GENERIC IDEMPOTENT MODULES FOR A FINITE GROUP

D. J. BENSON AND H. KRAUSE

Abstract. Let G be a finite group and k an algebraically closed field of characteristic p. Let F_{𝒰} be the Rickard idempotent kG-module corresponding to the set 𝒰 of subvarieties of the cohomology variety V_{G} which are not irreducible components. We show that F_{𝒰} is a finite sum of generic modules corresponding to the irreducible components of V_{G}. In this context, a generic module is an indecomposable module of infinite length over kG but finite length as a module over its endomorphism ring.

1. Introduction

The purpose of this paper is to draw attention to a particular infinite dimensional module for a finite group algebra, which plays the role of a generic module in a certain precise sense which will be described. The module in question is a particular case of Rickard’s construction [14] of idempotent modules in the stable category. These idempotent modules have played a pivotal role in some recent developments in modular representation theory, see for example [1, 3, 4, 5]. Theorem 1.1. Let 𝒰 denote the set of closed homogeneous irreducible subvarieties of the cohomology variety V_{G} which are not irreducible components, and let F_{𝒰} be the corresponding idempotent module. Then F_{𝒰} is endofinite.

2. Endofinite modules in the stable category

For any ring Λ, a Λ-module M is said to be endofinite if it has finite length as a module over End_{Λ}(M). This concept is investigated by Crawley–Boevey in [9]. The following are some properties of endofinite modules over a finite dimensional algebra; their proofs can be found in Section 4 of [9].

(1) If M has finite length then M is endofinite.

(2) An arbitrary direct sum or direct product of (possibly infinitely many) copies of an endofinite module is endofinite.

(3) Every finite direct sum of endofinite modules is endofinite.

(4) Every direct summand of an endofinite module is endofinite.

(5) Every projective module and every injective module is endofinite. This follows from (2) and (3), since there are finitely many isomorphism classes of indecomposable projectives, and every projective is a direct sum of indecomposable projectives; and similarly for injectives.

(6) Every endofinite module is pure injective. It is even Σ-pure injective, since this is equivalent to d.c.c. on additive subgroups of finite definition (in the first order language of Λ-modules). Endofinite is equivalent to a.c.c. and d.c.c. on additive subgroups of finite definition.

Date: January 21, 1999.

The first author is partly supported by a grant from the NSF.

A generic module is defined to be an indecomposable endofinite module of infinite length. As an example, let G = ⟨g_{1}, g_{2}⟩ ≅ ℤ/2 × ℤ/2 and let k have characteristic two. Let t be an indeterminate, and let M be the kG-module with underlying vector space k(t) ⊕ k(t), with G acting via

g_{1} ↦ ( \begin{matrix}I & I \\ 0 & I \end{matrix} ) g_{2} ↦ ( \begin{matrix}I & T \\ 0 & I \end{matrix} ) .

Here, I denotes the identity map on k(t) and T denotes multiplication by t on k(t). It is easy to see that M has infinite length over kG but length two over End_{kG}(M) ≅ k(t)[u]/(u^{2}). Since M is also indecomposable, it is a generic module. In fact, it is the module F_{𝒰} for this group, and there are no other generic modules.

More generally, Crawley–Boevey has shown that for finite dimensional algebras over an algebraically closed field there is a strong connection between generic modules and the representation type; see [9]. Of course, for an algebra of finite representation type, there are no generic modules. For a finite dimensional algebra of tame representation type, there is a natural correspondence between generic modules and one parameter families of finite length modules. A finite dimensional algebra has wild representation type if and only if there exists a generic module such that the endomorphism ring modulo its radical is the full division ring of fractions of the free ring in two variables k⟨x, y⟩. This division ring has infinite dimension over its center.

Let Λ be a finite dimensional self-injective algebra. We write Mod(Λ) for the category of all Λmodules and homomorphisms, and mod(Λ) for the full subcategory of finitely generated modules. The stable category StMod(Λ) has the same objects as Mod(Λ), but the arrows are Hom_{Λ}(M, N) modulo the homomorphisms which factor through a projective (= injective) module. The stable category of finitely generated modules is written stmod(Λ). The categories StMod(Λ) and stmod(Λ) are triangulated categories, in which the distinguished triangles correspond to short exact sequences in Mod(Λ). Recall from [11] that an object M in StMod(Λ) is said to be endofinite if \underline{Hom}_{Λ}(X, M) has finite length over \underline{End}_{Λ}(M) for all X in stmod(Λ). This does not conflict with the previous notion of endofiniteness, because of the following proposition.

Proposition 2.1. The following are equivalent for a Λ-module M:

(1) M is an endofinite Λ-module;

(2) M is an endofinite object in StModΛ;

(3) \underline{Hom}_{Λ}(S, M) is of finite length over \underline{End}_{Λ}(M) for every simple Λ-module S.

Proof. (1) ⇒ (2) Suppose that M is endofinite and let X ∈ modΛ. Clearly, Hom_{Λ}(X, M) has finite length over End_{Λ}(M). Therefore the quotient \underline{Hom}_{Λ}(X, M) has finite length over \underline{End}_{Λ}(M) since it is of finite length over End_{Λ}(M).

(2) ⇒ (3) Clear.

(3) ⇒ (1) Direct limits of injective modules are injective. Therefore we can apply Zorn’s lemma and find a maximal injective submodule I of M. Let M = I ⊕ M′. Given any simple Λ-module S, we have

\underline{Hom}_{Λ}(S, M) ≅ \underline{Hom}_{Λ}(S, M′) = Hom_{Λ}(S, M′).

Assuming (3), this module is of finite length over End_{Λ}(M′) since it is of finite length over \underline{End}_{Λ}(M′) ≅ \underline{End}_{Λ}(M). Thus Hom_{Λ}(Λ, M′) has finite length over End_{Λ}(M′) since Λ is of finite length over Λ. We conclude that M′ is endofinite. Every injective Λ-module is endofinite, so we conclude that M = I ⊕ M′ is endofinite and therefore the proof is complete.

For a Λ-module M, we denote by endol(M) the length of M as a module over End_{Λ}(M).

Lemma 2.2. Let H be a subgroup of a group G, and let k be a field.

(1) If M is a kG-module then endol(M ↓_{H}) ≤ endol(M).

(2) If M is a kH-module then endol(M ↑^{G}) ≤ endol(M) · |G : H|.

Proof. (1) The action of End_{kH}(M ↓_{H}) on M extends the action of End_{kG}(M). Therefore endol(M) is a bound for endol(M ↓_{H}).

(2) The endomorphism ring End_{kH}(M) acts on M ↑^{G}= kG ⊗_{kH} M via the ring homomorphism End_{kH}(M) → End_{kG}(M ↑^{G}) which sends an endomorphism α to the endomorphism id ⊗ α. The kH-module kG is free of rank |G : H|, and therefore the length of M ↑^{G} over End_{kH}(M) is equal to endol(M) · |G : H|. We conclude that the length of M ↑^{G} over End_{kG}(M ↑^{G}) is bounded by endol(M) · |G : H|.

Proposition 2.3. Let G be a finite group and k be a field of characteristic p. Then a kG-module M is endofinite if and only if its restriction to a Sylow p-subgroup of G is endofinite.

Proof. Let P be a Sylow p-subgroup of G. If M is endofinite then Lemma 2.2 (1) shows that M ↓_{P} is endofinite. Conversely, if M ↓_{P} is endofinite then by Lemma 2.2 (2), M ↓_{P} ↑^{G} is endofinite. But since |G : P | is coprime to p, M is a direct summand of M ↓_{P} ↑^{G}.

3. Idempotent modules

Definition 3.1. Let G be a finite group and k be a field.

(1) A kG-module E is called counital idempotent if there exists a map ε: E → k such that ε⊗id_{E} induces an isomorphism E ⊗_{k} E → E in StModkG.

(2) A kG-module F is called unital idempotent if there exists a map ϕ: k → F such that ϕ ⊗ id_{F} induces an isomorphism F → F ⊗_{k} F in StModkG.

Suppose that E is a counital idempotent kG-module. Then the kG-modules M such that ε⊗id_{M} induces an isomorphism E ⊗_{k} M → M in StModkG form a localizing subcategory ℒ_{E} of StModkG. Recall that a full subcategory of StModkG is localizing if it is a triangulated subcategory which is closed under arbitrary direct sums. Moreover, the inclusion functor ℒ_{E} → StModkG has a right adjoint which preserves arbitrary direct sums; it sends a kG-module M to E ⊗_{k} M. Note that ℒ_{E} does not depend on the map ε: E → k. In fact, if ε′: E → k is a second map inducing an isomorphism E ⊗_{k} E → E in StModkG then there exists an isomorphism α: E → E such that ε′ = εα in StModkG.

Lemma 3.2. Let ϕ: k → F be a map in StModkG such that ϕ ⊗ id_{F} is an isomorphism. Then ϕ induces an isomorphism \underline{Hom}_{kG}(F, F) → \underline{Hom}_{kG}(k, F) of \underline{End}_{kG}(F)-modules.

Proof. Complete the map ϕ to a triangle E → k → F → Ω^{−1}(E). Tensoring this triangle with F gives a new triangle which shows that F ⊗_{k} E = 0 (in the stable category). It follows that \underline{Hom}_{kG}(E, F) = 0 since any map E → F induces a commutative square

\xymatrix{k ⊗_{k} E \ar[r] \ar[d] & k ⊗_{k} F \ar[d]^{≅} \\ F ⊗_{k} E \ar[r] & F ⊗_{k} F}

The same argument shows that \underline{Hom}_{kG}(Ω^{−1}(E), F) = 0. Applying \underline{Hom}_{kG}(−, F) to the above triangle then shows that \underline{Hom}_{kG}(F, F) ≅ \underline{Hom}_{kG}(k, F).

Theorem 3.3. Let G be a p-group. Then a unital idempotent kG-module F is endofinite if and only if \underline{End}_{kG}(F) is an Artinian ring.

Proof. We may assume that the field k is of characteristic p, and therefore the trivial kG-module k is, up to isomorphism, the unique simple kG-module since G is a p-group. Lemma 3.2 shows that \underline{Hom}_{kG}(k, F) has finite length over \underline{End}_{kG}(F) if and only if \underline{End}_{kG}(F) is Artinian. Therefore the assertion follows from the above criterion for endofiniteness.

Remark 3.4. A unital idempotent kG-module F with Artinian endomorphism ring \underline{End}_{kG}(F) has a unique decomposition F = F_{1} ⊕ ⋯ ⊕ F_{n} into finitely many indecomposable objects in StModkG. Moreover, each F_{i} is a unital idempotent module and F_{i} ⊗_{k} F_{j} = 0 for all i ≠ j; see [11].

4. The module F_{𝒰}

Let G be a finite group and k an algebraically closed field of characteristic p. Let V_{G} be the maximal ideal spectrum of the graded cohomology ring H^{∗}(G, k) = Ext_{kG}^{∗}(k, k) (Note that this ring is only graded commutative; if this bothers you, notice that for p = 2 this means it is commutative; for p odd, elements of odd degree square to zero, so modulo the nil radical this ring is again commutative). Let V_{1}, …, V_{t} be the irreducible components of V_{G}, and let 𝒰 be the collection of closed homogeneous irreducible subvarieties of V_{G} which are not irreducible components. This collection 𝒰 is clearly closed under specialization, in the sense that if V ∈ 𝒰 and W ⊆ V then W ∈ 𝒰. Denote by 𝒞(𝒰) the thick subcategory of stmod(kG) consisting of finitely generated kG-modules M such that V_{G}(M) ∈ 𝒰. Let

E_{𝒰} → k → F_{𝒰} → Ω^{−1}E_{𝒰}

be the corresponding distinguished triangle of idempotent modules, as constructed by Rickard ([14], Section 5). Let 𝒞(𝒰)^{⊕} denote the triangulated subcategory of StMod(kG) generated by (possibly infinite) direct sums of objects in 𝒞(𝒰). Then passage to the quotient category StMod(kG) → StMod(kG)/𝒞(𝒰)^{⊕} has a left adjoint, which takes a module M to F (𝒰) ⊗_{k} M. Furthermore, the natural functor stmod(kG)/𝒞(𝒰) → StMod(kG)/𝒞(𝒰)^{⊕} is fully faithful.

Proposition 4.1. The stable endomorphism ring \underline{End}_{kG}(F_{𝒰}) is Artinian. It is a direct product of local rings, E_{1} × ⋯ × E_{t}, corresponding to the irreducible components V_{1}, …, V_{t}. The quotient E_{i}/J(E_{i}) of E_{i} by its radical is isomorphic to the function field k(V_{i}).

Proof. By the above remarks, the ring \underline{End}_{kG}(F_{𝒰}) is isomorphic to the endomorphism ring of k in the quotient category StMod(kG)/𝒞(𝒰)^{⊕}, or equivalently in stmod(kG)/𝒞(𝒰). So the method of Carlson, Donovan and Wheeler [7] can be used to give the following explicit description of \underline{End}_{kG}(F_{𝒰}). Let S_{𝒰} be the multiplicative subset of H^{∗}(G, k) consisting of elements which are not identically zero on any irreducible component of V_{G}, or equivalently, elements which do not belong to any minimal prime ideal in H^{∗}(G, k). Then \underline{End}_{kG}(F_{𝒰}) is isomorphic to the ring of degree zero elements in the homogeneous localization S_{𝒰}^{−1}H^{∗}(G, k). The effect of this localization is that the homogeneous prime ideals in S_{𝒰}^{−1}H^{∗}(G, k) correspond precisely to the minimal prime ideals in H^{∗}(G, k), namely to the irreducible components of V_{G}. The prime ideals in \underline{End}_{kG}(F_{𝒰}) are then obtained by intersecting these homogeneous primes with the degree zero part. It follows that in \underline{End}_{kG}(F_{𝒰}), every prime ideal is maximal. So this is a zero dimensional Noetherian graded commutative ring, and is hence Artinian.

A commutative Artinian ring is necessarily a direct product of local rings with nilpotent Jacobson radicals. For \underline{End}_{kG}(F_{𝒰}), these local rings correspond to the homogeneous maximal ideals in S_{𝒰}^{−1}H^{∗}(G, k). Furthermore, by definition, the function field of V_{i} is equal to the field of fractions of the integral domain of degree zero elements in the quotient of H^{∗}(G, k) by the corresponding minimal prime ideal. This is the same as the degree zero elements in the quotient of S_{𝒰}^{−1}H^{∗}(G, k) by the corresponding maximal ideal. This in turn is the same as the quotient of E_{i} by its Jacobson radical.

Remark 4.2. Greenlees and Lyubeznik [10] (Corollary 7.4) have shown that the ring S_{𝒰}^{−1}H^{∗}(G, k) is a finite product of Gorenstein local rings. It is tempting to suppose that it follows from this that the degree zero part \underline{End}_{kG}(F_{𝒰}) is a symmetric algebra. That this is not the case may be seen by examining the case where p is odd and G is an elementary abelian p-group of rank three. In this case, \underline{End}_{kG}(F_{𝒰}) is an Artinian local ring of length four and square zero radical, so it cannot be a symmetric algebra. The theorem of Greenlees and Lyubeznik should therefore be interpreted as applying to the graded endomorphism ring

\underline{End}_{kG}^{∗}(F_{𝒰}) = ⨁_{n∈ℤ} \underline{Hom}_{kG}(Ω^{n}F_{𝒰} , F_{𝒰}).

Remark 4.3. For other collections 𝒱 of closed homogeneous irreducible subvarieties of V_{G}, closed under specialization, the structure of the stable endomorphism ring of F_{𝒱} can be deduced from the work of Carlson and Wheeler [8].

Theorem 4.4. The module F_{𝒰} is endofinite.

Proof. By Proposition 2.3, it suffices to show that the restriction of F_{𝒰} to a Sylow p-subgroup P of G is endofinite. But the restriction of F_{𝒰} to P is stably isomorphic to the corresponding module F_{𝒰} there. By Proposition 4.1 this module has Artinian stable endomorphism ring, so by Theorem 3.3 it is endofinite.

5. Local control

In this section, we use the techniques of [2] to reduce the description of the summands of F_{𝒰} down to local subgroups, and there we give an explicit construction for this module.

By Proposition 4.1, F_{𝒰} decomposes into indecomposable pieces F_{1} ⊕ ⋯ ⊕ F_{t} corresponding to the irreducible components V_{1}, …, V_{t}. The correspondence has the property that

\underline{End}_{kG}(F_{i})/J\underline{End}_{kG}(F_{i}) ≅ k(V_{i}).

By a theorem of Quillen [12, 13], the irreducible components of V_{G} are in one-one correspondence with the conjugacy classes of elementary abelian p-subgroups of G. Let E_{i} be a representative of the conjugacy class corresponding to V_{i}, and set N_{i} = N_{G}(E_{i}).

Lemma 5.1. The module F_{i} is isomorphic to the module induced from the corresponding module F_{𝒰_{i}} for N_{i}.

Proof. We apply the methods of [2], but with ℳ_{r−1} replaced by 𝒞(𝒰), and ℳ_{r,r−1} replaced by stmod(kG)/𝒞(𝒰). The arguments work equally well in this context, and show the following. The category stmod(kG)/𝒞(𝒰) decomposes as a direct sum of triangulated categories ℳ_{G,E_{i}}, with the E_{i} as above. Induction and restriction between N_{i} and G establishes an equivalence of categories StMod(kN_{i})/𝒞(𝒰_{i}) ⇌ ℳ_{G,E_{i}}, under which the trivial module corresponds to the module called π_{G,E_{i}}(k) in [2].

Now we apply the adjunction mentioned before the proof of Proposition 4.1. This takes the trivial module in stmod(kG)/𝒞(𝒰) to F_{𝒰}, and the module π_{G,E_{i}}(k) to F_{i}. The corresponding adjunction at the level of N_{i} takes the trivial module in stmod(kN_{i})/𝒞(𝒰_{i}) to F_{𝒰_{i}} . Since passage to the quotient category commutes with induction, it follows that F_{𝒰_{i}} ↑^{G}≅ F_{i}.

Now N_{i} has a unique maximal elementary abelian p-subgroup, namely E_{i}. The map from the cohomology variety of N_{i} to the cohomology variety of G induces a birational equivalence V_{N_{i}} → V_{i}. In this situation, there is an explicit construction for F_{𝒰_{i}} which we now describe. This generalizes the construction given in Theorems 5.1.2 and 5.4.6 of [6].

So from now on, let G be a group with a unique maximal elementary abelian p-subgroup E, and assume that E is normal in G. Let r be the p-rank of E, and let X_{1}, …, X_{r−1} be indeterminates. Choose elements ζ_{1}, …, ζ_{r} forming a homogeneous system of parameters for H^{∗}(G, k) and having the same degree n. Set K = k(X_{1}, …, X_{r−1}), and for 1 ≤ j ≤ r − 1, set

ξ_{j} = X_{j}ζ_{r} − ζ_{j} ∈ H^{n}(G, K).

Represent ξ_{j} by a map ξ̂_{j} : K → Ω^{−n}K, and write L_{j} for the cokernel. Write ϕ_{j} : k → L_{j} for the

composite k ↪ K ⟶^{ζ̂_{r}} Ω^{−n}K → L_{j}.

Theorem 5.2. The kG-module F_{𝒰} is isomorphic to the tensor product L_{1} ⊗_{K} ⋯ ⊗_{K} L_{r−1}, with the coefficients restricted to k, and with the map ϕ_{1} ⊗ ⋯ ⊗ ϕ_{r−1} as ϕ : k → F_{𝒰}.

Proof. Since E is the unique maximal elementary abelian subgroup of G, it suffices to restrict to E = ⟨g_{1}, …, g_{r}⟩ and use the theory developed in [4]. Namely, let V be a closed homogeneous irreducible subvariety of the rank variety V_{E}^{r} ≅ 𝔸^{r}(k), and write L for the tensor product L_{1} ⊗_{K} ⋯ ⊗_{K} L_{r−1}. We need to complete the map ϕ : k → L to a triangle X → k → L → Ω^{−1}X, and then tensor with the function field k(V). We then need to check that the triangle restricts to k(V) → k(V) → 0 → Ω^{−1}k(V) if V is a proper subvariety of V_{E}^{r}, and to 0 → k(V) → k(V) → 0 otherwise. In other words, we need to show that the rank variety of L is {V_{E}^{r}}, while that of X is the complement of this.

Let H^{∗}(E, k)/(nilpotents) = k[x_{1}, …, x_{r}] with the x_{j} in degree one if p = 2 and two if p is odd. Let m denote n if p = 2 and n/2 if p is odd. Then res_{G,E}(ζ_{j}) (mod nilpotents) is equal to some homogeneous polynomial f_{j}(x_{1}, …, x_{r}) of degree m. The polynomials f_{1}, …, f_{r} form a regular sequence in k[x_{1}, …, x_{r}].

Denote by t_{1}, …, t_{r} the coordinate functions on V_{E}^{r}, regarded as elements of k(V). Thus if V = V_{E}^{r} then the t_{j} are algebraically independent, while if V is a proper subvariety of V_{E}^{r} then there exists a homogeneous irreducible polynomial in the elements t_{1}, …, t_{r} which equals zero in k(V).

The generic cyclic shifted subgroup of E corresponding to V is generated by

u_{V} = 1 + t_{1}(g_{1} − 1) + ⋯ + t_{r}(g_{r} − 1) ∈ k(V)E.

We must restrict the map k → L to u_{V}, being careful to observe that tensor product does not commute with restriction. To get around this difficulty, we form the tensor product of the complexes

0 → K ⟶^{ξ̂_{j}} Ω^{−n}K → 0

to form a Koszul-like complex whose homology is L. The point of this is that Yoneda composition of cohomology elements does commute with restriction. So we are restricting a tensor product of complexes of the form

0 → k(V) ⊗_{k} K ⟶^{1⊗ξ̂_{j}} k(V) ⊗_{k} Ω^{−n}K → 0

and then taking the homology. We can act as though tensor product commutes with restriction.

Let ζ be a nonzero element of H^{2}(⟨u_{V} ⟩, k(V)) if p is odd, and of H^{1}(⟨u_{V} ⟩, k(V)) if p = 2, chosen so that res_{E,⟨u_{V} ⟩}(x_{j}) = t_{j}ζ. Then

res_{G,⟨u_{V} ⟩}(ζ_{j}) = res_{E,⟨u_{V} ⟩}(f_{j}(x_{1}, …, x_{r})) = f_{j}(t_{1}, …, t_{r})ζ^{m},

and

res_{G,⟨u_{V} ⟩}(ξ_{j}) = (X_{j}f_{r}(t_{1}, …, t_{r}) − f_{j}(t_{1}, …, t_{r}))ζ^{m}.

These are the elements involved in the restriction to ⟨u_{V} ⟩ of the above Koszul-like complex. If f_{r}(t_{1}, …, t_{r}) = 0, so that V is contained in the corresponding hypersurface, then

res_{G,⟨u_{V} ⟩}(ξ_{j}) = −f_{j}(t_{1}, …, t_{r})ζ^{m}

does not involve any of the X_{j}. So we may perform the construction of the triangle over k(V) and restrict, and then tensor with K at the end. Since some f_{j}(t_{1}, …, t_{r}) is nonzero, the corresponding ξ̂_{j} restricts to a map with projective cokernel, and so k(V) ⊗_{k} L ↓_{⟨u_{V} ⟩} is projective. So from now on, assume that f_{r}(t_{1}, …, t_{r}) ≠ 0.

Using ζ̂^{m} to identify k(V) with Ω^{−m}k(V), it follows that up to irrelevant projective summands, the restriction is equal to the Koszul complex obtained using the elements X_{j}f_{r} − f_{j} on the ring k(V) ⊗_{k} K. Since f_{r}(t_{1}, …, t_{r}) ≠ 0, these elements form a regular sequence, in this ring, because each uses an X_{j} not used in any of the other terms. So the homology is equal to the quotient

k(V) ⊗_{k} K/(X_{1}f_{r} − f_{1}, …, X_{r−1}f_{r} − f_{r−1}).

If V = V_{E}, so that the t_{j} are algebraically independent, the map (given by restricting ϕ_{1}⊗⋯⊗ϕ_{r−1}) from k(V) to this quotient is an isomorphism, since each occurrence of X_{j} can be replaced by f_{j}/f_{r}. On the other hand, if V is a proper subvariety, so that there is a nontrivial homogeneous relation between the elements t_{1}, …, t_{r}, then there is also a nontrivial homogeneous relation between the elements f_{1}, …, f_{r}, say g(f_{1}, …, f_{r}) = 0. But then

1 ⊗ 1 = 1 ⊗ g(X_{1}, …, X_{r−1}, 1)/g(X_{1}, …, X_{r−1}, 1)
= g(f_{1}/f_{r}, …, f_{r−1}/f_{r}, 1) ⊗ 1/g(X_{1}, …, X_{r−1}, 1) = 0,

and so the quotient is zero, and the restriction of L to ⟨u_{V} ⟩ is projective.

The conclusion of this calculation is that the rank variety of L is equal to {V_{E}^{r}}, while the rank variety of X is the complement of this. Rickard’s characterization of F_{𝒰} (see the remark after Proposition 5.7 of [14]) now completes the proof of the theorem.

Remark 5.3. This theorem can be used as the basis for an alternative proof that F_{𝒰} is endofinite.

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

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

Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

   E-mail address: henning@mathematik.uni-bielefeld.de