GENERALIZED BENSON-CARLSON DUALITY

LEONARD EVENS AND STEPHEN F. SIEGEL

October 3, 1994

1. Introduction

1.1. Background. This paper deals with the landmark results of Benson and Carlson’s Projective resolutions and Poincaré duality complexes [6]. Our goal is to provide some necessary background for that work, and to prove some of the results of [6] in a more general setting. In particular, we analyze what happens when we replace Benson and Carlson’s complex C_{ζ} (in the notation of [6]) with an arbitrary Yoneda extension representing ζ.

Let Λ be a finite-dimensional cocommutative Hopf algebra over an algebraically closed field k. Because Λ is cocommutative, H^{∗}(Λ, k) is a graded-commutative kalgebra (i.e., xy = (−1)^{deg(x) deg(y)}yx for homogeneous x, y). Throughout this paper we will also assume that Λ has the following finiteness property in cohomology: Ext_{Λ}^{∗}(k, k) is a finitely generated k-algebra, and for any Λ-modules M and N, Ext_{Λ}^{∗}(M, N) is finitely generated as an Ext_{Λ}^{∗}(k, k)-module. (By Λ-module we always mean finitely generated left Λ-module.) The second condition is equivalent to requiring H^{∗}(Λ, M) to be finitely generated over H^{∗}(Λ, k) for any Λ-module M, because of the isomorphism Ext_{Λ}(M, N) ≅ H^{∗}(Λ, Hom_{k}(M, N)).

If Λ is the group algebra of a finite group then Λ certainly has the finiteness property in cohomology, by a theorem of Evens (cf. [7, Theorem 7.4.1]). The finiteness property also holds if Λ is a finite-dimensional cocommutative connected Hopf algebra, e.g. the restricted enveloping algebra of a p-restricted Lie algebra (Bajer and Sadofsky [1, Lemma 6.2], Wilkerson [10]). In fact, at present we know of no example of a finite-dimensional Hopf algebra without this property. It is known that any finitedimensional Hopf algebra is a Frobenius algebra (Larson and Sweedler [9]), which implies that a Λ-module is projective if and only if it is injective, and this property is fundamental for the proofs given here.

1.2. Preliminaries. We use the following notation for chain complexes of modules over a k-algebra R. If C is a complex of left or right R-modules, let Z_{n}C = Ker(∂ : C_{n} → C_{n−1}), B_{n}C = Im(∂ : C_{n−1} → C_{n}), and H_{n}C = Z_{n}C/B_{n}C. If C, D are complexes of right, left R-modules respectively, let C ⊗_{R} D denote the complex with (C ⊗_{R} D)_{n} = ⨁_{p+q=n} C_{p} ⊗_{R} C_{q}, with differential given by ∂(x ⊗ y) = ∂(x) ⊗ y + (−1)^{p}x ⊗ ∂(y) for x ∈ C_{p}, y ∈ C_{q}. If C and D are both complexes of left (resp. right) R-modules, then let Hom_{R}(C, D) denote the complex with Hom_{R}(C, D)_{n} = ∏_{q=p+n} Hom_{R}(C_{p}, D_{q}), with differential given by ∂(f) = ∂ ∘ f − (−1)^{n}f ∘ ∂, for f ∈ Hom_{R}(C, D)_{n}. If f − g ∈ B_{n}Hom_{R}(C, D) we write f ≃ g (f and g are chain homotopic). For any integer r, C[r] denotes the complex with C[r]_{p} = C_{p−r}, and with differential (−1)^{r}∂. The dual complex C^{∗} is defined to be Hom_{k}(C, k), where k is considered to be a complex concentrated in degree 0; if f ∈ Hom_{R}(C, D)_{r} then f^{∗} ∈ Hom_{R}(D^{∗}, C^{∗})_{r} is defined by f^{∗}(α) = (−1)^{rs}α ∘ f for α ∈ C_{s}^{∗}. We write C|D if C is isomorphic to a summand of D (as complexes). Finally, any chain complex may be considered a cochain complex, and vice-versa, by setting C^{n} = C_{−n}.

Let C denote the category of complexes of Λ-modules. Let C_{b}, _{b}C, C^{p}, ^{e}C denote, respectively, the full subcategories of C of complexes bounded below, complexes bounded above, complexes where each C_{n} is Λ-projective, and exact complexes. These adornments on the symbol C may be combined, so that, for example, the objects of _{b}^{e}C_{b}^{p} are bounded (above and below) exact complexes of projectives. The following are fundamental; for proofs, see Benson [4, Lemma 1.4.4, Theorem 1.4.3], replacing “module” with “bounded complex” throughout.

Lemma 1 (Fitting). Let M ∈ _{b}C_{b} and f ∈ End_{C}(M). Then M = Im(f^{n})⊕Ker(f^{n}) for n sufficiently large.

Theorem 2 (Krull-Schmidt). Let M ∈ _{b}C_{b} and let M = ⨁_{i=1}^{m} M^{(i)}, where each M^{(i)} is an indecomposable subcomplex of M. If M = ⨁_{i=1}^{n} M̃^{(i)} for some other indecomposable subcomplexes M̃^{(i)} of M, then n = m, and after renumbering if necessary, M̃^{(i)} ≅ M^{(i)} for each i.

2. Projective resolutions of complexes

2.1. Definition and basic properties. Let C ∈ C_{b}. A projective resolution of C is a complex P ∈ C_{b}^{p} together with a quasi-isomorphism ϵ : P → C.

Proposition 3. If C ∈ C_{b} then C has a projective resolution.

Proof. We construct P_{n} and ϵ_{n} by induction on n. These may be taken to be 0 for n sufficiently small since C is bounded below. So assume that P_{m} and ϵ_{m} have been constructed and satisfy ∂ϵ_{m} = ϵ_{m−1}∂ for m ≤ n. We refer to Diagram 1, leaving off subscripts of maps whenever this is unambiguous. Let P B_{n+1} be the pullback of (δ, θ) in the diagram, and let α_{n+1} be a surjection from a projective module P_{n+1}

[figure: \begin{aligned}P_{n+1} & & & P_{n} & & & P_{n−1} \\ & P B_{n+1} & Z_{n}P & & P B_{n} & Z_{n−1}P \\ & \frac{C_{n+1}}{B_{n+1}C} & Z_{n}C & & \frac{C_{n}}{B_{n}C} & Z_{n−1}C \\ C_{n+1} & & & C_{n} & & & C_{n−1} \end{aligned}; ϵ_{n+1}; α_{n+1}; ∂; ϵ_{n}; α_{n}; ∂; ϵ_{n−1}; β_{n+1}; γ_{n+1}; ι_{n}; θ_{n}; β_{n}; γ_{n}; ι_{n−1}; θ_{n−1}; δ_{n+1}; κ_{n}; δ_{n}; κ_{n−1}; η_{n+1}; ∂; η_{n}; ∂]

Diagram 1. Construction of a projective resolution.

to P B_{n+1}. By projectivity there is a map ϵ_{n+1} : P_{n+1} → C_{n+1} making the diagram commute. Define ∂ : P_{n+1} → P_{n} to be iγα; the diagram shows that ∂ϵ_{n+1} = ϵ_{n}∂.

To show that (ϵ_{∗})_{n} is surjective, let x ∈ Z_{n}C. Then η(x) ∈ Ker(δ), so it follows from the universal property of the pullback that there exists y ∈ P B_{n} such that β(y) = η(x) and γ(y) = 0. Choose z ∈ P_{n} such that α(z) = y. Then ∂(z) = ιγα(z) = 0, so z ∈ Z_{n}P, and ηϵ_{n}(z) = βα(z) = η(x), so ϵ_{n}(z) − x ∈ B_{n}C.

To show that (ϵ_{∗})_{n} is injective, let x ∈ Z_{n}P and suppose ϵ_{n}(x) is a boundary, say ϵ_{n}(x) = ∂(y), y ∈ C_{n+1}. Then θ(x) = δη(y), so there is a z ∈ P B_{n+1} such that β(z) = η(y) and γ(z) = x. Choose w ∈ P_{n+1} such that α(w) = z; it follows that ∂(w) = ιγα(w) = x, i.e., x is a boundary.

Remark 1. Any projective resolution of C must fit into a diagram like Diagram 1, for the universal property of pullbacks implies that there is a map α making the diagram commute at each stage. We need only check that such an α must be surjective. To see this, given any x ∈ P B_{n}, choose y ∈ C_{n} such that η(y) = β(x). Then by commutativity, ∂(y) = ϵιγ(x). Since ϵ_{∗} is an isomorphism, this implies ιγ(x) is a boundary, say ιγ(x) = ∂(w), w ∈ P_{n}. Now γα(w) = γ(x) and

∂ϵ(w) = ϵιγα(w) = ϵιγ(x) = ∂(y).

So ϵ(w) − y ∈ Z_{n}C, and since ϵ_{∗} is an isomorphism, there is a w′ ∈ Z_{n}P such that ηϵ(w′) = η(ϵ(w) − y) = ηϵ(w) − β(x). Now βα(w − w′) = ηϵ(w − w′) = β(x) and γα(w − w′) = γα(w) = γ(x), so α(w − w′) = x.

Remark 2. Observe that ∂(P) = Ker(α). For ιγα∂ = ∂^{2} = 0, hence γα∂ = 0. Moreover βα∂ = ηϵ∂ = η∂ϵ = 0. So by the universal property of pullbacks, α∂ = 0, i.e., ∂(P) ⊆ Ker(α). Conversely, if α(y) = 0 for y ∈ P, then y is a cycle and

[figure: \begin{aligned}& P_{n+1} & & P_{n} & & P_{n−1} & & P_{n−2} \\ & C_{n+1} & & C_{n} & & C_{n−1} & & C_{n−2} \\ X_{n+1} & & X_{n} & & X_{n−1} & & X_{n−2} \end{aligned}; g_{n+1}; g_{n}; s_{n}; f_{n}; g_{n−1}; s_{n−1}; f_{n−1}; g_{n−2}; s_{n−2}; f_{n−2}; ϵ_{n+1}; ϵ_{n}; ϵ_{n−1}; ϵ_{n−2}]

Diagram 2. Proof of the existence part of the Comparison Theorem

ηϵ(y) = 0, so ϵ(y) is a boundary, and since ϵ_{∗} is an isomorphism, y is a boundary; hence Ker(α) ⊆ ∂(P).

Theorem 4 (Comparison Theorem). Suppose C, X ∈ C_{b}, P ∈ C_{b}^{p}, and ϵ : X → C is a quasi-isomorphism. Then the map

H_{∗}(Hom_{Λ}(P, X)) ⟶ H_{∗}(Hom_{Λ}(P, C))
[f] ⟼ [ϵf]

is an isomorphism.

Remark 3. For ∗ = 0, the conclusion of the theorem may be expressed in the following form: given any Λ-chain map g : P → C, there exists a Λ-chain map f : P → X such that ϵf ≃ g; moreover if f′ : P → X is another such map, then f ≃ f′. In particular, if X →^{ϵ} C and P →^{ϵ̃} C are both projective resolutions, then there is a chain map f : P → X, unique up to chain homotopy, such that ϵf ≃ ϵ̃. Notice that the “comparison” f must also be a quasi-isomorphism since ϵ_{∗}f_{∗} = ϵ̃_{∗}, and both ϵ_{∗} and ϵ̃_{∗} are isomorphisms.

Proof. Suppose g ∈ Z_{r}Hom_{Λ}(P, C). By shifting the indices if necessary we may assume r = 0. We construct, by induction, maps f_{i} : P_{i} → X_{i} and s_{i} : P_{i} → X_{i+1} satisfying ∂f_{i} = f_{i−1}∂ and ϵf_{i} −g_{i} = ∂s_{i} +s_{i−1}∂, for all i ≤ n. For n sufficiently small these maps may all be taken to be 0, so we may assume that we have constructed them for i < n and complete the inductive step.

First note that f_{n−1}∂(P_{n}) ⊆ B_{n−1}X. For given x ∈ P_{n}, ∂f_{n−1}∂(x) = f_{n−2}∂^{2}(x) = 0, so f_{n−1}∂(x) ∈ Z_{n−1}X, and

ϵf_{n−1}∂(x) = g_{n−1}∂(x) + ∂s_{n−1}∂(x) = ∂g_{n}(x) + ∂s_{n−1}∂(x)

is a boundary, so since ϵ_{∗} is an isomorphism, f_{n−1}∂(x) is also a boundary.

Hence by the projectivity of P_{n}, there is a map f′_{n} : P_{n} → X_{n} such that ∂f′_{n} = f_{n−1}∂. Now

∂ ∘ (ϵf′_{n} − g_{n}) = (ϵf_{n−1} − g_{n−1})∂ = ∂s_{n−1}∂,

so ϵf′_{n} − g_{n} − s_{n−1}∂ has image contained in Z_{n}C. By considering the diagram

[figure: \begin{aligned}& & P_{n} \\ & & Z_{n}C \\ Z_{n}X & Z_{n}C & H_{n}C \end{aligned}; ϵf′_{n}−g_{n}−s_{n−1}∂; ϵ|_{ZnX}]

we see there is a map f′′_{n} : P_{n} → X_{n} with Im(f′′_{n}) ⊆ Z_{n}X and

Im(ϵf′_{n} + ϵf′′_{n} − g_{n} − s_{n−1}∂) ⊆ B_{n}C. (1)

Let f_{n} = f′_{n} + f′′_{n}. Then ∂f_{n} = f_{n−1}∂, as ∂f′′ = 0. Moreover, (1) guarantees the existence of a map s_{n} making

[figure: \begin{aligned}& P_{n} \\ C_{n+1} & C_{n} \end{aligned}; s_{n}; ϵf_{n}−g_{n}−s_{n−1}∂; ∂]

commute, completing the inductive step.

We must next show that if f : P → X is a chain map and ϵf ≃ 0 then f ≃ 0. So suppose ϵf = ∂s + s∂, where s ∈ Hom_{Λ}(P, C)_{1}. We show by induction on n that there are maps t_{i} : P_{i} → X_{i+1} and w_{i} : P_{i} → C_{i+2} such that

(2)  f_{i} = ∂t_{i} + t_{i−1}∂
(3)  ϵt_{i} − s_{i} = ∂w_{i} + w_{i−1}∂.

Assume we have such maps for i < n. Then

∂ ∘ (f_{n} − t_{n−1}∂) = (f_{n−1} − ∂t_{n−1})∂ = (∂t_{n−1} + t_{n−2}∂ − ∂t_{n−1})∂ = 0,

so Im(f_{n} − t_{n−1}∂) ⊆ Z_{n}X. Moreover,

ϵ ∘ (f_{n} − t_{n−1}∂) = ∂s_{n} + s_{n−1}∂ − s_{n−1}∂ + ∂w_{n−1}∂ = ∂ ∘ (s_{n} + w_{n−1}∂),

hence Im(f_{n} − t_{n−1}∂) ⊆ B_{n}X, as ϵ_{∗} is an isomorphism. So by the projectivity of P_{n}, there is a map t_{n} : P_{n} → X_{n+1} such that (2) holds with i = n.

[figure: \begin{aligned}& P_{n+2} & & P_{n+1} & & P_{n} & & P_{n−1} \\ & C_{n+2} & & C_{n+1} & & C_{n} & & C_{n−1} \\ X_{n+2} & & X_{n+1} & & X_{n} & & X_{n−1} \end{aligned}; f_{n+2}; f_{n+1}; s_{n+1}; f_{n}; s_{n}; w_{n}; t_{n}; f_{n−1}; s_{n−1}; w_{n−1}; t_{n−1}; ϵ; ϵ; ϵ; ϵ]

Diagram 3. Proof of the uniqueness part of the Comparison Theorem

We must next “adjust” t_{n} to show there is a map w_{n} satisfying (3) with i = n. Observe

∂ ∘ (ϵt_{n} − s_{n} − w_{n−1}∂) = ϵ ∘ (f_{n} − t_{n−1}∂) − ∂s_{n} − ∂w_{n−1}∂
= ∂s_{n} + s_{n−1}∂ − s_{n−1}∂ + ∂w_{n−1}∂ + w_{n−2}∂^{2} − ∂s_{n} − ∂w_{n−1}∂
= 0,

i.e., Im(ϵt_{n} − s_{n} − w_{n−1}∂) ⊆ Z_{n+1}C. So arguing as before, there is a map t′_{n} : P_{n} → Z_{n+1}X such that Im(ϵt′_{n} + ϵt_{n} − s_{n} − w_{n−1}∂) ⊆ B_{n+1}C. Replace t_{n} with t_{n} + t′_{n} (this does not affect (2) as ∂t′_{n} = 0). We then have, by the projectivity of P_{n}, a map w_{n} : P_{n} → C_{n+2} such that ∂w_{n} = ϵt_{n} −s_{n} −w_{n−1}∂, completing the inductive step.

2.2. Minimal resolutions. A projective resolution P →^{ϵ} C is said to be minimal if for any projective resolution X →^{ϵ′} C of C, P |X.

Theorem 5. Let C ∈ C_{b}. Then (i) C has a minimal projective resolution. (ii) Let X → C be an arbitrary projective resolution of C. Then the following are

equivalent: (a) X → C is minimal, (b) ∂(X) ⊆ rad(X), and (c) X has no nontrivial exact summands.

Because of Theorem 5(i), for any C ∈ C_{b} we may let MPR(C) denote a minimal projective resolution of C. Note that MPR(C) is unique up to isomorphism of complexes, for if P and P′ are both minimal, we have P |P′|P, and since each of these is finite-dimensional in each degree, P ≅ P′. The proof of the theorem requires two lemmas: the first is interesting in its own right, the second is a rather technical fact from homological algebra.

Lemma 6. Let C ∈ C_{b}. Suppose ∂(C) ⊆ rad(C) and f : C → C is a quasiisomorphism. Then f is an isomorphism.

Proof. Show by induction on n that f_{n} is an isomorphism. Since C_{n} = 0 for n sufficiently small, the initial step is trivial. So assume f_{n−1} is an isomorphism. By Fitting’s Lemma there is an m > 0 such that if we set g = f^{m}, A = Ker(g), and B = Im(g), then C_{i} = A_{i} ⊕ B_{i} for i ≤ n + 1. Clearly g_{n} is an isomorphism iff f_{n} is. Now g induces an automorphism of H_{n}C = H_{n}A ⊕ H_{n}B, but g is trivial on A, hence H_{n}A = 0, i.e., A_{n+1} → A_{n} → A_{n−1} is exact. But by the inductive hypothesis, A_{n−1} = 0. Therefore A_{n} ⊆ ∂(C_{n+1}) ⊆ rad(C_{n}). On the other hand, A_{n}|C_{n}. Hence A_{n} = 0.

Lemma 7. Let U, V, X, and Y be Λ-modules, and f : X → Y and g : U ⊕ V → Y maps of Λ-modules. Let i_{U} : U → U ⊕ V and i_{V} : V → U ⊕ V denote the inclusions. Suppose V is projective and gi_{V} (V) ⊆ f(X). Let τ : V → X be a map satisfying fτ = gi_{V}, and let

[figure: \begin{aligned}M & U \\ & U ⊕ V \\ X & Y \end{aligned}; ϕ; σ; i_{U}; g; f]

be a pullback of (f, gi_{U}). Then

[figure: \begin{matrix}M ⊕ V & U ⊕ V \\ X & Y \end{matrix}; ϕ⊕id_{V}; σ+τ; g; f]

is a pullback of (f, g).

Note that such a map τ exists by the definition of projective.

Proof. Suppose W is a Λ-module and δ : W → X and ϵ : W → U ⊕ V are maps satisfying fδ = gϵ. Then there is a commutative diagram

[figure: \begin{aligned}W & & U ⊕ V \\ & M & U \\ & X & Y \end{aligned}; δ−τπ_{V} ϵ; γ_{M}; ϵ; π_{U}; σ; ϕ; gi_{U}; f]

as

f ∘ (δ − τπ_{V} ϵ) = gϵ − fτπ_{V} ϵ = gϵ − gi_{V} π_{V} ϵ
= g ∘ (id_{V} − i_{V} π_{V})ϵ = gi_{U}π_{U}ϵ.

So by the definition of pullback there is a unique map γ_{M} making the diagram commute. Let γ_{V} = π_{V} ϵ, and define γ : W → M ⊕ V by γ(w) = (γ_{M}(w), γ_{V} (w)). We then have

(σ + τ)γ = σγ_{M} + τγ_{V} = δ − τπ_{V} ϵ + τπ_{V} ϵ = δ

and

(ϕ ⊕ id_{V})γ(w) = (ϕγ_{M}(w), γ_{V} (w)) = (π_{U}ϵ(w), π_{V} ϵ(w)) = ϵ(w),

as required.

Suppose γ′ : W → M ⊕ V also satisfies (σ + τ)γ′ = δ and (ϕ + id_{V})γ′ = ϵ. Write γ′(w) = (γ′_{M}(w), γ′_{V} (w)). From

(ϕ + id_{V})γ = ϵ = (ϕ + id_{V})γ′

we get ϕγ_{M} = ϕγ′_{M} and γ′_{V} = γ_{V}. Similarly, from

(σ + τ)γ = δ = (σ + τ)γ′

we get σγ_{M} = σγ′_{M}. So by the uniqueness of γ_{M}, γ′_{M} = γ_{M}, and therefore γ = γ′.

Proof of Theorem 5. Let X →^{ϵ} C be a projective resolution of C, and fix a diagram like Diagram 1 for X. We first show there is a decomposition X = P ⊕ W, with W exact and ∂(P) ⊆ rad(P).

To show this we produce, by induction on n, a decomposition X_{n} = P_{n} ⊕ W_{n} such that ∂(W_{n}) = Z_{n−1}W, ϵ(Z_{n}W) ⊆ B_{n}C, α|_{P_{n}} is a projective cover of α(P_{n}), and ∂(P_{n}) ⊆ rad(P_{n−1}). Since X_{n} = 0 for n sufficiently small, the initial step is trivial. So assume decompositions with these properties have been constructed through degree n − 1.

Now W_{n−1} → ⋯ → W_{0} → 0 is an exact sequence of projectives, so Z_{n−1}W is projective. Moreover, Z_{n−1}X = Z_{n−1}P ⊕ Z_{n−1}W, and θ(Z_{n−1}W) ⊆ δ(C_{n}/B_{n}C) (as ϵ_{n−1}(Z_{n−1}W) ⊆ B_{n−1}C). Hence by Lemma 7 and the uniqueness of pullbacks, there is a decomposition P B_{n} = P B′_{n} ⊕ V such that γ = γ′ ⊕ γ′′ and β = β′ + μ, where γ′′ : V → Z_{n−1}W is an isomorphism and

[figure: \begin{matrix}P B′_{n} & Z_{n−1}P \\ \frac{C_{n}}{B_{n}C} & Z_{n−1}C \end{matrix}; β′; γ′; θ′:=θ|_{Z_{n−1}P}; δ]

is a pullback.

Since α : X_{n} → P B′_{n} ⊕V is surjective, there is a decomposition X_{n} = P_{n} ⊕W_{n} such that α′ := α|_{P_{n}} is a projective cover of P B′_{n} and α(W_{n}) = V. Hence ∂(W_{n}) = Z_{n−1}W. To see that ∂(P_{n}) ⊆ rad(P_{n−1}), recall from the remarks following Proposition 3 that Ker(α_{n−1}) = ∂(X_{n}), so in particular the composite

[figure: P_{n}  P B′_{n}  Z_{n−1}P  P_{n−1}  α(P_{n−1}); α′_{n}; γ′; ι′; α′_{n−1}]

is trivial, so it certainly induces the trivial map modulo radicals. But α′_{n−1} is a projective cover, and therefore induces an isomorphism modulo radicals, whence we conclude ∂(P_{n}) = ι′γ′α′(P_{n}) ⊆ rad(P_{n−1}).

To complete the inductive step we must show ϵ(Z_{n}W) ⊆ B_{n}C. But ιγ′′α(Z_{n}W) = ∂(Z_{n}W) = 0, so since ι and γ′′ are monomorphisms, α(Z_{n}W) = 0. Hence ηϵ(Z_{n}W) = βα(Z_{n}W) = 0, as required.

We now claim that P →^{ϵ|_{P}} C is a minimal projective resolution. It is certainly a projective resolution, since H_{∗}W = 0. Suppose X̃ →^{ϵ̃} C is any projective resolution of C. Then we have just shown that X̃ = P̃ ⊕ W̃, where ∂(P̃) ⊆ rad(P̃) and P̃ →^{ϵ̃|P̃} C is a projective resolution. By the Comparison Theorem, there are quasi-isomorphisms f : P → P̃ and g : P̃ → P. So fg and gf are quasi-isomorphisms, and by Lemma 6, fg and gf are isomorphisms. Hence f and g are isomorphisms, i.e., P ≅ P̃. So P |X̃, establishing the claim, and completing the proof of (i).

The proof of (ii) is now immediate. For if X → C is minimal then by uniqueness X ≅ P, and P has property (b). Hence (a) ⇒ (b). If (b) holds and X = C ⊕ D, with D exact, then X ↠ C ↪ X is a quasi-isomorphism, so by Lemma 6 it is an isomorphism, and therefore D = 0. Hence (b) ⇒ (c). Finally, the statement (c) ⇒ (a) follows directly from the definition of minimal.

If D ∈ _{b}C, an injective resolution of D is a complex I ∈ _{b}C^{p} together with a quasiisomorphism η : D → I (recall that a Λ-module is projective iff it is injective). Since η : D → I is an injective resolution iff η^{∗} : I^{∗} → D^{∗} is a projective resolution, all of the statements above concerning projective resolutions have dual versions concerning injective resolutions.

For C ∈ C_{b}, D ∈ _{b}C, we now define Ext_{Λ}^{∗}(C, D) = H^{∗}Hom_{Λ}(P, D), where P is any projective resolution of C. By Theorem 4, this is independent of the projective resolution chosen, in the usual sense. We also have the usual canonical isomorphisms Ext_{Λ}^{∗}(C, D) ≅ H^{∗}Hom_{Λ}(P, I) ≅ H^{∗}Hom_{Λ}(C, I), where I is any injective resolution of D. If P is minimal and D is a complex with trivial differential and rad(D) = 0, then by Theorem 5(ii)(b), Ext_{Λ}^{∗}(C, D) = Hom_{Λ}(P, D), which is one of the advantages of using minimal resolutions.

2.3. The hypercohomology spectral sequence. Recall that a doubly-indexed collection of modules E_{0}^{p,q} together with maps d′ : E_{0}^{p,q} → E_{0}^{p+1,q}, d′′ : E_{0}^{p,q} → E_{0}^{p,q+1} forms a double complex if (d′)^{2} = (d′′)^{2} = d′d′′ + d′′d′ = 0. A double complex yields a spectral sequence {E_{r}, d_{r}} in which the differentials are easy to describe explicitly: x ∈ E_{0}^{p,q} lives to E_{n}^{p,q} iff there exist x_{i} ∈ E_{0}^{p+i,q−i} (0 ≤ i < n) with x_{0} = x, d′′(x_{0}) = 0, and d′(x_{i−1}) + d′′(x_{i}) = 0 for 1 ≤ i < n. If this is the case and x represents ζ ∈ E_{n}^{p,q} then d_{n}(ζ) is represented by d′(x_{n−1}).

Now let C ∈ C_{b}, D ∈ _{b}C, and let I_{D} be an injective resolution of D. The hypercohomology spectral sequence arises from the double complex defined by E_{0}^{p,q} = Hom_{Λ}(C_{q}, I_{D}^{p}), d′(f) = ∂ ∘ f, d′′(f) = (−1)^{p+q+1}f ∘ ∂. We have

E_{2}^{p,q} ≅ Ext_{Λ}^{p}(H_{q}(C), D) ⇒ Ext_{Λ}^{p+q}(C, D).

Moreover, if I is an injective resolution of k then the tensor product induces a map of double complexes

Hom_{Λ}(k, I_{D}) ⊗ Hom_{Λ}(C_{ζ}, I) → Hom_{Λ}(C_{ζ}, I_{D} ⊗ I),

and since I_{D} ⊗ I is also an injective resolution of D, this map yields a pairing of spectral sequences

Ext_{Λ}^{p}(k, D) ⊗ Ext_{Λ}^{p′}(H_{q}(C_{ζ}), k) Ext_{Λ}^{p+p′}(H_{q}(C_{ζ}), D)

Ext_{Λ}^{p}(k, D) ⊗ Ext_{Λ}^{p′+q}(C_{ζ}, k)  Ext_{Λ}^{p+p′+q}(C_{ζ}, D). [figure]

[figure]

[figure]

[figure]

We could also start with the double complex Hom_{Λ}(P_{p}, D^{q}), where P is a projective resolution of C, to obtain the second hypercohomology spectral sequence

E_{2}^{p,q} ≅ Ext_{Λ}^{p}(C, H^{q}(D)) ⇒ Ext_{Λ}^{p+q}(C, D).

As a first consequence of these two spectral sequences, we obtain

Lemma 8. Suppose C, D ∈ _{b}C_{b}. Then Ext_{Λ}^{∗}(C, D) is a finitely generated Ext_{Λ}^{∗}(k, k)- module.

Proof. We first reduce to the case where C is a module concentrated in degree 0 as follows. In the hypercohomology spectral sequence we have E_{2} is a finite direct sum ⨁_{q} Ext_{Λ}^{∗}(H_{q}(C), D), so E_{2} is finitely generated over Ext_{Λ}^{∗}(k, k). The spectral sequence stops (it has a finite number of non-zero rows), so E_{∞} is also finitely generated over Ext_{Λ}^{∗}(k, k), and this implies that Ext_{Λ}^{∗}(C, D) is as well by [7, Lemma 7.4.5]. Now if C is a module concentrated in degree 0 then apply the second spectral sequence to reduce to the case where both C and D are modules.

2.4. Complexes of finite projective dimension. Let C ∈ C_{b}. We say that C has finite projective dimension if C has a bounded projective resolution, i.e., a projective resolution P → C with P ∈ _{b}C_{b}^{p}. Dually, for C ∈ _{b}C, we say that C has finite injective dimension if C has a bounded injective resolution.

Lemma 9. Let C ∈ _{b}C_{b}. Then the following are equivalent:

(i) C has finite projective dimension.

(ii) C has finite injective dimension.

(iii) Ext_{Λ}^{n}(C, C) = 0 for n sufficiently large.

(iv) C = C′ ⊕ C′′, where C′ ∈ _{b}C_{b}^{p}, C′′ ∈ _{b}^{e}C_{b}, and C′ has no exact summands.

Note that for such a C it follows from Theorem 5(ii) that the inclusion C′ ↪ C is a minimal projective resolution of C.

Proof. It is clear that (i)⇒(iii). Conversely, if (iii) holds, then for any simple Λmodule S, Lemma 8 implies Ext_{Λ}^{∗}(C, S) is finitely generated over Ext_{Λ}^{∗}(k, k), so it is certainly finitely generated over Ext_{Λ}^{∗}(C, C) (here we are using the fact that the cup and Yoneda products are compatible, cf. [6, Lemma 2.2]). Hence Ext_{Λ}^{∗}(C, S) is finite-dimensional, i.e., bounded. Let P → C be a minimal resolution of C. Since there are only finitely many simple Λ-modules, this means that there is an integer N such that Hom_{Λ}(P_{n}, S) = Ext_{Λ}^{n}(C, S) = 0 for all simples S and n > N. Hence P_{n} = 0 for all n > N, i.e. (i) holds. Proceeding dually, we obtain (ii)⇔(iii).

Now suppose (i), (ii), and (iii) hold. Let P →^{ϵ} C be a minimal projective resolution of C, C →^{η} I a minimal injective resolution, and let f = ηϵ. Then f : P → I is a quasi-isomorphism, and to prove (iv) it suffices to show that f is an isomorphism. But one may extend f to a surjective map P ⊕ W → I, for some W ∈ _{b}^{e}C_{b}^{p}. Let K be the kernel of this map. Then K is a bounded complex of projectives, and by considering the long exact sequence in homology arising from the short exact sequence

0 → K → P ⊕ W → I → 0,

we conclude that H_{∗}K = 0, i.e. K ∈ _{b}^{e}C_{b}^{p}. This means that K is isomorphic to the direct sum of complexes of the form ⋯ → 0 → N →^{=} N → 0 → ⋯, where N is an injective (i.e. projective) Λ-module, each of which is easily seen to be an injective object in C. Hence K is an injective object in C, so the short exact sequence above splits, and P ⊕ W ≅ K ⊕ I. Now K and W contain only exact summands, while P and I have no exact summands, so by the Krull-Schmidt Theorem, P ≅ I. Fix an isomorphism θ : I → P. Then θf : P → P is a quasi-isomorphism, so by Lemma 6, θf, and therefore f, is an isomorphism, and we have (iv). Finally, (iv)⇒(i), as C′ ↪ C is a (minimal) projective resolution of C.

The following is a direct generalization of Lemma 5.1 of [6].

Proposition 10. Suppose C, D ∈ _{b}C_{b} have finite projective dimension, and f : C → D is a quasi-isomorphism. Then there are decompositions C = C′ ⊕ C′′ and D = D′ ⊕ D′′, with C′, D′ ∈ _{b}C_{b}^{p} and C′′, D′′ ∈ _{b}^{e}C_{b}, such that f|_{C′} is an isomorphism onto D′.

Proof. Write C = C′ ⊕ C′′, D = D′ ⊕ D′′ as in Lemma 9 (iv), and let i : C′ ↪ C denote inclusion and π : D → D′ the projection. It suffices to show there exists a map g : D → C′ such that gfi = id_{C′} (by replacing D′ with fi(C′) and D′′ with Ker(g)). Now h = πfi : C′ → D′ is a quasi-isomorphism. Moreover, C′ and D′ are both minimal projective resolutions of D, and are therefore isomorphic. So we may apply Lemma 6 to conclude that h is an isomorphism. Let g = h^{−1}π. Then gfi = h^{−1}πfi = h^{−1}h = id_{C′}.

Now suppose C, D ∈ _{b}C_{b}. We write C ∼_{d} D if there is a sequence of quasiisomorphisms

C → X_{1} ← X_{2} → ⋯ ← X_{n} → D

with each X_{i} ∈ _{b}C_{b}. (This just means that C and D are isomorphic objects in the derived category.) It is clear that ∼_{d} is an equivalence relation.

Proposition 11. Suppose C, D ∈ _{b}C_{b} and C ∼_{d} D. If C has finite projective dimension then so does D, and MPR(C) ≅ MPR(D).

Proof. If X → Y is a quasi-isomorphism and P → X is a bounded projective resolution of X then P → X → Y is a bounded projective resolution of Y, and by Proposition 10, MPR(X) ≅ MPR(Y). Dually, if X ← Y is a quasi-isomorphism and I ← X is a bounded injective resolution of X, then I ← X ← Y is a bounded injective resolution of Y, and by Proposition 10, MPR(X) ≅ MPR(Y). So the proof follows by induction on the length n of a chain of quasi-isomorphisms joining C and D.

3. Generalized Benson-Carlson duality

3.1. The main theorem. Let U and V be Λ-modules and n a positive integer. An n-extension of U by V is a complex e ∈ _{b}^{e}C_{b} with e_{−1} = U, e_{n} = V, and e_{r} = 0 if r > n or r < −1. If ẽ is also an n-extension of U by V we write e ⇝ ẽ if there exists f ∈ Z_{0}Hom_{Λ}(e, ẽ) with f_{−1} = id_{U}, f_{n} = id_{V}. We complete ⇝ to an equivalence relation and let YExt_{Λ}^{n}(U, V) denote the set of equivalence classes. There is a functorial bijection YExt_{Λ}^{n}(U, V) ⟶ Ext_{Λ}^{n}(U, V) (cf. Hilton and Stammbach [8, IV.9]).

Let n ≥ 2. A truncated n-extension of U by V is a complex C ∈ _{b}C_{b} with C_{i} = 0 if i > n − 1 or i < 0, and with H_{i}(C) isomorphic to U if i = 0, V if i = n − 1, and 0 otherwise. If U and V are simple, then it is easily seen that the bijection referred to above induces a bijection between equivalence classes of truncated n-extensions of U by V under ∼_{d} and P (Ext_{Λ}^{n}(U, V)), the projective space of Ext_{Λ}^{n}(U, V). This follows from Schur’s Lemma, which states that the endomorphism ring of U or V consists of scalar multiples of the identity, which implies that a truncated n-extension determines an n-extension only up to non-0 scalar multiple.

We can now state the main theorem, which generalizes Theorem 5.5 of [6]. Let D ∈ _{b}C_{b} and let R be the quotient of Ext_{Λ}^{∗}(k, k) by the annihilator of Ext_{Λ}^{∗}(D, D). Recall that non-zero homogeneous elements of positive degree ζ_{1}, …, ζ_{d} ∈ Ext_{Λ}^{∗}(k, k) are said to form a homogeneous system of parameters (h.s.o.p.) for Ext_{Λ}^{∗}(D, D) if their images in R generate a subring over which R is finitely generated as a module.

Theorem 12. Let k be an algebraically closed field and Λ a finite-dimensional cocommutative Hopf algebra over k with the finiteness property in cohomology. Let D ∈ _{b}C_{b}, and ζ_{1}, …, ζ_{r} a h.s.o.p. for Ext_{Λ}^{∗}(D, D). Assume n_{i} = deg(ζ_{i}) ≥ 2 for all i. Let C_{ζ_{i}} be a truncated n_{i}-extension representing [ζ_{i}], and let C = ⨂_{i} C_{ζ_{i}} ⊗ D. Then

(i) There is a decomposition C = N ⊕ Q, where N ∈ _{b}C_{b}^{p}, Q ∈ _{b}^{e}C_{b}, and N has no exact summands.

(ii) If D^{∗}[t] ∼_{d} D for an integer t, then N^{∗}[s] ≅ N, where s = t + ∑_{i=1}^{r}(n_{i} − 1). (iii) Up to isomorphism of complexes, N is independent of the choices C_{ζ_{i}} of trun-

cated n_{i}-extension representing [ζ_{i}]

A key observation is the following Lemma, which is essentially [6, Proposition 5.2]. Given ζ ∈ Ext_{Λ}^{n}(k, k), let e_{ζ} be an n-extension of k by k representing ζ. Then e_{ζ}^{∗}[n − 1] is also an n-extension of k by k, and therefore represents some element ζ^{∗} of Ext_{Λ}^{n}(k, k). Since equivalent extensions are taken to equivalent extensions by this operation, we have a well-defined operation ζ ↦ ζ^{∗} on Ext_{Λ}^{n}(k, k).

Lemma 13. For n ≥ 1 and ζ ∈ Ext_{Λ}^{n}(k, k), ζ^{∗} = ±ζ. In particular, if C_{ζ} is a truncated n-extension representing [ζ], then (C_{ζ})^{∗}[n − 1] ∼_{d} C_{ζ}.

Proof. Let P → k be a projective resolution of k. After identifying P with P^{∗∗} in the usual way, there is an endomorphism θ of Hom_{Λ}(P, P^{∗}) defined by θ(f) = f^{∗} (cf. Section 1.2). We claim that for f ∈ ZHom_{Λ}(P, P^{∗}), f ≃ f^{∗}. To see this, reason as follows: Hom_{Λ}(P, P^{∗}) ≅ Hom_{Λ}(P ⊗_{k} P, k), and θ is induced by the twisting endomorphism τ of P ⊗_{k} P. By the Comparison Theorem, τ is homotopic to the identity on P ⊗_{k} P, hence θ is homotopic to the identity on Hom_{Λ}(P, P^{∗}). Say θ −1 = ∂s+s∂. Then given f ∈ ZHom_{Λ}(P, P^{∗}), ∂(f) = 0, hence θ(f)−f = ∂(s(f)), i.e., θ(f) ≃ f.

Now Ext_{Λ}^{∗}(k, k) ≅ H^{∗}(Hom_{Λ}(P, P^{∗})), as k → P^{∗} is an injective resolution of k. Fix an n-extension e representing ζ. Then by projectivity and injectivity, there exist maps f_{i}, g_{i} making the following diagram, in which the middle row is e, commute:

[figure: \begin{matrix}P_{n+1} & P_{n} & P_{n−1} & P_{n−2} & ⋯ & P_{1} & P_{0} & k & 0 \\ 0 & k & M_{n−1} & M_{n−2} & ⋯ & M_{1} & M_{0} & k & 0 \\ 0 & k & P_{0}^{∗} & P_{1}^{∗} & ⋯ & P_{n−2}^{∗} & P_{n−1}^{∗} & P_{n}^{∗} & P_{n+1}^{∗} \end{matrix}; f_{n}; f_{n−1}; f_{n−2}; f_{1}; ϵ; f_{0}; g_{n−1}; g_{n−2}; g_{1}; g_{0}; g_{−1}; ϵ^{∗}]

Let f = f_{n}, g = g_{−1}. Now ϵ^{∗}f ∈ Z^{n}Hom_{Λ}(P, P^{∗}) represents ζ. But the diagram shows that ±gϵ ≃ ϵ^{∗}f, hence ±gϵ also represents ζ. Taking the dual diagram, we see that ϵ^{∗}g^{∗} represents ±ζ^{∗}. But as was shown in the previous paragraph, ϵ^{∗}g^{∗} = ±(gϵ)^{∗} ≃ gϵ.

The next lemma involves the hypercohomology spectral sequence (see Section 2.3). Let n ≥ 2, ζ ∈ Ext_{Λ}^{n}(k, k), and C_{ζ} a truncated n-extension representing [ζ]. Extend C_{ζ} to an n-extension representing ζ with maps η : k → (C_{ζ})_{n−1} and ϵ : (C_{ζ})_{0} → k. This yields identifications H_{n−1}(C_{ζ}) = k and H_{0}(C_{ζ}) = k and therefore also E_{2}^{0,n−1} = Ext_{Λ}^{0}(H_{n−1}(C_{ζ}), k) = k and E_{2}^{n,0} = Ext_{Λ}^{n}(H_{0}(C_{ζ}), k) = Ext_{Λ}^{n}(k, k). Define ζ̃ ∈ E_{2}^{0,n−1} to be the element corresponding to 1 ∈ k.

Lemma 14. Let D ∈ _{b}C. Then in the two-row spectral sequence

E_{2}^{pq}(ζ) = Ext_{Λ}^{p}(H_{q}(C_{ζ}), D) ⇒ Ext_{Λ}^{p+q}(C_{ζ}, D)

the differential d_{n} is given by

d_{n}(αζ̃) = ±αζ ∈ E_{n}^{p+n,0}(ζ) = Ext_{Λ}^{p+n}(k, D).

Proof. If we can show that d_{n}(ζ̃) = ζ in E_{n}(k) then we are done, using the fact that d_{n} is a derivation. So without loss of generality we assume D = k. By injectivity there are maps f_{i}, ζ̂ making the following diagram commute:

[figure: \begin{matrix}0 & k & C_{n−1} & C_{n−2} & ⋯ & C_{1} & C_{0} & k & 0 \\ 0 & k & I_{0} & I_{1} & ⋯ & I_{n−2} & I_{n−1} & I^{n} & I_{n+1} & ⋯ \end{matrix}; η; f_{n−1}; f_{n−2}; f_{1}; f_{0}; ϵ; ζ̂]

Now ζ̂ represents ζ (this is just the bijection between YExt_{Λ}^{n}(k, k) and Ext_{Λ}^{n}(k, k)) and ζ̃ is represented in E_{0}^{0,n−1} by f_{n−1}. By definition of the differential, ±d_{n}(ζ̃) is represented by ±∂ ∘ f_{0}. By commutativity this is ζ̂∘ ϵ, which represents ζ under the identification Ext_{Λ}^{n}(k, k) = Ext_{Λ}^{n}(H_{0}(C), k).

Proof of Theorem 12. To prove (i), show Ext_{Λ}^{n}(C, C) = 0 for n sufficiently large and apply Lemma 9. To do this proceed as in the proof of Theorem 4.1 of [6] and show that the E_{∞}-term of the appropriate spectral sequence is a finite module over Ext_{Λ}^{∗}(k, k)/(ζ_{1}, …, ζ_{d}), using Lemma 14.

Now if A, A′, B, B′ ∈ _{b}C_{b} and A ∼_{d} A′ and B ∼_{d} B′ then A ⊗ B ∼_{d} A′ ⊗ B′. So Lemma 13 implies C^{∗}[s] ∼_{d} C. Hence by Proposition 11, the minimal projective resolutions of C^{∗}[s] and C are isomorphic. Now C ≅ N ⊕Q, so C^{∗}[s] ≅ N^{∗}[s]⊕Q^{∗}[s]. On the other hand, C^{∗}[s] ≅ N ⊕ Q′, for some Q′ ∈ _{b}^{e}C_{b}. So by the Krull-Schmidt Theorem, N ≅ N^{∗}[s], proving (ii).

If we choose different truncated extensions C′_{ζ_{i}} representing the [ζ_{i}], then C′_{ζ_{i}} ∼_{d} C_{ζ_{i}} for each i, and therefore ⨂_{i} C′_{ζ_{i}} ∼_{d} C. Hence their minimal projective resolutions are isomorphic, proving (iii).

Example. Suppose k has characteristic 2, G is the alternating group on 4 letters, and Λ = kG. We have (cf. [3, p. 197])

H^{∗}(G, k) = k[u, v, w | deg(u) = 2, deg(v) = deg(w) = 3, u^{3} + v^{2} + vw + w^{2} = 0].

The simple Λ-modules are k, S, T where S and T also are one-dimensional (corresponding to the third roots of 1 in k) and we have S^{∗} ≅ T. The projective covers of the simples have Loewy structures

P_{k} = \begin{matrix}k \\ S T \\ k \end{matrix} , P_{S} = \begin{matrix}S \\ k T \\ S \end{matrix} , P_{T} = \begin{matrix}T \\ k S \\ T \end{matrix} .

For a h.s.o.p. we may take u and v, and these are represented by extensions of the form

\begin{aligned}u : & & & 0 & → & k & → & \begin{matrix}S \\ k \end{matrix} & → & \begin{matrix}k \\ S \end{matrix} & → & k & → & 0 \\ v : & 0 & → & k & → & \begin{matrix}S T \\ k \end{matrix} & → & \begin{matrix}T \\ S \end{matrix} ⊕ \begin{matrix}S \\ T \end{matrix} & → & \begin{matrix}k \\ S T \end{matrix} & → & k & → & 0 \end{aligned}

One then gets C = C_{u} ⊗C_{v} decomposes as a direct sum of a complex N of projectives

P_{k} → P_{S} ⊕ P_{T} → P_{T} ⊕ P_{S} → P_{k},

which indeed satisfies N^{∗}[3] ≅ N, and an exact complex

\begin{matrix}T \\ S \end{matrix} → \begin{matrix}T \\ S \end{matrix} ⊕ \begin{matrix}T \\ k \end{matrix} ⊕ \begin{matrix}k \\ T \end{matrix} → \begin{matrix}T \\ k \end{matrix} ⊕ \begin{matrix}k \\ T \end{matrix} ⊕ \begin{matrix}S \\ T \end{matrix} → \begin{matrix}S \\ T \end{matrix} .

3.2. Further applications. One may also apply the techniques described above to Benson and Carlson’s work on the construction of projective resolutions as tensor products of periodic complexes [5]. We review this briefly here.

Let n ≥ 2, ζ ∈ Ext_{Λ}^{n}(k, k), and P → k the minimal resolution. One constructs a special n-extension é_{ζ} representing ζ as the bottom row in the commutative diagram

[figure: \begin{matrix}⋯ & P_{n} & P_{n−1} & P_{n−2} & ⋯ & P_{0} & k & 0 \\ 0 & k & M_{ζ} & P_{n−2} & ⋯ & P_{0} & k & 0 \end{matrix}; ζ]

where the square with M_{ζ} in the bottom right is a pushout. Let Ć_{ζ} denote the truncated n-extension arising from this, and let Ć_{ζ}^{(∞)} denote the complex in C_{b} formed by splicing together infinitely many copies of Ć_{ζ}.

Now let M be a Λ-module and ζ_{1}, …, ζ_{r} a h.s.o.p. for Ext_{Λ}^{∗}(M, M) with n_{i} = deg(ζ_{i}) ≥ 2 for each i. Using variety theory, one can show that M ⊗ M_{ζ_{1}} ⊗ ⋯ ⊗ M_{ζ_{r}} is projective. Since the tensor product of a projective Λ-module with any Λ-module is also projective, it follows that X́ = M ⊗ Ć_{ζ_{1}}^{(∞)} ⊗ ⋯ ⊗ Ć_{ζ_{r}}^{(∞)} ∈ C_{b}^{p}. Moreover, the Künneth Theorem implies that X́ is exact in positive degrees, and that X́_{1} → X́_{0} → M → 0 is exact (where the map X́_{0} → M is formed by tensoring the identity on M with the augmentations from Ć_{ζ_{i}} → k). Hence

Theorem 15 (Benson-Carlson). M ⊗ Ć_{ζ_{1}}^{(∞)} ⊗ ⋯ ⊗ Ć_{ζ_{r}}^{(∞)} is a projective resolution of M.

We can generalize this as follows.

Theorem 16. Let D ∈ _{b}C_{b}, ζ_{1}, …, ζ_{r} a h.s.o.p. for Ext_{Λ}^{∗}(D, D) such that n_{i} = deg(ζ_{i}) ≥ 2 for all i, and for each i let C_{ζ_{i}} be a truncated n_{i}-extension representing [ζ_{i}]. Let X = D ⊗ C_{ζ_{1}}^{(∞)} ⊗ ⋯ ⊗ C_{ζ_{r}}^{(∞)}. Then there is a decomposition X = Y ⊕ Z, where Y is exact and Z is a projective resolution of D.

Proof. It is easily seen that the special extension é_{ζ} has the following property: if e_{ζ} is any n-extension representing ζ then é_{ζ} ⇝ e_{ζ}. On the truncated level, this means that if C_{ζ} is any truncated n-extension representing [ζ] then there is a quasi-isomorphism Ć_{ζ} → C_{ζ}. Now let C̀_{ζ} = Ć_{ζ}^{∗}[n − 1]. By Lemma 13, C̀_{ζ} also represents [ζ], and it is equally easy to see that C̀_{ζ} enjoys the following dual property: if C_{ζ} is any truncated n-extension representing ζ then there is a quasi-isomorphism C_{ζ} → C̀_{ζ}. Clearly, Theorem 15 remains true if each Ć_{ζ} is replaced by C̀_{ζ}.

Now we may splice together quasi-isomorphisms to get quasi-isomorphisms Ć_{ζ_{i}}^{(∞)} → C_{ζ_{i}}^{(∞)} → C̀_{ζ_{i}}^{(∞)}. Tensoring over i yields quasi-isomorphisms X́ → X → X̀. But both X́ and X̀ are projective resolutions of D, so there exist quasi-isomorphisms P_{D} → X́ and X̀ → P_{D}, where P_{D} is the minimal resolution of D. Composing, we have quasi-isomorphisms P_{D} → X → P_{D}. By Lemma 6, this composition must be an isomorphism. Hence P_{D} splits off of X.

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Department of Mathematics, Northwestern University, Evanston IL 60208-2730 E-mail address: len@math.nwu.edu, siegel@math.nwu.edu