ALMOST SPLIT SEQUENCES FOR COMODULES

WILLIAM CHIN, MARK KLEINER, AND DECLAN QUINN

Abstract. Let Γ be a coalgebra over a field k. We introduce an operator Tr that takes a right quasi-finitely copresented Γ-comodule M to a left quasi-finitely copresented Γ-comodule Tr M. If M is indecomposable not injective and Tr M is finite-dimensional over k, we prove the existence of an almost split sequence 0 → M → E → DTr M → 0 in the category of all right Γ-comodules, where D = Hom_{k} ( , k). If Γ is right semiperfect and the embedding of each simple right comodule S into its injective envelope I(S) has the property that the socle of I(S)/S is finite-dimensional, the above almost split sequence exists for each finite-dimensional M, and DTr M is also finite-dimensional.

1. Introduction

Throughout the paper, k is a fixed field and D = Hom_{k} ( , k). Almost split sequences were discovered by Auslander and Reiten for finitely

generated modules over a finite-dimensional (artin) algebra [ARS], and play

1991 Mathematics Subject Classification. 16G10, 16G70. Key words and phrases. Coalgebra, comodule, almost split sequence, transpose,

semiperfect.

The second-named author’s research described in this publication was made possible in part by Award No. UM1-314 of the U.S. Civilian Research & Development Foundation for the Independent States of the Former Soviet Union.

2 WILLIAM CHIN, MARK KLEINER, AND DECLAN QUINN

an important role in other settings as well. We deal with almost split sequences for comodules over an infinite-dimensional coalgebra; the finitedimensional case by duality reduces to that of a finite-dimensional algebra. Comodules arise naturally in several areas of mathematics, for example, in the representation theory of algebraic groups and quantum groups.

In our investigation, it was natural to ask whether one of the known existence proofs for almost split sequences over an artin algebra could be adapted, and we concentrated on the most elegant one, which uses the notion of transpose [ARS]. The problem then became to define the transpose over an arbitrary coalgebra, and we solved it using the cohom functor introduced in [T]. Although we define the transpose in a different setting, it has properties very similar to those of the classical one. These considerations are presented in Section 3. The main result is in Section 4, where we obtain a six-term exact sequence involving the transpose, cohom, and cotensor product, which resembles the well-known six-term exact sequence for modules, with the transpose, Hom, and tensor product. The six-term sequence implies the existence of an almost split sequence in the category of all comodules whose left-end term is any indecomposable noninjective quasi-finitely copresented comodule with a finite-dimensional transpose.

The last three sections contain applications of the main result to certain classes of coalgebras. We show in Section 5 that if a coalgebra is right semiperfect [L] and every simple right comodule S embeds into its injective envelope I(S) so that the socle of I(S)/S is finite-dimensional, then almost

ALMOST SPLIT SEQUENCES 3

split sequences exist in the category of finite-dimensional right comodules. If the coalgebra is a Hopf algebra, almost split sequences are strongly influenced by the behavior of the trivial comodule.

Section 5 has several applications. If G is a virtually reductive affine group scheme, then its coordinate algebra k[G] is by definition a left and right semiperfect coalgebra so that, by our results, the category of finitedimensional k[G]-comodules has almost split sequences. The latter, among other theorems, was proved in [D1, D2] using special features of group schemes. These two interesting papers also have general results on almost split sequences for finite-dimensional comodules obtained by restricting to the finite-dimensional subcoalgebras of the given coalgebra. We intend to investigate possible connections between this and our approaches in the future.

Let k_{ζ}[G] be the quantized coordinate algebra of the semisimple algebraic group G, where char k = 0 and the parameter q is specialized to a p-th-power primitive root of unity ζ ∈ k. Since k_{ζ}[G] is a left and right semiperfect coalgebra [APW, Section 9], almost split sequences for finite-dimensional k_{ζ}[G]-comodules exist. Another example of a semiperfect coalgebra is the coordinate bialgebra of n × n matrices; the details for n = 2 are in [CK].

Section 6 provides many more examples by giving a combinatorial characterization of the quivers whose path coalgebra satisfies the conditions of Section 5, and by interpreting comodules over the path coalgebra in terms of the more familiar representations of quivers. Since a subcoalgebra of a

4 WILLIAM CHIN, MARK KLEINER, AND DECLAN QUINN

right semiperfect coalgebra is right semiperfect, subcoalgebras of path coalgebras satisfying the conditions of Section 5 also satisfy these conditions. Recall that every pointed coalgebra is Morita-Takeuchi equivalent to a subcoalgebra of the path coalgebra of its quiver [CM]. Section 7 establishes the existence of almost split sequences in the category of finite-dimensional modules (functors) over a left locally bounded category as a consequence of the results of Sections 5 and 6 by passing to the category of finite-dimensional comodules over a right semiperfect coalgebra. The latter existence result is a generalization of that for modules over a locally bounded category, which was obtained in [BG] by arguments that do not extend to our situation.

We now fix the terminology for the rest of the paper, using freely the definitions, results, and notation of [AF, ARS, Gr, Mo, T]. Denote by Γ a k-coalgebra with comultiplication Δ : Γ → Γ ⊗ Γ and counit ϵ : Γ → k, where ⊗ = ⊗_{k}. A right Γ-comodule M is given by a structure map ρ : M → M ⊗ Γ, the category M^{Γ} of right Γ-comodules is an abelian category with enough injectives, I(M) denotes an injective envelope of M, and Soc M stands for the socle, i.e., the sum of simple subcomodules, of M. We identify the category ^{Γ}M of left Γ-comodules with the category M^{Γ^{op}}, where Γ^{op} is the opposite coalgebra of Γ. If Θ is another k-coalgebra, ΘM^{Γ} denotes the category of Θ − Γ-bicomodules. A comodule M ∈ M^{Γ} is quasi-finite if dim_{k} Hom_{Γ} (F, M) < ∞ for all finite-dimensional F ∈ M^{Γ}. In what follows, M_{q}^{Γ} denotes the full subcategory of M^{Γ} determined by the quasi-finite comodules, and I^{Γ} denotes the full subcategory determined

ALMOST SPLIT SEQUENCES 5

by the quasi-finite injectives. If A is a full subcategory of M^{Γ}, then A_{I} is the full subcategory determined by those comodules in A that have no non-zero injective direct summand. A comodule M ∈ M^{Γ} is quasi-finitely copresented if its minimal injective copresentation 0 → M → I_{0} → I_{1} satisfies I_{j} ∈ I^{Γ} for j = 0, 1; in the sequel, M_{qc}^{Γ} denotes the full subcategory of M^{Γ} determined by the quasi-finitely copresented comodules.

Recall from [T] that if X ∈ M_{q}^{Γ} and Y ∈ M^{Γ}, then h_{Γ} (X, Y) = lim_{→_{λ}} D Hom_{Γ} (Y_{λ}, X) where {Y_{λ}} is the set of finite-dimensional subcomodules of Y. Here h_{Γ} ( , ) is an additive bifunctor characterized by the property that h_{Γ} (X, ) is a left adjoint of the functor ⊗X; h_{Γ} ( , ) is called the cohom functor. The coalgebra of coendomorphims of X is e_{Γ} (X) = h_{Γ} (X, X), and X is a left e_{Γ} (X)-comodule. For M ∈ M^{Γ} and N ∈ ^{Γ}M, the cotensor product is M□_{Γ}N.

2. Direct sum decomposition

Our presentation is inspired by [ARS, II.2].

Let M ∈ M_{q}^{Γ} and let add M be the full subcategory of M_{q}^{Γ} determined by the direct summands of arbitrary direct sums of copies of M. Put B = e_{Γ} (M), note that I^{B} = add B in M^{B}, and consider the functors

F_{M} = h_{Γ} (M, −) : add M → I^{B} and G_{M} = −□_{B}M : I^{B} → add M.

Theorem 2.1. F_{M} and G_{M} are inverse equivalences of categories.

6 WILLIAM CHIN, MARK KLEINER, AND DECLAN QUINN

Proof. Dropping the subscript M, it is immediate that G(F (M)) = G(B) ≅ M and F (G(B)) ≅ F (M) = B. The rest follows from the additivity of F and G.

Corollary 2.2. (a) If M ∈ M_{q}^{Γ} is indecomposable then End_{Γ} M is a local

ring.

(b) The Krull-Remak-Schmidt-Azumaya Theorem holds for arbitrary direct sums of quasi-finite comodules in M^{Γ}.

Proof. (a) In view of Theorem 2.1, it suffices to show that End_{Γ} I is local if I ∈ I^{Γ} is indecomposable, which follows immediately from the fact that I is the injective hull of a simple Γ-comodule [Gr, assertion (1.5g)(i), pp. 152-153].

(b) Since M^{Γ} is an abelian category, this follows from (a) and [AF, Th. 12.6].

3. The transpose

We define and establish basic properties of the operator of transpose on M_{qc}^{Γ}. Let ^{∗} denote the contravariant functor ^{∗} = h_{Γ} ( , Γ) : M_{q}^{Γ} → M^{Γ^{op}}, as well as h_{Γ^{op}} ( , Γ^{op}) : M_{q}^{Γ^{op}} → M^{Γ}; it will be usually clear from the context which of the two functors is meant. We say that an X ∈ M_{q}^{Γ} is strongly quasi-finite if X^{∗} ∈ M_{q}^{Γ^{op}}, and denote by M_{sq}^{Γ} the full subcategory of M^{Γ} determined by all strongly quasi-finite comodules. Given an X ∈ M_{q}^{Γ} and a U ∈ M_{q}^{Γ^{op}}, [T, Prop. 1.3 and Remark 1.11] gives the adjunctions

ALMOST SPLIT SEQUENCES 7

ϕ_{X} : Hom_{Γ^{op}} (h_{Γ} (X, Y), Z) ≅ Hom_{Γ−Γ} (Y, Z ⊗ X),

where Y ∈ ^{Γ}M^{Γ} and Z ∈ M^{Γ^{op}}, and

ψ_{U} : Hom_{Γ} (h_{Γ^{op}} (U, V), W) ≅ Hom_{Γ−Γ} (V, U ⊗ W),

where V ∈ ^{Γ}M^{Γ} and W ∈ M^{Γ}. For X ∈ M_{sq}^{Γ} we may choose U = Z = X^{∗} and V = Y = Γ so that the above adjunction isomorphisms specialize to

ϕ_{X} : Hom_{Γ^{op}} (X^{∗}, X^{∗}) ≅ Hom_{Γ−Γ} (Γ, X^{∗} ⊗ X)  (3.1)

and

ψ_{X^{∗}} : Hom_{Γ} (X^{∗∗}, X) ≅ Hom_{Γ−Γ} (Γ, X^{∗} ⊗ X).  (3.2)

Since the right-hand sides of (3.1) and (3.2) coincide, we denote by η_{X} : X^{∗∗} → X a unique morphism of Γ-comodules satisfying ψ_{X^{∗}}(η_{X}) = ϕ_{X}(1_{X^{∗}}).

Proposition 3.1. (a) The map η : ^{∗∗} = h_{Γ^{op}} (h_{Γ} ( , Γ), Γ) → E given by η_{X} : X^{∗∗} → X is a natural transformation of functors, where E : M_{sq}^{Γ} → M^{Γ} is the natural embedding.

(b) The restriction of η to I^{Γ} is a natural isomorphism ^{∗∗} → 1_{I^{Γ}}.

(c) ^{∗} : I^{Γ} → I^{Γ^{op}} and ^{∗} : I^{Γ^{op}} → I^{Γ} are dualities.

Proof. (a) Given a morphism s : X_{1} → X_{2} of Γ-comodules X_{1}, X_{2} ∈ M_{sq}^{Γ}, we have to show that the diagram

X_{1}^{∗∗} ⟶^{η_{X_{1}}} X_{1}
s^{∗∗}↓  ^{s}↓
X_{2}^{∗∗} ⟶^{η_{X_{2}}} X_{2}

8 WILLIAM CHIN, MARK KLEINER, AND DECLAN QUINN

commutes. According to [T, 1.5] and [M, formula (3), p. 79], we have

ϕ_{X_{2}}(s^{∗}) = (s^{∗} ⊗ 1_{X_{2}^{∗}})ϕ_{X_{2}}(1_{X_{2}^{∗}}) = (1_{X_{1}^{∗}} ⊗ s)ϕ_{X_{1}}(1_{X_{1}^{∗}})

and

ψ_{X_{1}^{∗}}(s^{∗∗}) = (s^{∗} ⊗ 1_{X_{2}^{∗∗}})ψ_{X_{2}^{∗}}(1_{X_{2}^{∗∗}}),

so that

(s^{∗} ⊗ 1_{X_{2}^{∗}})ϕ_{X_{2}}(1_{X_{2}^{∗}}) = (1_{X_{1}^{∗}} ⊗ s)ψ_{X_{1}^{∗}}(η_{X_{1}}) = ψ_{X_{1}^{∗}}(s ∘ η_{X_{1}}),

as well as

ψ_{X_{1}^{∗}}(η_{X_{2}} ∘ s^{∗∗}) = (1_{X_{1}^{∗}} ⊗ η_{X_{2}})ψ_{X_{1}^{∗}}(s^{∗∗})
= (1_{X_{1}^{∗}} ⊗ η_{X_{2}})(s^{∗} ⊗ 1_{X_{2}^{∗∗}})ψ_{X_{2}^{∗}}(1_{X_{2}^{∗∗}})
= (s^{∗} ⊗ 1_{X_{2}})(1_{X_{2}^{∗}} ⊗ η_{X_{2}})ψ_{X_{2}^{∗}}(1_{X_{2}^{∗∗}})
= (s^{∗} ⊗ 1_{X_{2}})ψ_{X_{2}^{∗}}(η_{X_{2}})
= (s^{∗} ⊗ 1_{X_{2}})ϕ_{X_{2}}(1_{X_{2}^{∗}}).

Therefore ψ_{X_{1}^{∗}}(s ∘ η_{X_{1}}) = ψ_{X_{1}^{∗}}(η_{X_{2}} ∘ s^{∗∗}), and the diagram commutes because ψ is a bijection.

(b) If X ∈ I^{Γ} then X^{∗} ∈ I^{Γ^{op}} according to [T, Prop. 3.1 and 3.2, Th. 2.5(2)], so that the restriction of η to I^{Γ} makes sense. We view X as an e_{Γ} (X) − Γ-bicomodule, so that X^{∗} ∈ ^{Γ}M^{e_{Γ} (X)} and X^{∗∗} ∈ ^{e_{Γ} (X)}M^{Γ}. Similar to (3.1) and (3.2), we have the adjunction isomorphisms

\overline{ϕ}_{X} : Hom_{Γ−e_{Γ} (X)} (X^{∗}, X^{∗}) ≅ Hom_{Γ−Γ} (Γ, X^{∗}□_{e_{Γ} (X)}X)

and

\overline{ψ}_{X^{∗}} : Hom_{e_{Γ} (X)−Γ} (X^{∗∗}, X) ≅ Hom_{Γ−Γ} (Γ, X^{∗}□_{e_{Γ} (X)}X).

ALMOST SPLIT SEQUENCES 9

If κ : X^{∗}□_{e_{Γ} (X)}X → X^{∗} ⊗ X is the natural inclusion, [T, 1.8] implies

ϕ_{X}(1_{X^{∗}}) = κ\overline{ϕ}_{X}(1_{X^{∗}}).  (3.3)

According to [T, Remark 1.11], the diagram

Hom_{e_{Γ} (X)−Γ} (X^{∗∗}, X) ⟶^{\overline{ψ}_{X^{∗}}} Hom_{Γ−Γ} (Γ, X^{∗}□_{e_{Γ} (X)}X)
ζ↓ ↓^{Hom_{Γ−Γ} (Γ,κ)}
Hom_{Γ} (X^{∗∗}, X) ⟶^{ψ_{X^{∗}}} Hom_{Γ−Γ} (Γ, X^{∗} ⊗ X)

commutes, where ζ is the natural inclusion. By the commutativity and (3.3), we have \overline{ψ}_{X^{∗}}(η_{X}) = ϕ_{X}(1_{X∗}). By [T, Prop. 3.1 and 3.2, Th. 2.5(4)], η_{X} is an isomorphism because X ∈ I^{Γ}.

(c) This follows immediately from (a) and (b) and their duals.

Definition 1. If 0 → M → I_{0} →^{f} I_{1} is a minimal injective copresentation of M ∈ M_{qc}^{Γ}, we define Tr M as a Γ^{op}-comodule making the sequence 0 → Tr M → I_{1}^{∗} →^{f^{∗}} I_{0}^{∗} exact. By Proposition 3.1(c), Tr M ∈ M_{qc}^{Γ^{op}}; it is determined uniquely up to isomorphism.

We note that Proposition 3.1 allows us to transfer from finitely presented modules to quasi-finitely copresented comodules most of the considerations of [ARS, IV.1] concerning the transpose. For each M ∈ M_{q}^{Γ} there is a unique up to isomorphism decomposition M = M_{I} ⊕ M′ where M_{I} ∈ (M_{q}^{Γ})_{I} and M′ ∈ I^{Γ}. The following result is an analog of [ARS, IV Prop. 1.7].

Proposition 3.2. Let M ∈ M_{qc}^{Γ}.

(a) If M = ⊕_{α∈A}M_{α} then M_{α} ∈ M_{qc}^{Γ} and Tr M ≅ ⊕_{α∈A} Tr M_{α}.

10 WILLIAM CHIN, MARK KLEINER, AND DECLAN QUINN

(b) Tr M = 0 if and only if M is injective.

(c) Tr Tr M ≅ M_{I}.

(d) If M, N ∈ (M_{qc}^{Γ})_{I}, then Tr M ≅ Tr N if and only if M ≅ N.

(e) Tr : M_{qc}^{Γ} → M_{qc}^{Γ^{op}} induces a bijection between the isomorhism classes of indecomposable comodules in (M_{qc}^{Γ})_{I} and in (M_{qc}^{Γ^{op}})_{I}.

Proof. Using Proposition 3.1, it is not hard to modify the arguments of [ARS, p. 105, top] and produce a proof of this proposition. We prove (c) as an illustration and leave the rest to the reader.

(c) In view of (a) and (b), we show that if M ∈ (M_{qc}^{Γ})_{I} is indecomposable, then Tr Tr M ≅ M. By assumption, the morphism f in the minimal injective copresentation 0 → M → I_{0} →^{f} I_{1} of M is an indecomposable map that is not an isomorphism. By Proposition 3.1(c), the map f^{∗} in the exact sequence 0 → Tr M → I_{1}^{∗} →^{f^{∗}} I_{0}^{∗} is also indecomposable and not an isomorphism, whence Tr M is indecomposable and the latter sequence is a minimal injective copresentation of Tr M. Repeating the procedure, we obtain the exact sequence 0 → Tr Tr M → I_{0}^{∗∗} →^{f^{∗∗}} I_{1}^{∗∗} and, using parts (a) and (b) of Proposition 3.1, the exact commutative diagram

0 ⟶ Tr Tr M ⟶ I_{0}^{∗∗} ⟶^{f^{∗∗}} I_{1}^{∗∗}
η_{I_{0}} ↓ ↓^{η_{I_{1}}}
0 ⟶ M ⟶ I_{0} ⟶^{f} I_{1}

in which the vertical arrows are isomorphisms. Hence there exists a unique arrow Tr Tr M → M making the diagram commute, and the arrow must be an isomorphism.

ALMOST SPLIT SEQUENCES 11

4. Almost split sequences

Theorem 4.1. Let M ∈ M_{qc}^{Γ} and let 0 → U →^{f} V →^{g} W → 0 be an exact sequence in M^{Γ}.

(a) There exists an exact sequence

0 → U□_{Γ} Tr M ⟶^{f□_{Γ} Tr M} V □_{Γ} Tr M ⟶^{g□_{Γ} Tr M} W □_{Γ} Tr M ⟶

h_{Γ} (M, U) ⟶^{h_{Γ} (M,f)} h_{Γ} (M, V) ⟶^{h_{Γ} (M,g)} h_{Γ} (M, W) → 0

of k-spaces.

(b) The sequence

0 → U□_{Γ} Tr M ⟶^{f□_{Γ} Tr M} V □_{Γ} Tr M ⟶^{g□_{Γ} Tr M} W □_{Γ} Tr M → 0

is exact if and only if so is the sequence

0 → Hom_{Γ} (W, M) ⟶^{Hom_{Γ} (g,M)} Hom_{Γ} (V, M) ⟶^{Hom_{Γ} (f,M)} Hom_{Γ} (U, M) → 0.

(c) If dim_{k} Tr M < ∞, then every morphism U → M factors through f if and only if every morphism DTr M → W factors through g.

Proof. The proof is inspired by [ARS, IV.4].

(a) Applying the contravariant functor h_{Γ} ( , Y) for Y = U, V, W to a minimal injective copresentation 0 → M → I_{0} → I_{1} of M, we obtain the

12 WILLIAM CHIN, MARK KLEINER, AND DECLAN QUINN

commutative diagram

0  0
↓ ↓
h_{Γ} (I_{1}, U) ⟶ h_{Γ} (I_{0}, U) ⟶ h_{Γ} (M, U) ⟶ 0
↓ ↓ ↓
h_{Γ} (I_{1}, V) ⟶ h_{Γ} (I_{0}, V) ⟶ h_{Γ} (M, V) ⟶ 0
↓ ↓ ↓
h_{Γ} (I_{1}, W) ⟶ h_{Γ} (I_{0}, W) ⟶ h_{Γ} (M, W) ⟶ 0
↓  ↓  ↓
0  0  0

of k-spaces, where the rows are exact because the contravariant functor h_{Γ} is right exact, and the columns are exact because the covariant functor h_{Γ} is right exact and I_{0}, I_{1} ∈ I^{Γ} [T, 1.6 and 1.12]. By [T, Prop. 1.14(b)], we have a natural isomorphism h_{Γ} (I_{j}, U) ≅ U□_{Γ}I_{j}^{∗} for j = 0, 1 because U ≅ U□_{Γ}Γ, and similar isomorphisms for V and W. Since □_{Γ} is a left exact functor, the sequence 0 → U□_{Γ} Tr M → U□_{Γ}I_{1}^{∗} → U□_{Γ}I_{0}^{∗} is exact, and we extend the above diagram to the following exact commutative diagram of k-spaces.

0  0  0 ↓  ↓  ↓ 0 ⟶ U□_{Γ} Tr M ⟶ h_{Γ} (I_{1}, U) ⟶ h_{Γ} (I_{0}, U) ⟶ h_{Γ} (M, U) ⟶ 0 ↓  ↓  ↓  ↓ 0 ⟶ V □_{Γ} Tr M ⟶ h_{Γ} (I_{1}, V) ⟶ h_{Γ} (I_{0}, V) ⟶ h_{Γ} (M, V) ⟶ 0 ↓  ↓  ↓  ↓ 0 ⟶ W □_{Γ} Tr M ⟶ h_{Γ} (I_{1}, W) ⟶ h_{Γ} (I_{0}, W) ⟶ h_{Γ} (M, W) ⟶ 0 ↓  ↓  ↓ 0  0  0 .

Now the desired result follows from the Snake Lemma.

ALMOST SPLIT SEQUENCES 13

(b) By (a), the sequence

0 → U□_{Γ} Tr M ⟶^{f□_{Γ} Tr M} V □_{Γ} Tr M ⟶^{g□_{Γ} Tr M} W □_{Γ} Tr M → 0

is exact if and only if so is the sequence

0 → h_{Γ} (M, U) ⟶^{h_{Γ} (M,f)} h_{Γ} (M, V) ⟶^{h_{Γ} (M,g)} h_{Γ} (M, W) → 0.

According to [T, 1.12], there is a natural isomorphism D h_{Γ} (X, Y) ≅ Hom_{Γ} (Y, X) for X ∈ M_{q}^{Γ} and Y ∈ M^{Γ}. Thus the latter sequence is exact if and only if so is the sequence 0 → Hom_{Γ} (W, M) ⟶^{Hom_{Γ} (g,M)} Hom_{Γ} (V, M) ⟶^{Hom_{Γ} (f,M)} Hom_{Γ} (U, M) → 0, and (b) is proved.

(c) Since h_{k} (Tr M, k) ≅ DTr M and, according to [T, Prop. 1.10], there is an isomorphism Hom_{Γ} (h_{k} (Tr M, k), Y) ≅ Hom_{k} (k, Y □_{Γ} Tr M) natural in Y, we conclude that the first sequence in the statement of (b) is exact if and only if so is the sequence 0 → Hom_{Γ} (DTr M, U) ⟶^{f^{∗}} Hom_{Γ} (DTr M, V) ⟶^{g^{∗}} Hom_{Γ} (DTr M, W) → 0. The rest follows from (b).

Theorem 4.2. If M ∈ (M_{qc}^{Γ})_{I} is indecomposable and dim_{k} Tr M < ∞, there exists an exact sequence 0 → M →^{f} E →^{g} DTr M → 0 where g is a right almost split morphism in M^{Γ}.

Proof. The argument is a slight simplification of the proof of [ARS, V Th. 1.15(a), pp. 145-146].

Since M is not injective, there exists a non-split exact sequence

α : 0 → M →^{s} V →^{t} W → 0

14 WILLIAM CHIN, MARK KLEINER, AND DECLAN QUINN

in M^{Γ}. Therefore 1_{M} : M → M does not factor through s, and Theorem 4.1 implies that some morphism q : DTr M → W does not factor through t. Then the pullback sequence

αq : 0 → M →^{h} F →^{j} DTr M → 0

does not split, and the exact sequence

Hom_{Γ} (DTr M, F) ⟶^{Hom_{Γ} (DTr M,j)} Hom_{Γ} (DTr M, DTr M) → Coker Hom_{Γ} (DTr M, j) → 0

satisfies Coker Hom_{Γ} (DTr M, j) ≠ 0. Since Λ = (End_{Γ} DTr M)^{op} is a finite-dimensional k-algebra, there is a morphism l : DTr M → DTr M whose image in the Λ-module Coker Hom_{Γ} (DTr M, j) generates a simple Λ-submodule. We claim that the pullback sequence

β = αql : 0 → M →^{f} E →^{g} DTr M → 0

is the desired exact sequence.

First, β does not split because its congruence class in Coker Hom_{Γ} (DTr M, j) ⊂ Ext_{Γ}^{1} (DTr M, M) generates a non-zero submodule. It remains to show that if Y ∈ M^{Γ} and w : Y → DTr M is not a split epimorphism, then the pullback sequence βw : 0 → M → G → Y → 0 splits, which according to Theorem 4.1 is equivalent to proving that the pullback sequence βwv splits for all morphisms v : DTr M → Y. Since M is indecomposable not injective, so is the Γ^{op}-comodule Tr M by Proposition 3.2, whence the Γcomodule DTr M is indecomposable because Tr M is finite-dimensional over k. By Corollary 2.2(a), Λ is a local ring. Since w is not a split epimorphism, wv ∈ Λ is not invertible whence wv ∈ rad Λ. By Nakayama’s Lemma, the

ALMOST SPLIT SEQUENCES 15

Λ-submodule of Coker Hom_{Γ} (DTr M, j) generated by the congruence class of βwv is a proper submodule of the simple submodule generated by the congruence class of β. Hence the congruence class of βwv is zero, i. e., βwv is a split exact sequence.

Corollary 4.3. (a) If M ∈ (M_{qc}^{Γ})_{I} is indecomposable and dim_{k} Tr M < ∞, then there exists an almost split sequence 0 → M → E → DTr M → 0 in M^{Γ}.

(b) Let N ∈ M^{Γ} be indecomposable not projective with dim_{k} N < ∞. If D N ∈ M_{qc}^{Γ^{op}}, then there exists an almost split sequence 0 → TrD N → F → N → 0 in M^{Γ}.

Proof. (a) By Theorem 4.2, we have an exact sequence β : 0 → M →^{f} E →^{g} DTr M → 0 where g is a right almost split morphism in M^{Γ}. To show f is a left almost split morphism in M^{Γ}, one can use [A, Prop. 4.4, Ch. II]; we present the following easy argument of the proof of [KP, Lemma 5.2].

It suffices to prove that a morphism h : M → Y that does not factor through f is a split monomorphism. Since the pushout sequence hβ does not split, we obtain the exact commutative diagram

β : 0 ⟶ M ⟶^{f} E ⟶^{g} DTr M ⟶ 0
h↓ ↓ ∥
hβ : 0 ⟶ Y ⟶ Z ⟶ DTr M ⟶ 0
u↓ ↓ ∥
β : 0 ⟶ M ⟶^{f} E ⟶^{g} DTr M ⟶ 0

16 WILLIAM CHIN, MARK KLEINER, AND DECLAN QUINN

because g is right almost split. It follows that β̄ = (uh)β̄, where β̄ is the congruence class of β. Since M is indecomposable, End_{Γ} M is local by Corollary 2.2(a). Since β̄ ≠ 0, morphism uh must be an invertible element of End_{Γ} M by Nakayama’s Lemma. Thus h is a split monomorphism.

(b) Since N is indecomposable finite-dimensional not projective, D N is an indecomposable finite-dimensional not injective Γ^{op}-comodule by [L, Lemma 15(a)], so that Tr D N ∈ M_{qc}^{Γ} is indecomposable not injective by Proposition 3.2(e). Since DTr(Tr D N) ≅ N in view of Proposition 3.2(c) and the fact that D is a duality, (a) implies the existence of an almost split sequence 0 → Tr D N → F → N → 0 in M^{Γ}.

If Γ is a Hopf algebra over k, the existence of an almost split sequence with a finite-dimensional right-end term depends on the behavior of the trivial comodule, as explained by the next statement.

Corollary 4.4. Let Γ be a Hopf algebra over k, let W be the trivial Γ^{op}- comodule, and suppose that dim_{k} Soc(I(W)/W) < ∞. If N ∈ M^{Γ} is indecomposable not projective and dim_{k} N < ∞, then there exists an almost split sequence 0 → TrD N → F → N → 0 in M^{Γ} with TrD N ∈ M_{qc}^{Γ}.

Proof. By assumption, a minimal injective copresentation of W is of the form 0 → W → J_{0} → J_{1} where J_{0}, J_{1} ∈ I^{Γ^{op}}. Since dim_{k} D N < ∞, the exact sequence 0 → D N ≅ D N ⊗ W → D N ⊗ J_{0} → D N ⊗ J_{1} in M^{Γ^{op}} satisfies D N ⊗J_{0}, D N ⊗J_{1} ∈ I^{Γ^{op}} by arguments similar to those in the proof of the fundamental theorem of Hopf modules [Mo, pp. 15-16]. Therefore D N ∈ M_{qc}^{Γ^{op}}, and the statement follows from Corollary 4.3(b).

5. Right semiperfect coalgebras

Let Γ be an arbitrary k-coalgebra and let M ∈ (M_{qc}^{Γ})_{I} be indecomposable. If dim_{k} Tr M < ∞, Corollary 4.3(a) says that there exists an almost split sequence 0 → M → E → DTr M → 0 in M^{Γ}. However, no information is given about dim_{k} Tr DTr M, so that if DTr M is not injective, it is not clear whether there exists an almost split sequence with the left-end term DTr M. The experience of the well-developed theory of almost split sequences over artin algebras shows that interesting applications arise not so much from the consideration of a single almost split sequence, but from studying the collection of all almost split sequences, i.e., the Auslander-Reiten quiver. It is therefore desirable to describe coalgebras for which the above construction of an almost split sequence can be iterated and thus the Auslander-Reiten quiver constructed. A sufficient condition for this is that for all M ∈ M^{Γ}, dim_{k} M < ∞ imply dim_{k} Tr M < ∞. We describe a class of right semiperfect coalgebras for which the condition holds. Recall [L] that a coalgebra Γ is called right semiperfect if every indecomposable injective Γ^{op}-comodule is finite-dimensional.

Theorem 5.1. Let Γ be a right semiperfect coalgebra. If N ∈ M^{Γ} is indecomposable not projective and dim_{k} N < ∞, then there exists an almost split sequence 0 → TrD N → F → N → 0 in M^{Γ} with dim_{k} Soc TrD N < ∞ and dim_{k} Soc(I(TrD N)/ TrD N) < ∞.

Proof. By assumption, a minimal injective copresentation 0 → D N → J_{0} → J_{1} of the finite-dimensional Γ^{op}-comodule D N has the property that both J_{0} and J_{1} are direct sums of finitely many indecomposable injectives. Now the existence of the almost split sequence follows from Corollary 4.3(b). The rest is left to the reader.

Proposition 5.2. Let M ∈ M_{qc}^{Γ}.

(a) dim_{k} Soc Tr M < ∞ if and only if dim_{k} Soc(I(M)/M) < ∞.

(b) If Γ is a right semiperfect coalgebra, then dim_{k} Tr M < ∞ if and only if dim_{k} Soc(I(M)/M) < ∞.

Proof. (a) Let 0 → M → I_{0} → I_{1} be a minimal injective copresentation of M, then 0 → Tr M → I_{1}^{∗} → I_{0}^{∗} is a minimal injective copresentation of the Γ^{op}-comodule Tr M by Proposition 3.1(c). We note that dim_{k} Soc(I(M)/M) < ∞ if and only if I_{1} is a direct sum of finitely many indecomposable comodules, if and only if so is I_{1}^{∗}, using that I_{1} ≅ I_{1}^{∗∗}.

(b) Since every indecomposable injective Γ^{op}-comodule is finite-dimensional, dim_{k} Tr M < ∞ if and only if dim_{k} Soc Tr M < ∞. Now the statement follows from (a).

Corollary 5.3. The following are equivalent for a right semiperfect coalgebra Γ.

ALMOST SPLIT SEQUENCES 19

(a) For all finite-dimensional Γ-comodules M, dim_{k} Tr M < ∞.

(b) For all simple Γ-comodules S, dim_{k} Soc(I(S)/S) < ∞.

(c) For all finite-dimensional Γ-comodules M, dim_{k} Soc(I(M)/M) < ∞.

Proof. (a)⇒(b) and (c)⇒(a) Since every simple comodule is finite-dimensional, this follows from Proposition 5.2(b).

(b)⇒(c) An exact sequence 0 → M_{1} → M → M_{2} → 0 gives rise to the exact commutative diagram

0  0  0
↓  ↓  ↓
0 ⟶ M_{1} ⟶  M  ⟶ M_{2} ⟶ 0
↓  ↓  ↓
0 ⟶ I(M_{1}) ⟶ I(M_{1}) ⊕ I(M_{2}) ⟶ I(M_{2}) ⟶ 0
↓  ↓  ↓
0 ⟶ N_{1} ⟶  N  ⟶ N_{2} ⟶ 0
↓  ↓  ↓
0  0  0

in M^{Γ}. Proceeding by induction on the length of M (the base of induction is given by (b)), we choose M_{1}, M_{2} ≠ 0 and note that dim_{k} Soc N_{i} < ∞, i = 1, 2, by the induction hypothesis. Since I(M) is a direct summand of I(M_{1}) ⊕ I(M_{2}), and Soc N is a direct summand of Soc N_{1} ⊕ Soc N_{2}, we have dim_{k} Soc(I(M)/M) ≤ dim_{k} Soc N ≤ dim_{k} Soc N_{1} + dim_{k} Soc N_{2} < ∞.

Corollary 5.4. Let Γ be a right semiperfect coalgebra such that for all simple Γ-comodules S, dim_{k} Soc(I(S)/S) < ∞. If M ∈ M^{Γ} is indecomposable not injective and dim_{k} M < ∞, then there exists an almost split sequence 0 → M → E → DTr M → 0 in M^{Γ} and dim_{k} DTr M < ∞.

Proof. This follows from Corollaries 4.3(a), 5.3 and Proposition 5.2(b).

6. Path coalgebras

We recall the notion of path coalgebra of an arbitrary quiver [CM], and characterize the quivers whose path coalgebra Γ is either right semiperfect, or right semiperfect with the property that dim_{k} Soc(I(S)/S) < ∞ for all simple S ∈ M^{Γ}, so that Theorem 5.1 and Corollary 5.4 apply. We also relate Γ-comodules to representations of the quiver.

Let Q = (Q_{0}, Q_{1}) be a quiver (oriented graph with no finiteness assumption) with the set of vertices Q_{0} and the set of arrows Q_{1}. A path p : u → v on Q starting at u and ending at v, where u, v ∈ Q_{0}, is:

(1) a path of length 0 if p = u = v, i.e., it is just a vertex v ∈ Q_{0};

(2) a path of length t > 0 if it is a finite sequence a_{t}a_{t−1} … a_{1} of arrows a_{j} ∈ Q_{1} such that u is the starting vertex of a_{1}, v is the ending vertex of a_{t}, and the starting vertex of a_{j} is the ending vertex of a_{j−1} for 1 < j ≤ t.

In the sequel, |p| denotes the length of path p, and kQ denotes the k-space whose basis is the set of paths.

The composed path of q : w → x, p : u → v is the concatenated path qp : u → x if v = w (with vp = pu = p), and 0 ∈ kQ otherwise. This composition extended by k-linearity and distributivity turns kQ into an

ALMOST SPLIT SEQUENCES 21

associative k-algebra without identity (if Q_{0} is a finite set, then ∑_{v∈Q_{0}}v is the identity). The comultiplication Δ : kQ → kQ ⊗ kQ and counit ϵ : kQ → k are first defined on the paths and then extended to all of kQ. For all paths p, Δ(p) = ∑_{st=p}s ⊗ t where s and t are paths. If |p| = 0 then ϵ(p) = 1, if |p| > 0 then ϵ(p) = 0. For the rest of the paper, we denote by Γ the k-coalgebra with the underlying k-space kQ; Γ is called the path coalgebra of Q.

Denote by Rep Q the category of representations of Q over k [Ga]. An object (V, f) ∈ Rep Q consists of a collection of arbitrary (not-necessarily finite-dimensional) k-spaces V_{u}, u ∈ Q_{0}, and a collection of arbitrary linear transformations f_{a} : V_{x} → V_{y} for each arrow a : x → y. A morphism α : (V, f) → (W, g) consists of a collection of linear transformations α_{x} : V_{x} → W_{x}, x ∈ Q_{0}, satisfying g_{a}α_{x} = α_{y}f_{a} for each arrow a : x → y. Given a representation (V, f) of Q and a path p, we define f_{p} as follows. If |p| = 0, then p = u ∈ Q_{0} and we put f_{p} = 1_{V_{u}}. If |p| > 0 then p = a_{t}a_{t−1} … a_{1} with a_{j} ∈ Q_{1}, and we put f_{p} = f_{a_{t}}f_{a_{t−1}} … f_{a_{1}}. A representation (V, f) is locally nilpotent if for all u ∈ Q_{0} and all x ∈ V_{u}, there exist at most finitely many paths p satisfying f_{p}(x) ≠ 0; we denote by lnRep Q the full subcategory of Rep Q determined by the locally nilpotent representations.

The following result stated in [C] makes a connection between representations of Q and Γ-comodules.

Proposition 6.1. The categories lnRep Q and M^{Γ} are equivalent.

22 WILLIAM CHIN, MARK KLEINER, AND DECLAN QUINN

Proof. We indicate how to construct on objects functors F : lnRep Q → M^{Γ} and G : M^{Γ} → lnRep Q satisfying F G ≅ id and GF ≅ id; the rest is left to the reader.

Given a (V, f) ∈ lnRep Q, we construct M = F (V, f) ∈ M^{Γ} by putting M = ⊕_{u∈Q_{0}}V_{u} and defining comultiplication ρ : M → M ⊗ Γ by ρ(x) = ∑ f_{p}(x)⊗p, where x ∈ V_{u} and p runs through the set of paths on Q starting at u. Since f_{p}(x) ≠ 0 for at most finitely many paths p, the sum is finite and ρ(x) is well defined.

Let now M ∈ M^{Γ}. For each path p, denote by ϵ_{p} the element of the convolution algebra D Γ determined by ϵ_{p}(q) = δ_{pq} (Kronecker’s symbol) for all paths q; the set {ϵ_{u}|u ∈ Q_{0}} consists of pairwise orthogonal idempotents of D Γ. For u ∈ Q_{0}, the set M_{u} = {x ∈ M | (1⊗ϵ_{u})ρ(x) = x} = ϵ_{u} ⇀ M [Mo] is a k-subspace of M, and M = ⊕_{u∈Q_{0}}M_{u} as k-space. We construct (V, f) = GM by putting V_{u} = M_{u} and defining, for each arrow a : u → v, the map f_{a} : V_{u} → V_{v} as f_{a}(x) = x_{a}, where x_{a} is determined by the formula ρ(x) = ∑ x_{p} ⊗ p with the summation taken over all paths p starting at u. Since at most finitely many p’s satisfy x_{p} ≠ 0, we see that (V, f) ∈ lnRep Q.

The next statement is an easy consequence of well-known facts, see for example [Gr, Mo].

Proposition 6.2. Let Q be a quiver.

(a) Γ = ⊕_{u∈Q_{0}}E_{u} is a direct sum of right Γ-comodules, where E_{u} = Γ ↼ ϵ_{u} is the subspace spanned by all paths ending at u.

ALMOST SPLIT SEQUENCES 23

(b) The set {ku|u ∈ Q_{0}} is a complete set of pairwise non-isomorphic simple right Γ-comodules, where ku is the span of the path u of length 0. The comodule E_{u} is an injective envelope of ku.

We now characterize right semiperfect path coalgebras.

Corollary 6.3. Let Q be a quiver.

(a) The coalgebra Γ is right semiperfect if and only if for all u ∈ Q_{0}, there exist only finitely many paths starting at u.

(b) If S = ku, u ∈ Q_{0}, then dim_{k} Soc(I(S)/S) equals the cardinality of the set of arrows of Q ending at u; in particular, dim_{k} Soc(I(S)/S) < ∞ if and only if there exist at most finitely many arrows ending at u.

Proof. (a) In view of [L, Th.10], this an immediate consequence of the dual statement to Proposition 6.2(a) .

(b) It is clear from Proposition 6.2 that E_{u}/ku as k-space can be viewed as the span of all paths p of length > 0 ending at u, i.e., E_{u}/ku ≅ ⊕_{a}E_{s(a)} as right Γ-comodules, where a runs through the set of arrows ending at u, and s(a) is the starting vertex of a. It remains to apply Proposition 6.2(b).

According to Corollary 6.3, if at most finitely many paths start at each vertex, Theorem 5.1 applies to the path coalgebra Γ; if in addition at most finitely many arrows end at each vertex, Corollary 5.4 also applies.

24 WILLIAM CHIN, MARK KLEINER, AND DECLAN QUINN

7. Left locally bounded categories

We prove the existence of almost split sequences for finitely generated modules over a left locally bounded category. Recall that Λ is a k-category if the morphism sets Λ(x, y) are k-spaces and composition of morphisms is k-bilinear. A k-category Λ is locally finite-dimensional [BG, pp. 336-337] if it satisfies the following three conditions:

(a) For each x ∈ Λ, the endomorphism algebra Λ(x, x) is local, and Λ(x, x)/rad Λ(x, x) ≅ k.

(b) Distinct objects of Λ are not isomorphic.

(c) For all x, y ∈ Λ, dim_{k} Λ(x, y) < ∞. Λ is locally bounded if (c) is replaced by (c′) For all x, y ∈ Λ, ∑_{y∈Λ} dim_{k} Λ(x, y) < ∞ and ∑_{y∈Λ} dim_{k} Λ(y, x) <

∞.

We call Λ left locally bounded if (c) is replaced by (c′′) For all x, y ∈ Λ, ∑_{y∈Λ} dim_{k} Λ(x, y) < ∞. A locally bounded category is left locally bounded, but the converse is

false.

Given a quiver Q, denote by kQ^{+2} the ideal of the path category kQ such that for all x, y ∈ Q_{0}, kQ^{+2}(x, y) is the subspace of kQ(x, y) spanned by the paths of length ≥ 2. The proof of the next statement is a minor modification of the argument in [BG, p. 337, top], and is omitted.

ALMOST SPLIT SEQUENCES 25

Proposition 7.1. A k-category Λ is left locally bounded if and only if there exists a quiver Q and an ideal I ⊂ kQ^{+2} satisfying the following conditions:

(a) For each x ∈ Q_{0}, the number of arrows starting at x is finite.

(b) For each x ∈ Q_{0}, there is a natural number N_{x} such that I contains each path of length ≥ N_{x} starting at x.

(c) Λ ≅ kQ/I.

By definition, a Λ-module is a k-linear functor from Λ into the category of k-spaces. Such an object can also be viewed as a representation of the quiver Q with relations, i.e., as a representation (V, f) such that for all x, y ∈ Q_{0} and all linear combinations of paths a_{1}p_{1} + a_{2}p_{2} + ⋯ + a_{t}p_{t} ∈ I(x, y), where p_{j} is a path and a_{j} ∈ k, we have a_{1}f(p_{1}) + a_{2}f(p_{2}) + ⋯ + a_{t}f(p_{t}) = 0. We identify the category of Λ-modules with the category of representations of Q with the relations. A Λ-module (V, f) is finitedimensional if ∑_{x∈Q_{0}} dim_{k} V (x) < ∞.

Proposition 7.2. Let Λ = kQ/I be a left locally bounded category, where Q is a quiver and I ⊂ kQ^{+2} an ideal. Denote by Γ the path coalgebra.

(a) The set I^{⊥} = {γ ∈ Γ|⟨I, γ⟩ = 0} is a subcoalgebra of Γ.

(b) The category of (finite-dimensional) Λ-modules is equivalent to the category of (finite-dimensional) right I^{⊥}-comodules.

(c) I^{⊥} is a right semiperfect coalgebra.

(d) Every simple right I^{⊥}-comodule S satisfies dim_{k} Soc(I(S)/S) < ∞ if and only if for each x ∈ Q_{0} the number of arrows ending at x is finite.

26 WILLIAM CHIN, MARK KLEINER, AND DECLAN QUINN

Proof. (a) We identify a path p on Q with the element ϵ_{p} ∈ D Γ satisfying ⟨ϵ_{p}, q⟩ = 0 for all paths q. In this way the path category kQ (which can also be viewed as a k-algebra without identity) embeds into the convolution algebra D Γ. Although kQ ≠ D Γ in general, the proof of [Mo, Lemma 5.1.4(2)] works because kQ contains the dual basis to the path basis for Γ.

(b) For each right I^{⊥}-comodule M, the coefficient space cf(M) is a subcoalgebra of I^{⊥} so that the structure map is ρ : M → M ⊗ I^{⊥}. Viewing M as a module over D Γ and, hence, over its subalgebra kQ, we have f ⇀ m = (1 ⊗ f)ρ(m) = 0 for all f ∈ I and m ∈ M, so M is a Λ-module. On the other hand, any Λ-module is a locally nilpotent representation (V, f) of Q (because Λ is left locally bounded) on which I acts as zero. By Proposition 6.1, (V, f) gives rise to a right Γ-comodule M on which I acts as zero. Therefore, the coefficient space cf(M) is contained in I^{⊥}, and M ∈ M^{I^{⊥}}.

(c) By Yoneda’s Lemma or [BG, 2.2, p. 337], the category of Λ-modules has enough projectives (direct sums of representable functors). By (b), so does the equivalent category M^{I^{⊥}}, whence I^{⊥} is right semiperfect by [L, Th. 10]. One can also use Proposition 7.1(b) to prove that indecomposable injective (I^{⊥})^{op}-comodules are finite-dimensional.

(d) Since M^{I^{⊥}} ⊂ M^{Γ}, a simple right I^{⊥}-subcomodule S of I^{⊥} is a simple right Γ-subcomodule of Γ, i.e., S = ku for some u ∈ Q_{0} by Proposition 6.2(b). An indecomposable direct summand of I^{⊥} in M^{I^{⊥}} containing S, which we denote by F_{u}, is an injective envelope of S, whence S = Soc F_{u}. It follows that F_{u} is a Γ-subcomodule of E_{u}, where the latter is described

ALMOST SPLIT SEQUENCES 27

in Proposition 6.2. Therefore F_{u}/S ⊂ E_{u}/S and dim_{k} Soc_{I^{⊥}} (F_{u}/S) ≤ dim_{k} Soc_{Γ} (E_{u}/S). Since I ⊂ kQ^{+2}, I^{⊥} contains every path of length ≤ 1, so that dim_{k} Soc_{I^{⊥}} (F_{u}/S) is no less than the cardinality of the set of arrows of Q ending at u. By Proposition 6.3(b), we have dim_{k} Soc_{I^{⊥}} (F_{u}/S) = dim_{k} Soc_{Γ} (E_{u}/S), and the latter is finite if and only if at most finitely many arrows of Q end at u.

Corollary 7.3. Let Λ = kQ/I be a left locally bounded category, where Q is a quiver and I ⊂ kQ^{+2} an ideal.

(a) For each indecomposable finite-dimensional not projective Λ-module N, there exists an almost split sequence 0 → L → M → N → 0 in the category of Λ-modules.

(b) Suppose that for all x ∈ Q_{0}, the number of arrows of Q ending at x is finite. For each indecomposable finite-dimensional not injective Λmodule L, there exists an almost split sequence 0 → L → M → N → 0 in the category of Λ-modules with M and N finite-dimensional.

Proof. The statement follows from Corollary 7.2, Theorem 5.1, and Corollary 5.4.

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[A] M. Auslander, Functors and morphisms determined by objects, Representation Theory of Algebras, Proceedings of the Philadelphia Conference, Lecture Notes in Pure and Applied Mathematics, vol. 37, Dekker, New York, 1978, pp. 1-244.

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Department of Mathematics, DePaul University, Chicago, Illinois 60614

E-mail address: wchin@condor.depaul.edu

Department of Mathematics, Syracuse University, Syracuse, New York 13244-1150

E-mail address: mkleiner@syr.edu

Department of Mathematics, Syracuse University, Syracuse, New York 13244-1150

E-mail address: dpquinn@syr.edu