lemma 2.7. Let A be a finite-dimensional selfinjective k-algebra. Let M ∈ A -Mod be indecomposable. Then M ∈ A -Proj if and only if soc(A)M ≠ 0.

Proof. Because A is selfinjective the left and right annihilators of J, the Jacobson radical of A, coincide and is the socle, soc(A), of A. It follows that soc(A) has J as both a left and right annihilator and so soc(A)M ≠ 0 for any projective A-module M. Conversely consider the injective hull I(M) of M, so we have

0 → M → I(M).

As A is selfinjective I(M) is a direct sum of indecomposable projective modules. Consider the projection of M (as a submodule of I(M)) onto any summand P say. If M is not projective the projection must be contained in JP = J(P) as otherwise it would be surjective and the map would split giving M = P as M indecomposable. Hence M ⊆ JI(M) and so soc(A)M = 0. □

Corollary 2.8. Let A be a finite-dimensional selfinjective k- algebra and let M ∈ A -Mod. Then M ∈ A -Proj if and only if soc(A)M = soc(M).

Proof. If M is projective then soc(A)M = soc(M). As soc(M_{1}⊕M_{2}) = soc(M_{1})⊕ soc(M_{2}) for M_{i} ∈ A -mod (i = 1, 2), the result follows from Lemma 2.7 by considering the indecomposable summands of M. □

Lemma 2.9. Let A be a finite-dimensional selfinjective k-algebra. Let A_{1} and A_{2} be selfinjective subalgebras of A and let J_{i} denote the Jacobson radical of A_{i} (for 1 = 1, 2). Suppose that

(i) J_{1}A_{2} = A_{2}J_{1} and (ii) soc(A_{1}) soc(A_{2}) ⊆ soc(A).

If M ∈ A -Mod is such that M ↓_{A_{1}} ∈ A_{1} -Proj and (M/J_{1}M)↓_{A_{2}} ∈ A_{2} -Proj then M ∈ A -Proj.

Proof. It suffices to assume M is indecomposable and then show that soc(A)M ≠ 0 by Lemma 2.7. As M/J_{1}M ∈ A_{2} -proj, we have soc(A_{2})M + J_{1}M ≠ J_{1}M by Lemma 2.7. In particular soc(A_{2})M ⊈ J_{1}M. Hence soc(A_{1}) soc(A_{2})M ≠ 0, again by Lemma 2.7 because M is projective as an A_{1}-module, and so by (ii) soc(A)M ≠ 0. □

We can now apply these observations to the situation of interest to us and establish.

Corollary 2.10. Suppose M ∈ A_{m}^{n} -Mod is such that

M ↓_{k[X_{m}]} ∈ k[X_{m}] -Proj, and (M/X_{m}M)↓_{A_{m−1}^{n}} ∈ A_{m−1}^{n} -Proj .

Then M is projective.

Here A_{m−1}^{n} is identified with the subalgebra of A_{m}^{n} generated by {X_{i} | 1 ≤ i ≤ m − 1}.

Proof. This follows immediately from Lemma 2.9 on taking A_{1} = k[X_{m}] and A_{2} = A_{m−1}^{n}, and noting that (i) and (ii) of Lemma 2.9 are easily seen to hold. □

We can now prove our main result.

Proof of theorem 2.6. Suppose M is an A_{m}^{n}-module with no projective summand such that V_{A_{m}^{n}}^{r}(M) = {0}. So for all 0 ≠ x in the k-span{X_{i} | 1 ≤ i ≤ m}, we know that M as a module for k[x] is projective.

We want to show that M = 0 and we’ll use induction on m. We start with the case m = 2, which is the main step in the proof.

Case m = 2. Let

M_{1} := J^{n}M, M_{2} := J^{n−1}M, and M_{3} := {b ∈ M : Jb ∈ M_{2}}.

Then M_{1} ⊆ M_{2} ⊆ M_{3} ⊆ M and J annihilates M_{3}/M_{2} and also M_{2}/M_{1}.

For 0 ≠ λ ∈ k^{2} define by \overline{u}_{λ} the induced map

\overline{u}_{λ} : M_{3}/M_{2} → M_{2}/M_{1}

given by multiplication by u_{λ}. We claim that \overline{u}_{λ} is an isomorphism. Assuming that this is the case the result then follows because \overline{u}_{(1,0)}\overline{u}_{(0,1)}^{−1} is an automorphism of a finitedimensional vector space and hence has an eigenvalue, μ ∈ k, say. It would then follow that \overline{u}_{(1,−μ)} could not be an isomorphism contradicting the claim, unless M_{2}/M_{1} = 0. But if M_{2}/M_{1} = 0 then by Nakayama’s lemma M_{2} = J^{n−1}M = 0 and it would follow that u_{λ}^{n−1}M = 0. But M is projective as a k[u_{λ}]-module, hence we must have M = 0 as required.

To show that \overline{u}_{λ} is an isomorphism, we first prove that it is injective. Let b + M_{2} ∈ Ker(\overline{u}_{λ}), so u_{λ}b ∈ M_{1}. We now use the observation that u_{λ}J^{n−1} = J^{n} (since m = 2), so u_{λ}b = u_{λ}b′ for some b′ ∈ M_{2}. Hence b − b′ ∈ Ker(u_{λ}). But M ↓_{k[u_{λ}]} is projective by assumption so b − b′ = u_{λ}^{n−1}b′′ ∈ M_{2}, for some b′′ ∈ M. Hence b ∈ M_{2} and \overline{u}_{λ} is injective.

To show that \overline{u}_{λ} is surjective, and hence an isomorphism, it suffices to show that

dim(M_{3}/M_{2}) = dim(M_{2}/M_{1}).

To see this we only have to show that \overline{u}_{λ} is an isomorphism for a particular choice of λ ∈ k^{2}. Take λ = (1, 0), so u_{λ} = X_{1}. We assume u_{λ} is not surjective, i.e. X_{1}M_{3} ⊊ M_{2}. Since J^{n−1} = kX_{2}^{n−1} + X_{1}J^{n−2}, this means that X_{2}^{n−1}b ∉ X_{1}M_{3}, for some b ∈ M. Because M is assumed not to be projective we have 0 = X_{1}^{n−1}X_{2}^{n−1}M = soc(M). In particular X_{1}^{n−1}(X_{2}^{n−1}b) = 0 and hence X_{2}^{n−1}b = X_{1}b′, for some b′ ∈ M, because M ↓_{k[X_{1}]} is projective. To finish it suffices to show b′ ∈ M_{3} as then X_{2}^{n−1}b = X_{1}b′ ∈ X_{1}M_{3} contradicting our assumption. But X_{1}(X_{2}b′) = ζX_{2}X_{1}b′ = 0, so X_{2}b′ = X_{1}^{n−1}b′′ ∈ M_{2} showing b′ ∈ M_{3}. This completes the proof for the case m = 2.

Case m ≥ 3. Fix m and assume that the theorem is true for all A_{r}^{n} with r < m. Using Corollary 2.10 it suffices to prove \overline{M} := M/X_{m}M is projective as an A_{m−1}^{n}-module (A_{m−1}^{n} being identified with its embedding in A_{m}^{n} as the subalgebra generated by X_{1}, …, X_{m−1}). By our inductive assumption this will follow if we can show V_{A_{m−1}^{n}}^{r}(\overline{M}) = \{0\}. That is we

need to show that \overline{M} is projective as a k[u_{λ}]-module for arbitrary λ = (λ_{1}, …, λ_{m−1}) ∈ k^{m−1}. To see this, first observe that the subalgebra A′ generated by u_{λ} and X_{m} is isomorphic to A_{2}^{n}. Moreover V_{A′}^{r}(M) ⊆ V_{A_{m}^{n}}^{r}(M) = 0. Hence M is projective as an A′-module. It now easily follows that \overline{M} must be projective as a k[u_{λ}]-module. □

2.2. Here we look at an application of our rank variety for A_{m}^{n}-modules, in particular Theorem 2.6, to the structure of tree classes appearing in components of the stable Auslander-Reiten quiver of A_{m}^{n}. Before doing that we recall some terminology (good references for more background on stable Auslander-Reiten quivers are [3] and [6] ).

Definition 2.11. If A is a finite-dimensional selfinjective k-algebra, then A is said to have enough periodic modules if given any 0 ≠ M ∈ A -mod which is non-projective then there exists some W ∈ A -mod such that the following two properties hold.

(i) W ≅ Ω^{r}W ⊕ (proj) for some r > 0, in which case W is said to be Ω-periodic. (ii) \underline{Hom}_{A}(W, M) ≠ 0.

The motivation behind this definition is that it ensures the existence of suitable subadditive functions on the stable Auslander-Reiten quiver of A. The connection of subadditve functions with tree classes of components of the stable Auslander-Reiten quiver comes from a generalisation by Happel, Preiser and Ringel (see [17]) of earlier work characterising Dynkin diagrams. The following theorem, which is a generalisation of Webb’s theorem (see [24]) as interpreted by Okuyama (see [22]), is from [13].

Theorem 2.12 (Theorem 5.6 in [13]). Let A be a finite-dimensional symmetric k-algebra and let T be the tree class of a connected component of the stable Auslander-Reiten quiver of A. Suppose A has enough periodic modules then T is one of the following.

(i) A finite Dynkin diagram (of type A_{n}, D_{n}, E_{6}, E_{7} or E_{8}).

(ii) An infinite Dynkin diagram (of type A_{∞}, D_{∞}, A_{∞}^{∞}).

(iii) A Euclidean diagram.

Remark 2.13. Theorem 5.6 in [13] actually has slightly weaker hypotheses: it’s enough that A be a finite-dimensional selfinjective k-algebra with enough periodic modules provided that the Nakayama functor has finite order for any indecomposable module in A -mod.

We will now see that A_{m}^{n} has enough periodic modules.

Lemma 2.14. Let 0 ≠ λ ∈ k^{m} and 1 ≤ i < n, then a minimal projective resolution of the left A_{m}^{n}-module A_{m}^{n}u_{λ}^{i} is given by

⋯ → A_{m}^{n} ⟶^{u_{λ}^{n−i}} A_{m}^{n} ⟶^{u_{λ}^{i}} A_{m}^{n} ⟶^{u_{λ}^{n−i}} A_{m}^{n},

where the differentials are induced by right multiplication by u_{λ}^{i} and u_{λ}^{n−i} as shown. In particular Ω(A_{m}^{n}u_{λ}^{i}) ≅ A_{m}^{n}u_{λ}^{n−i} and hence Ω^{2}(A_{m}^{n}u_{λ}^{i}) ≅ A_{m}^{n}u_{λ}^{i}.

Proof. By Lemma 2.3 it is clear that the maps are differentials and minimality of the resolution is obvious once the complex has been shown to be exact. Since λ ≠ 0 we may assume that say λ_{1} ≠ 0, and then note that A_{m}^{n} is generated as a k-algebra by {X_{2}, ⋯, X_{m}, u_{λ}} and that any element in A_{m}^{n} can be uniquely expressed as a linear combination of monomials in these ordered generators (if λ_{1} = 0 we make the obvious modifications). In particular A_{m}^{n} viewed as a right k[u_{λ}]-module is free of rank dim A_{m−1}^{n}. Hence dim(im(u_{λ}^{i} : A_{m}^{n} → A_{m}^{n})) = dim A_{m}^{n}u_{λ}^{i} = n^{m−1}(n − i), for 1 ≤ i ≤ n, and exactness of the complex clearly follows from the rank-nullity formula. □

The following is essentially Lemma 3.7 in [12].

Lemma 2.15. Suppose M ∈ A_{m}^{n} -mod, 0 ≠ λ ∈ k^{m} and for 1 ≤ i < n let C_{i} denote the complex

M ⟶^{u_{λ}^{n−i}} M ⟶^{u_{λ}^{i}} M

where the differentials are induced by left multiplication by u_{λ}^{i} and u_{λ}^{n−i}. Then the following are equivalent.

(i) λ ∈ V^{r}(M).

(ii) \underline{Hom}_{A_{m}^{n}}(Au_{λ}^{i}, M) ≠ 0 for some 1 ≤ i < n.

(ii′) \underline{Hom}_{A_{m}^{n}}(Au_{λ}^{i}, M) ≠ 0 for all 1 ≤ i < n.

(iii) The complex C_{i} is not exact for some 1 ≤ i < n.

(iii′) The complex C_{i} is not exact for all 1 ≤ i < n.

Proof. The exactness of the complex C_{i} can be examined after restricting to the Nakayama subalgebra k[u_{λ}]. Then the equivalence of (iii), (iii′) and (i) follows from Remarks 2.5 (4). To complete the proof we will show that (iii), say for i, is equivalent to (ii) for n − i. To that aim fix i (1 ≤ i < n) and, to simplify notation, set A := A_{m}^{n} and u := u_{λ}.

By Lemma 2.14 we have an exact sequence of A-modules

0 → Au^{n−i} → A ⟶^{u^{i}} Au^{i} → 0

Applying the functor Hom_{A}(−, M) to this exact sequence we obtain the exact sequence

0 → Hom_{A}(Au^{i}, M) → Hom_{A}(A, M) ⟶^{δ} Hom_{A}(Au^{n−i}, M) → Ext_{A}^{1}(Au^{i}, M) → 0,

where δ is the map induced by the inclusion map Au^{n−i} → A. Now it is easy to see that, for 0 ≤ j < n, Hom_{A}(Au^{j}, M) ≅ ker(u^{n−j} : M → M) and that under these identifications the map δ corresponds to left multiplication by u^{n−i} : M → ker(u^{i} : M → M). These identifications, together with the standard result that Ext_{A}^{1}(X, M) ≅ \underline{Hom}_{A}(ΩX, M) and Lemma 2.14 show that the above exact sequence can be identified as

0 → ker(u^{n−i}) → M ⟶^{u^{n−i}} ker(u^{i}) → \underline{Hom}_{A}(Au^{n−i}, M) → 0.

From which it immediately follows that

H^{∗}(C_{i}) = ker(u^{i})/ im(u^{n−i}) ≅ \underline{Hom}_{A}(Au^{n−i}, M).  □

Bringing all of this together we have the following Corollary.

Corollary 2.16. A_{m}^{n} has enough periodic modules and if T is a tree class for a connected component of the stable Auslander-Reiten quiver of A_{m}^{n} then T is either a (finite or infinite) Dynkin diagram or a Euclidean diagram.

Proof. This is immediate from Lemma 2.15, Theorem 2.6 and Theorem 2.12. □

Remark 2.17. In fact T will be finite Dynkin if and only if A_{m}^{n} has finite representation type which occurs only if m = 1, and T will be Euclidean if and only if A_{m}^{n} has tame representation type which occurs only when n = m = 2.

3. Extending rank varieties to related algebras.

In this section we are interested in the following question. Suppose we have two finitedimensional selfinjective k-algebras and suppose we already have a rank variety theory for one of the algebras, in particular a rank variety characterising projectivity (i.e. satisfying Dade’s lemma). What kind of relationship should the two algebras have with one another in order to be able to use the rank variety of one to define a rank variety (again satisfying Dade’s lemma) for the other algebra? Clearly if this is to be possible we will need to be able relate projective modules between the two algebras.

We begin by considering A and B, two finite-dimensional slefinjective k-algebras, with A having a rank variety, V_{A}^{r}, satisfying Dade’s lemma, and suppose we have an additive functor

H : B -mod → A -mod

satisfying (i) HB ∈ A -proj. (ii) If X ∈ B -mod and HX ∈ A -proj then X ∈ B -proj.

Note that through additivity (i) is equivalent to, for X ∈ B -mod, X ∈ B -proj ⟹ HX ∈ A -proj. Hence taken together (i) and (ii) are equivalent to, for X ∈ B -mod,

(3.1)  X ∈ B -proj ⟺ HX ∈ A -proj .

Then we might define a rank variety for B by defining, for M ∈ B -mod,

V_{B}^{r}(M) := V_{A}^{r}(HM).

It follows immediately from (3.2) that V_{B}^{r} will satisfy Dade’s lemma, since we assumed V_{A}^{r} does.

A common situation where the above setup occurs is when we have a k-algebra monomorphism i : A → B. Since the above discussion is concerned with relating projectivity between A and B let’s first recall some standard (see [23]) conditions on A and B that achieve this. Through i, B becomes an A-bimodule and we have the induction functor

−↑^{B} = B ⊗_{A} − : A -mod → B -mod .

It has a right adjoint −↓_{A}, the restriction functor, and we will require the functors −↑^{B} and −↓_{A} to send the full subcategories A -proj and B -proj to each other. It’s clear that −↑^{B} already does and in order for −↓_{A} to do so we need B↓_{A} = _{A}B ∈ A -proj.

We will denote by (Res) either of the following equivalent conditions holding. (Res1) The counit of the adjunction ϵ : −↓_{A}↑^{B} → Id is a split epimorphism. (Res2) The natural map, induced by multiplication, B ⊗_{A} B → B is a split epimorphism

of B-bimodules. If (Res) holds and B ↓_{A} ∈ A -proj then it is clear that taking H := −↓_{A} will satisfy (3.1).

For completeness we recall the analogous conditions on − ↑^{B}, although we shall not use these. Denote by (Ind) either of the following equivalent conditions. (Ind1) The unit of the adjunction η : Id → −↑^{B} ↓_{A} is a split monomorphism. (Ind2) The map i : A → B is a split monomorphism of A-bimodules. That is A is a

summand of B as an A-bimodule. If (Ind) holds and X ∈ A -mod then

(3.2)  X ∈ A -proj ⟺ X ↑^{B} ∈ B -proj .

Remark 3.1. An example (see [23]) where all the conditions; B ↓_{A} ∈ A -proj, (Res) and (Ind), hold is when we have a finite group G acting on a finite-dimensional k-algebra A, with the order of G invertible in k, and we take B = AG to be the skew group ring, i : A → B the natural inclusion.

In §4 when we wish to use our rank variety for A_{m}^{n} to define rank varieties for other classes of algebras we will need a slightly more flexible set up than that given just by considering algebra monomorphisms as above. The greater flexibility we need will come from allowing Morita equivalent algebras to enter the mix. For the sake of clarity we gather the necessary conditions and results (for applications also) in the following lemma.

Lemma 3.2. Let A and A′ be two finite-dimensional selfinjective k-algebras and suppose the following conditions hold.

(1) There exists a finite-dimensional k-algebra B and a Morita equivalence between A′ and B induced by a functor G : A′ -mod → B -mod.

(2) There is an algebra monomorphism i : A → B, through which we identify A with

i(A) as a subalgebra of B, such that B↓_{A} ∈ A -proj and (Res) hold. Then H := (G−) ↓_{A} : A′ -mod → A -mod satisfies (3.1). Moreover if A has enough periodic modules then so does A′.

Proof. We first establish the last statement concerning periodic modules. Through the Morita equivalence G it suffices to establish that B has enough periodic modules. Let 0 ≠ N be a non-projective B-module. By hypothesis (Res) holds and in particular the equivalence (3.1) holds. So N ↓_{A} is a non-projective A-module and hence, since A has enough periodic modules there exists an Ω-periodic A-module V such that \underline{Hom}_{A}(V, N ↓ A) ≠ 0 and therefore, by Shapiro’s lemma, \underline{Hom}_{B}(V ↑^{B}, N) ≠ 0. Since − ↑^{B} preserves projectivity it’s clear that V ↑^{B} is an Ω-periodic B-module as required.

That H satisfies (3.1) follows immmediately from hypothesis (2) which implies, from the preceding discussion, that −↓_{A} : B -mod → A -mod satisfies (3.2). That G : A′ -mod → B -mod also satisfies (3.2) is obvious, being a Morita equivalence and hence so does H. □

Remark 3.3. With the notation of Lemma 3.2 it follows from the discussion in this section that if V_{A}^{r} is a rank variety that satisfies Dade’s lemma then, for M ∈ A′ -mod, V_{A′}^{r}(M) := V_{A}^{r}(GM ↓_{A}) defines a rank variety for A′ that also satisfies Dade’s lemma.

4. Quantum complete intersections.

In this section we consider a certain generalisation (a quantum complete intersection), A_{q,m}^{n}, of the k-algebra A_{m}^{n}, and develop a rank variety theory for this algebra using V_{A_{m}^{n}}^{r}. In §4.1 we define the algebra A_{q,m}^{n} and in §4.2 we define a finite group, E_{q} whose order is invertible in k, together with a primitive central idempotent e, that yields a matrix algebra ekE_{q}. Finally in §4.3 we use the results from §4.2 and apply the analysis from §3 to define a rank variety for A_{q,m}^{n}. Recall that k is assumed to be an algebraically closed field with char(k) = p ≥ 0, unless stated otherwise.

4.1. Let q = (q_{ij}) ∈ Mat_{m}(k) be a commutation matrix (so by definition q_{ii} = 1 and q_{ij}q_{ji} = 1, ∀ i ≠ j) such that the q_{ij} are also roots of unity for all 1 ≤ i, j ≤ m. We define the k-algebra A_{q,m}^{n} as follows.

Definition 4.1.

A_{q,m}^{n} := k⟨Z_{1}, …, Z_{m}⟩/⟨Z_{i}^{n}, Z_{i}Z_{j} − q_{ij}Z_{j}Z_{i}(i < j)⟩.

Remarks 4.2.

(1) The algebra A_{q,m}^{n} is a finite-dimensional, symmetric k-algebra and dim A_{q,m}^{n} = n^{m}.

(2) Clearly if q_{ij} = ζ for all i < j with ζ a primitive n′th root of unity (recall n′ is the p′ part of n), then A_{q,m}^{n} = A_{m}^{n}.

(3) As noted in [9] we can view A_{q,m}^{n} as a quotient of the quantum symmetric algebra k_{q}[X] by the ideal of a quantum commuting k_{q}[X]-regular sequence ϕ = X_{1}^{n}, …, X_{m}^{n} (for details on these concepts see [4], where they were first introduced). Hence the algebra A_{q,m}^{n} may properly be called a quantum complete intersection.

4.2. We consider a construction that is a central extension of abelian groups. Similar constructions have been considered in [1] and [9].

Let q ∈ Mat_{m}(k) be a commutation matrix as defined at the begining to §4.1 and let r ≥ 1 be such that q_{ij}^{r} = 1 for all 1 ≤ i, j ≤ m with HCF(r, p) = 1 (this is clearly possible since char(k) = p). Let E_{q} be the central extension given by

(4.1)  1 → ⟨ν⟩ → E_{q} → \overbrace{ℤ/rℤ × ⋯ × ℤ/rℤ}^{m} → 1,

where ⟨ν⟩ ≅ ℤ/rℤ. To define E_{q} properly let e_{i} be a preimage of a generator of the ith factor appearing in the right hand side of (4.1). We require the following relations to hold.

(1) ν^{r} = 1 and e_{i}^{r} = 1 for 1 ≤ i ≤ m.

(2) e_{i}e_{j} = ν_{ij}e_{j}e_{i} for 1 ≤ i < j ≤ m.

(3) e_{i}ν = νe_{i} for 1 ≤ i ≤ m. To complete the definition of E_{q} we need to specify the elements ν_{ij} ∈ ⟨ν⟩. To do that we need to fix an isomorphism ⟨ν⟩ ≅ U_{r} (where U_{r} denotes the group of rth roots of unity in k). That is we need to fix a character ϕ ∈ Irr(⟨ν⟩). Once this has been done, we define ν_{ij} by ϕ(ν_{ij}) = q_{ij}. Clearly |E_{q}| = r^{m+1} and, since char(k) = p does not divide r, kE_{q} is a semisimple k-algebra.

To define the primitive central idempotent e we follow the same construction as in [9]. Let E := E_{q} and E_{0} := Z(E) then we have a well defined nonsingular symplectic form

E/E_{0} × E/E_{0} → k
(eE_{0}, e′E_{0}) → ϕ([e, e′]).

In particular, since it is nonsingular, we must have |E/E_{0}| = d^{2}, for some positive integer d. If E_{1} is the subgroup that is the preimage in E of a maximal isotropic subgroup of E/E_{0}, then |E : E_{1}| = |E_{1} : E_{0}| = d (E_{1} can be taken to be any maximal abelian subgroup of E).

Let χ be any irreducible character of E_{1} that extends ϕ. That is χ ∈ Irr(E_{1}) and χ ↓_{⟨ν⟩}= ϕ. With χ_{0} := χ ↓_{E_{0}}∈ Irr(E_{0}), it is easy to see that {χ^{g} | g ∈ E : E_{1}} (here E : E_{1} denotes a set of coset representatives) enumerates all the irreducible characters of E_{1} that give χ_{0} upon restriction to E_{0}. Then χ_{0} ↑^{E_{1}}= ∑_{g∈E:E_{1}} χ^{g} and it follows from all this that ψ := χ ↑^{E} is an irreducible character and hence determines an irreducible representation

ρ : ekE_{q} ⟶^{≅} Mat_{d}(k).

Moreover the primitive central idempotent, e, associated to ρ is given by

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

In particular we have

ρ(e_{i})ρ(e_{j}) = q_{ij}ρ(e_{j})ρ(e_{i}).

Remarks 4.3.

(1) If r = n is coprime to char(k) = p and, for i < j, q_{ij} = ζ a primitive nth root of unity, then the construction of E_{q} and isomorphism ρ : ekE_{q} → Mat_{d}(k) gives a representation of the generalised Clifford algbera C_{m}^{(n)} (see [20], [21]). In particular if n = 2 then we have a Clifford algebra and the representation is a Clifford module (see [2] or [1]). These representations of generalised Clifford algebras are constructed in the same spirit as that used in [1] for Clifford modules.

(2) In the construction of the irreducible character ψ (and hence also the idempotent e), and irreducible representation ρ, some choices were made. We can split these choices up as follows. First a choice of character χ_{0} ∈ Irr(E_{0}) that restricted to ⟨ν⟩ to give ϕ; different choices here would yield different idempotents e. Once χ_{0} has been fixed and hence the block algebra ekE_{q}, we have a further choice of irreducible character χ ∈ Irr(E_{1}) that must then restrict to E_{0} to give χ_{0}. Different choices here then give different choices for the algebra isomorphism ρ : ekE_{q} → Mat_{d}(k).

(3) If r = n (coprime to char(k) = p) and, for i < j, q_{ij} = ζ then, as already remarked, we have constructed modules for the generalised Clifford algebra C_{m}^{(n)}. For m = 2r or m = 2r + 1, let g_{i} := e_{2i−1}^{−1}e_{2i}, for 1 ≤ i ≤ r. For m = 2r + 1, let γ := g_{1}g_{2} ⋯ g_{r}e_{m}^{−1}. Then it is not difficult to show

E_{0} = Z(E) ≅ \begin{cases}⟨ν⟩ & if m = 2r, \\ ⟨ν, γ⟩ & if m = 2r + 1. \end{cases}

Analysis of the choice for χ_{0} shows that, for m = 2r, we have only one choice and hence, up to isomorphism, only one C_{m}^{(n)}-module. For m = 2r + 1 we have n different choices for χ_{0} and hence n nonisomorphic C_{m}^{(n)}-modules. Thus we can recover some of the results in [20].

(4) In general it is not hard to see that different choices for χ_{0} yield matrices for ρ(e_{i}) (1 ≤ i ≤ m) that differ up to scalars.

4.3. Using the notation of Lemma 3.2 let A′ := A_{q,m}^{n}, for q a commutation matrix as in §4.1. Define a new commutation matrix q′ = (q′_{ij}) ∈ Mat_{m}(k) by

q′_{ij}q_{ij} = ζ, for i < j.

Note that the q′_{ij} are also clearly roots of unity. Choosing a primitive central idempotent e as in §4.2 for the group E_{q′}, we define a k-algebra B by

B := ekE_{q′} ⊗_{k} A′.

B is clearly Morita equivalent to A′, so condition (1) of Lemma 3.2 holds, and B has a subalgebra, A, generated by {ee_{i} ⊗_{k} Z_{i} | 1 ≤ i ≤ m}. There is an obvious surjection from A_{m}^{n} (see Definition 2.1) onto this subalgebra; mapping X_{i} to ee_{i} ⊗_{k} Z_{i}. This is easily seen to be an algebra isomorphism, and through this isomorphism we identify the subalgebra A with A_{m}^{n}. Now note that E_{q′} acts on A via conjugation (that is e_{i} ∈ E_{q′} acts on a ∈ A(= A′) as (e_{i}^{−1} ⊗_{k} 1)a(e_{i} ⊗_{k} 1)). It is then clear that B is in fact the skew group algebra AE_{q′}. Since |E_{q′}| is coprime to p, by Remark 3.1, condition (2) of Lemma 3.2 holds. Hence Lemma 3.2 holds and Definition 2.4 and Theorem 2.6 allow us to apply Remark 3.3 and define a rank variety for the algebra A_{q,m}^{n}.

Definition 4.4. Let S denote a simple ekE_{q′}-module and let H : A_{q,m}^{n} -mod → A_{m}^{n} -mod denote the functor given by HM := S ⊗_{k} M ↓_{A_{m}^{n}}. For M ∈ A_{q,m}^{n} -mod define

V_{A_{q,m}^{n}}^{r}(M) := V_{A_{m}^{n}}^{r}(HM).

Corollary 4.5. For M ∈ A_{q,m}^{n} -mod then V_{A_{q,m}^{n}}^{r}(M) = 0 if and only if M ∈ A_{q,m}^{n} -proj.

Proof. Immediate from Remark 3.3. □

Corollary 4.6. A_{q,m}^{n} has enough periodic modules. Moreover if T is a tree class for a connected component of the stable Auslander-Reiten quiver of A_{q,m}^{n} then T is either a (finite or infinite) Dynkin diagram or a Euclidean diagram.

Proof. This is immediate from Lemma 3.2 and Theorem 2.12. □

Remarks 4.7.

(1) In making this definition of V_{A_{q,m}^{n}}^{r} we had choices in the construction of the blocks ekE_{q′}. However as noted in Remarks 4.3 (4), different choices would yield isomorphic affine varieties.

(2) If q_{ij} = 1, for all 1 ≤ i, j ≤ m. Then A_{q,m}^{n} would be the truncated polynomial algebra

A_{q,m}^{n} = Λ_{m}^{n} := k[Z_{1}, …, Z_{m}]/(Z_{i}^{n}),

and so we have a rank variety for such algebras. In particular, for the case n = 2, we recover (via Clifford modules) the definition of rank variety for Λ_{m} := Λ_{m}^{2}-modules given in [12].

(3) If we take char(k) = p > 0, n = p and {q_{ij}} such that q_{ij}^{p−1} = 1 then A_{q,m}^{n} becomes the algebra A considered by Benson and Green in [9]. In [9] Benson and Green utilise an explicitly constructed isomorphism between a matrix algebra over (their) A and a certain block of a group algebra to define a rank variety for (their) A by using Carlson’s original rank variety for the group algebra. The construction above makes it clear that their rank variety is isomorphic to the rank variety for A given by Definition 4.4, although in terms of computation our approach may be slightly simpler (see Remarks 2.5 (5)).

5. Rank varieties for infinite-dimensional A_{m}^{n}-modules.

The aim of this section is to show that the ideas and approach used in [8] to deal with infinite-dimensional modules over elementary abelian p-groups can easily be adapted to cover the algebra A_{m}^{n} considered in §2. As in [8] one of the main tasks will be to verify Dade’s lemma in the infinite-dimensional module case and in §5.2 we achieve this for the algebra A_{m}^{n}, focusing on those parts of the argument that must be modified due to both infinite-dimensionality and noncommutativity. We then imitate the definition of the rank variety from [8] to define the rank variety V_{A_{m}^{n}}^{r} (which is not really a variety but a collection of closed sets) in §5.3. A brief background and set up for the infinite-dimensional case is given in §5.1, the reader is referred to [8] for more details.

5.1. As pointed out in [8] one of the motivations for a new definition of a rank variety for infinitely generated modules was the existence of infinite-dimensional modules whose rank variety (either Carlson’s original definition for group algebras or Definition 2.4 for A_{m}^{n}), over k, was just {0}, but these modules are definitely not projective. The problem, as identified in [8], is that the field k is not large enough to find suitable linear combinations of the generators which expose the module as not being projective. One must consider proper field extensions, K ⊋ k.

Denote by V^{r}(k) the underlying affine m-space, 𝔸^{m}, of the vector space J(A_{m}^{n})/J^{2}(A_{m}^{n}), with coordinate vectors {x_{i} | 1 ≤ i ≤ m} the images of the generators {X_{i}}. Then the effect of considering proper field extensions K ⊋ k; K is assumed algebraically closed and has nontrivial transcendence degree over k, is to provide us with generic points for our varieties. Recall that if A is a closed subset of a topological space X, then x ∈ A is called a generic point if A = \overline{\{x\}}, the closure of x in X.

If {y_{i} | 1 ≤ i ≤ m} are the coordinate functions on V^{r}(k) (so y_{i}(x_{j}) = δ_{ij}) with coordinate ring

k[V^{r}(k)] = k[y_{1}, …, y_{m}],

and we are given a closed irreducible subvariety V in V^{r}(k) (with 𝔭 ⊆ k[V^{r}(k)] the corresponding prime ideal), then the coordinate ring is k[V] = k[V^{r}(k)]/𝔭, with function field k(V), the fraction field of k[V]. If now K is some algebraically closed extension field of k with tr-deg(K/k) ≥ tr-deg(k(V)/k) then we can embed k(V) ↪ K and the point

y′_{1}x_{1} + ⋯ + y′_{m}x_{m} ∈ V^{r}(K),

is a generic point for V. Here y′_{i} is the image in K of y_{i} under the composite map k[V^{r}(k)] → k[V] → K.

One can also consider generic points for projective varieties (see [8]). We will simply record the result and accompanying remark from [8].

Proposition 5.1. Provided tr-deg(K/k) ≥ m, each line through the origin in V^{r}(K) contains a generic point for a uniquely determined closed homogeneous irreducible subvariety of V^{r}(k). For any point on such a line, this is the smallest homogeneous subvariety containing the subvariety for which the point is generic.

Remark 5.2. For V^{r}(K) to have generic points for all closed irreducible subvarieties of V^{r}(k) it is necessary and sufficient that tr-deg(K/k) ≥ m. For V^{r}(K) to have lines through the origin corresponding to all the closed homogeneous irreducible subvarieties of V^{r}(k) it suffices that tr-deg(K/k) ≥ m − 1. The extra transcendence degree simply provides enough generic points on these lines.

In terms of rank varieties and Dade’s lemma the reason for considering K ⊋ k is to provide a replacement to the eigenvalue argument in Theorem 2.6. The problem with that argument is that it is possible for V and W to be k vector spaces that are not finitedimensional and there exist f, g : V → W, k-linear maps, such that for all λ, μ ∈ k, not both zero, the map

λf + μg : V → W

is an isomorphism. With V (and hence also W) being finite-dimensional, then k being algebraically closed guarantees the existence of eigenvalues (and hence eigenvectors) for fg^{−1}, which then rules out this type of behaviour.

For infinite-dimensional vector spaces the substitute for eigenvalues is Lemma 4.1 of [8], which we recall.

Lemma 5.3. Suppose V and W are vector spaces over an algebraically closed field k and K is a nontrivial field extension. Suppose f, g : V → W are k-linear maps such that for all λ, μ ∈ K, not both zero, the linear map λf + μg : K ⊗_{k} V → K ⊗_{k} W is an isomorphism. Then V = W = 0.

5.2. Recall k is assumed to be an algebraically closed field with char(k) = p ≥ 0, n′ is the p′ part of n and ζ ∈ k is some fixed primitive n′th root of unity. Given K ⊇ k an algebraically closed field extension and λ = (λ_{i}) ∈ K^{m} we define

u_{λ} := λ_{1} ⊗_{k} X_{1} + ⋯ + λ_{m} ⊗_{k} X_{m} ∈ K ⊗_{k} A_{m}^{n}.

Just as in Lemma 2.3, we have u_{λ}^{n} = 0 and hence for M ∈ A_{m}^{n} -Mod we can consider the restriction of K ⊗_{k} M to K[u_{λ}] and ask whether it is projective. We define

V_{A_{m}^{n}}^{r}(K ⊗_{k} M) := {0} ∪ {λ ∈ K^{m} | K ⊗_{k} M ↓_{K[u_{λ}]} is not projective}.

Following [8] we can now prove Dade’s lemma for arbitrary modules.

Theorem 5.4 (Dade’s lemma for A_{m}^{n} and infinite-dimensional modules). Suppose K ⊇ k is an algebraically closed field extension with tr-deg(K/k) ≥ m−1, and let M ∈ A_{m}^{n} -Mod. Then V_{A_{m}^{n}}^{r}(K ⊗_{k} M) = {0} if and only if M is projective.

Proof. Clearly if M ∈ A_{m}^{n} -Proj then V_{A_{m}^{n}}^{r}(K ⊗_{k} M) = {0}. So we may suppose that M ∈ A_{m}^{n} -Mod has no projective summands and is such that V_{A_{m}^{n}}^{r}(K ⊗_{k} M) = {0} and we must then show that M = 0.

Just as in Theorem 2.6 we will argue by induction on m and, as before, the case m = 2 will be the main case.

Case m = 2. Again we define

M_{1} := J^{n}M, M_{2} := J^{n−1}M, and M_{3} := {b ∈ M : Jb ∈ M_{2}},

and let M̂ := K ⊗_{k} M, M̂_{1} := K ⊗_{k} M_{1} and M̂_{3} := K ⊗_{k} M_{3} be their extensions to K-spaces. We have M̂_{1} ⊆ M̂_{2} ⊆ M̂_{3} ⊆ M̂ and Ĵ:= J(K ⊗_{k} A_{m}^{n}) annihilates M̂_{3}/M̂_{2} and also M̂_{2}/M̂_{1}. For 0 ≠ λ ∈ K^{2} define

\overline{u}_{λ} : M̂_{3}/M̂_{2} → M̂_{2}/M̂_{1}

to be the induced map given by multiplication by u_{λ} ∈ K ⊗_{k} A_{m}^{n}. We claim that \overline{u}_{λ} is an isomorphism.

To show that \overline{u}_{λ} is injective is exactly the same argument as that used in the proof of Theorem 2.6. However in order to show that \overline{u}_{λ} is surjective, we can no longer rely on dimension arguments (as used in the proof of Theorem 2.6). Instead we modify the argument used in the proof of Theorem 5.2 in [8] to account for noncommutativity. We must show u_{λ}M̂_{3} = M̂_{2}. Suppose that λ = (λ_{1}, λ_{2}) ∈ K^{2} with λ_{1} ≠ 0 and let y := 1⊗_{k} X_{2}. Then Ĵ^{n−1} is K linearly spanned by y^{n−1} and u_{λ}Ĵ^{n−2} and Ĵ^{n−2}M̂ ⊆ M̂_{3}. We have

M̂_{2} = y^{n−1}M̂ + u_{λ}M̂_{3}.

If u_{λ}M̂_{3} ⊊ M̂_{2}, then there exists b ∈ M̂ with y^{n−1}b ∉ u_{λ}M̂_{3}. Because M̂ has no projective summand, soc(K ⊗_{k} A_{m}^{n}) annihilates it. In particular u_{λ}^{n−1}y^{n−1}b = 0. But M̂ ↓_{K[u_{λ}]} is a free module so there exists b′ ∈ M̂ such that y^{n−1}b = u_{λ}b′. Then, using the fact that u_{λ′}y = yu_{λ} where λ′ = (ζ^{−1}λ_{1}, λ_{2}), we have u_{λ′}yb′ = y^{n}b = 0. But M̂ ↓_{K[u′_{λ}]} is also a free module so there exists b′′ ∈ M̂ such that u_{λ′}^{n−1}b′′ = yb′. So we can conclude that both u_{λ}b′ and yb′ belong to M̂_{2} and this means b′ ∈ M̂_{3}. Then y^{n−1}b = u_{λ}b′ ∈ u_{λ}M̂_{3}, a contradiction. If λ_{1} = 0 then we choose y = 1 ⊗_{k} X_{1} and make the obvious modifications to the previous argument. Hence \overline{u}_{λ} is an isomorphism.

By Lemma 5.3, applied to the maps x_{1}, x_{2} : M_{3}/M_{2} → M_{2}/M_{1} induced by multiplication by X_{1}, X_{2}, we get M_{1} = M_{2} = M_{3}. Hence J^{n−1}M = 0 but M ↓_{k[X_{1}]} is assumed free and so we must have M = 0 which finishes the case m = 2.

Case m ≥ 3. As before we assume the theorem true for all A_{r}^{n} with r < m. Using Corollary 2.10 it suffices to prove \overline{M} := M/X_{m}M is projective as an A_{m−1}^{n}-module (A_{m−1}^{n} being identified with its embedding in A_{m}^{n} as the subalgebra generated by X_{1}, … X_{m−1}).

For 0 ≠ λ = (λ_{1}, …, λ_{m−1}) ∈ K^{m−1}, let K_{0} denote the algebraic closure in K of the subfield generated by k and the ratios between λ_{1}, …, λ_{m−1} (so e.g. if, for some 1 ≤ i ≤ m − 1, λ_{i} ≠ 0, then the subfield is generated by k and λ_{1}/λ_{i}, …, λ_{m−1}/λ_{i}). We clearly have tr-deg(K_{0}/k) ≤ m − 2 and so tr-deg(K/K_{0}) ≥ 1. If A′ ≅ K_{0} ⊗_{k} A_{2}^{n} is the K_{0} subalgebra generated by u_{λ} := λ_{1}/λ_{i} ⊗_{k} X_{1} + ⋯ + λ_{m−1}/λ_{i} ⊗_{k} X_{m−1} and 1 ⊗_{k} X_{m} then we have V_{A′}^{r}(K_{0} ⊗_{k} M) ⊆ V_{A_{m}^{n}}^{r}(K ⊗_{k} M) = {0} and so by the theorem for the case m = 2 we have M projective as an A′-module. It now follows that K ⊗_{k} \overline{M} is projective as a K[u_{λ}]-module and hence V_{A_{m−1}^{n}}^{r}(K ⊗_{k} \overline{M}) = \{0\}. Now, by the inductive hypothesis,

we have \overline{M} a projective module and hence by Corollary 2.10 M ∈ A_{m}^{n} -Proj. □

5.3. With Theorem 5.4 we could just define a rank variety for M ∈ A_{m}^{n} -Mod to be just V_{A_{m}^{n}}^{r}(K ⊗_{k} M), for some algebraically closed field K ⊇ k of sufficient transcendence degree. However, as pointed out in [8], this would appear to depend upon a choice of K and is hence a somewhat unnatural definition. To get a good definition we will repeat the approach taken in [8]. First we must address the dependence on a choice of K and so we next recall Proposition 5.2 of [8], whose proof for A_{m}^{n} is identical to that given in [8].

Proposition 5.5. Let M ∈ A_{m}^{n} -Mod and let V ⊆ V^{r}(k) be a closed homogeneous irreducible variety. Suppose K and K′ are algebraically closed field extensions of k with both tr-deg(K/k) ≥ dim V and tr-deg(K′/k) ≥ dim V. Suppose α ∈ V^{r}(K) and β ∈ V^{r}(K′) are generic points for V. Then

α ∈ V_{A_{m}^{n}}^{r}(K ⊗_{k} M) ⟺ β ∈ V_{A_{m}^{n}}^{r}(K′ ⊗_{k} M).

We can now define, following [8], the new rank variety (more accurately a collection of closed sets) for arbitrary A_{m}^{n}-modules. Recall that V^{r}(k) denotes the affine m-space, 𝔸^{m}(k), of the vector space J(A_{m}^{n})/J^{2}(A_{m}^{n}).

Definition 5.6. Let V^{r}(k) be the set of all nonzero closed homogeneous irreducible subvarieties of V^{r}(k). For M ∈ A_{m}^{n} -Mod, let V_{A_{m}^{n}}^{r}(M) be the set of all V ∈ V^{r}(k) such that for any algebraically closed field extension K ⊇ k with tr-deg(K/k) ≥ dim V and any generic point α ∈ V^{r}(K) for V, we have α ∈ V_{A_{m}^{n}}^{r}(K ⊗_{k} M).

Remark 5.7. All of this section (principally Definition 5.6 and Theorem 5.4) can clearly be generalised to the algebra A_{q,m}^{n} considered in §4 in a way entirely analogous to the generalisation, in the finite-dimensional case, of V_{A_{m}^{n}}^{r} to V_{A_{q,m}^{n}}^{r}.

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D. J. Benson, Department of Mathematical Sciences, University of Aberdeen, Meston Building, King’s College, Aberdeen, AB24 3UE, UK

E-mail address: /b\/e\/n\/s\/o\/n\/d\/j\ at maths dot abdn dot ac dot uk

Karin Erdmann, Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LB, UK E-mail address: /e\/r\/d\/m\/a\/n\/n\ at at maths dot ox dot ac dot uk

Miles Holloway, Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LB, UK E-mail address: /h\/o\/l\/l\/o\/w\/a\/y\ at at maths dot ox dot ac dot uk