REALIZING MAPS BETWEEN MODULES OVER TATE COHOMOLOGY RINGS

APOSTOLOS BELIGIANNIS AND HENNING KRAUSE

Abstract. Let G be a finite group and k be a field. Given two representations A and B of G, we investigate when all homomorphisms Ĥ^{∗}(G, A) → Ĥ^{∗}(G, B) over the Tate cohomology ring Ĥ^{∗}(G, k) are of the form Ĥ^{∗}(G, α) for some morphism α: A → B. We construct an extended Milnor sequence which computes the obstruction for homomorphisms Ĥ^{∗}(G, A) → Ĥ^{∗}(G, B) to be realizable.

Introduction

Let k be a field and G be a finite group. Given a graded module over the Tate cohomology ring Ĥ^{∗}(G, k), it is natural to ask when this module is of the form Ĥ^{∗}(G, A) for some module A over the group algebra kG. Recently, this question has been answered, at least for modules which are direct summands of modules of the form Ĥ^{∗}(G, A); see [5]. In this paper we continue this program and ask when a map Ĥ^{∗}(G, A) → Ĥ^{∗}(G, B) of Ĥ^{∗}(G, k)-modules is of the form Ĥ^{∗}(G, α) for some map α: A → B of kG-modules. We do not give a complete answer to this question. However, we give a global answer for all maps Ĥ^{∗}(G, A) → Ĥ^{∗}(G, B) for a fixed kG-module A.

Theorem 1. Let A be a kG-module. Then the following are equivalent:

(i) For every kG-module B, every map Ĥ^{∗}(G, A) → Ĥ^{∗}(G, B) of graded Ĥ^{∗}(G, k)- modules is of the form Ĥ^{∗}(G, α) for some map α: A → B of kG-modules.

(ii) The projective dimension of Ĥ^{∗}(G, A) over Ĥ^{∗}(G, k) is at most 1 and there exists a decomposition A = A′ ⊕ A′′ such that A′ belongs to the localizing subcategory of the stable module category \underline{Mod}(kG) which is generated by k, and Ĥ^{∗}(G, A′′) = 0.

For a p-group G this result is quite satisfactory because every kG-module belongs to the localizing subcategory generated by k. However, in general one cannot expect a simple homological condition on Ĥ^{∗}(G, A) which ensures that every map out of Ĥ^{∗}(G, A) is realizable by a map of kG-modules.

Theorem 1 is really a lot more general. We can replace the stable category of kG-modules by any triangulated category with arbitrary coproducts, and Ĥ^{∗}(G, −) can be replaced by any homology theory which is represented by a compact object. In this context we prove our result; see Theorem 3.3.

There is also a dual result which characterizes the fact that for a fixed kG-module B every map Ĥ^{∗}(G, A) → Ĥ^{∗}(G, B) is realizable by a map A → B. This is stated in Theorem 4.2.

The proof of Theorem 1 and its generalization is based on a five-term exact sequence which is closely related to some generalized and extended variants of Milnor’s sequence, appearing for example in [6, 4]. Our sequence describes the obstruction for the realizability of maps of the form Ĥ^{∗}(G, A) → Ĥ^{∗}(G, B). The obstruction is by definition the cokernel of the natural map

\underline{Hom}_{kG}(A, B) ⟶ Hom_{Ĥ^{∗}(G,k)}(Ĥ^{∗}(G, A), Ĥ^{∗}(G, B))

and we use for its description the ideal of maps α: A → B satisfying Ĥ^{∗}(G, α) = 0. Such maps are known as phantom maps in algebraic topology and their analogue in group representation theory has been studied before by Benson and Gnacadja [3].

Next we discuss in this paper the realizability of extensions. It turns out that the long exact sequence which describes the realizability obstruction for maps Ĥ^{∗}(G, A) → Ĥ^{∗}(G, B), has a higher dimensional analogue which computes

Ext_{Ĥ^{∗}(G,k)}^{n}(Ĥ^{∗}(G, A), Ĥ^{∗}(G, B))

for n ≥ 1. We use this to describe the exact sequences 0 → Ĥ^{∗}(G, B) → E → Ĥ^{∗}(G, A) → 0 which are of the form Ĥ^{∗}(G, ε) for some exact sequence ε: 0 → B → C → A → 0 of kG-modules.

The final part of this paper is devoted to various applications of our general theory. Most of these applications take place in the stable category of kG-modules.

Our motivating problem of realizing modules and maps over the Tate cohomology ring is closely related to the problem when Brown representability holds for homology theories. Recently, some progress has been made in [1, 8, 7], and we use some of the ideas which have been developed in these papers.

In contrast to [5], we do not exploit the A_{∞}-structure of Tate cohomology.

Acknowldegement. The work on this paper started while the second named author visited the island Lefkada. It is a pleasure to thank Apostolos Beligiannis for his generous hospitality.

1. Exact triangles and phantom maps

We begin this paper with a brief introduction into relative homological algebra for triangulated categories. Throughout we fix a triangulated category 𝒯 containing all small coproducts. Given two objects A and B in 𝒯, we denote by 𝒯(A, B)_{∗} the graded Hom-group 𝒯(A, B)_{∗} = ⨁_{i∈ℤ} 𝒯(Σ^{i}A, B). In particular, the graded Homgroup 𝒯(A, A)_{∗} is a ℤ-graded ring.

We fix a homology theory H: 𝒯 → 𝒜b, that is, H is a functor which sends triangles to exact sequences and preserves all coproducts. We make the additional assumption that H is represented by an object T ∈ 𝒯, that is, H = 𝒯(T, −)_{∗}. Therefore the endomorphism ring Γ = 𝒯(T, T)_{∗} acts in a natural way on H(A) for each object A ∈ 𝒯. Thus we have a functor

H: 𝒯 ⟶ Mod(Γ), A ↦ 𝒯(T, A)_{∗}

into the category Mod(Γ) of ℤ-graded Γ-modules.

There is an obvious way to generalize our setting. Instead of representing the homology theory H by a single object T, one could take a set of compact objects in 𝒯 which is closed under the shift Σ. Recall that an object A ∈ 𝒯 is compact if the functor 𝒯(A, −) preserves all coproducts. Viewing the full subcategory Γ of objects in such a set of compact objects as a ℤ-graded ring with several objects, we obtain a homology theory H: 𝒯 → Mod(Γ) by sending an object A to 𝒯(−, A)|_{Γ}.

The examples we have in mind include the following.

Example 1.1. 1. Fix a finite group G and a field k. Let ℋ be the homotopy category of projective modules over the group algebra kG. Setting T to be a Tate resolution of the trivial representation k, we have that T is compact in ℋ and Γ = ℋ(T, T)_{∗} is the Tate cohomology ring Ĥ^{∗}(G, k). We can use instead of ℋ the full subcategory of ℋ consisting of acyclic complexes of projective kG-modules, which is equivalent to the stable module category \underline{Mod}(kG) of the group algebra kG. The equivalence identifies T with k.

2. Let 𝒟(Mod(Λ)) be the derived category of unbounded complexes over a ring Λ. We fix a finitely presented Λ-module T of finite projective dimension and view T as a stalk complex concentrated in degree zero. Then T is a compact object in 𝒟(Mod(Λ)) and the graded endomorphism ring Γ of T equals the Yoneda algebra Ext_{Λ}^{∗}(T, T). For instance, we can take T to be a tilting module, or simply T = Λ.

3. Let Ho(𝒮) be the stable homotopy category of spectra, and let T be the sphere spectrum S^{0}. Then Mod(Γ) is the module category of the stable homotopy ring of spheres.

4. Let 𝒯 be a compactly generated triangulated category and let 𝒫 be the set of isoclasses of compact objects in 𝒯. Viewing the full subcategory of objects in 𝒫 as a ring Γ with several objects, we obtain a homology theory H: 𝒯 → Mod(Γ) which is the basis for the well-studied theory of purity in compactly generated triangulated categories.

We call a triangle A ⟶ B ⟶ C ⟶ ΣA in 𝒯 exact if the induced sequence 0 ⟶ H(A) ⟶ H(B) ⟶ H(C) ⟶ 0 is exact in 𝒜b. An object P in 𝒯 is called projective if for any exact triangle as above, any morphism P ⟶ C factors through the map B ⟶ C. A projective presentation of the object A is an exact triangle triangle K ⟶ P ⟶ A ⟶ ΣA where P is projective. The concept of an injective object and an injective presentation are defined dually.

Closely related to the notion of exactness is the following concept. We call a map f: A ⟶ B phantom if the induced map H(f) is zero. Plainly the collection Ph(A, B) of all phantom maps between A and B is a subgroup of 𝒯(A, B) and it is easy to see that in this way we obtain an ideal Ph(−, −) of 𝒯, i.e. an additive subfunctor of 𝒯(−, −), which is Σ-stable in the sense that Σ^{n}f is phantom for all n ∈ ℤ, provided that f is phantom. Obviously, a triangle A ⟶ B ⟶ C ⟶ ΣA in 𝒯 is exact if and only if the third map C → ΣA is phantom.

Any object A of 𝒯 admits a projective presentation

Ω(A) ⟶^{g} P (A) ⟶^{f} A ⟶^{h} ΣΩ(A) .

Indeed let P (A) = ∐_{i∈ℤ} ∐_{Σ^{i}T⟶A} Σ^{i}T and let f: P (A) ⟶ A be the induced canonical morphism. Then the triangle in 𝒯 with base f is plainly exact, hence it is a projective presentation of A since by construction P (A) is projective. Observe that h: A ⟶ ΣΩA is a weakly universal phantom map out of A in the sense that any other phantom map out of A factors through h. It follows from the above construction that the full subcategory of projective objects of 𝒯 is identified with Add(Σ^{∗}T), the full subcategory of 𝒯 consisting of all direct summands of arbitrary coproducts of objects in {Σ^{i}T | i ∈ ℤ}.

A projective resolution of an object A in 𝒯 is a complex

⋯ ⟶^{δ_{3}} P_{2} ⟶^{δ_{2}} P_{1} ⟶^{δ_{1}} P_{0} ⟶ A ⟶ 0  (1.1)

in 𝒯, where each P_{n} is projective and the induced complex

⋯ ⟶ 𝒯(P, P_{2}) ⟶ 𝒯(P, P_{1}) ⟶ 𝒯(P, P_{0}) ⟶ 𝒯(P, A) ⟶ 0

is exact for any projective P. Any object A admits a projective resolution. It is obtained by splicing together exact triangles

Ω^{n+1}(A) ⟶^{g_{n}} P_{n} ⟶^{f_{n}} Ω^{n}(A) ⟶^{h_{n}} ΣΩ^{n+1}(A)

where each P_{n} is projective and Ω^{0}(A) = A. Defining δ_{n} = g_{n−1} ^{∘} f_{n} for all n ≥ 1 gives a resolution of the form (1.1). Actually any projective resolution is of this form. Injective resolutions are defined dually.

Finally we define the projective dimension pd A of an object A in 𝒯 to be the smallest integer n ≥ 0 such that there exists a projective resolution (1.1) with P_{i} = 0 for all i > n. We put pd A = ∞ if pd A ≠ n for all n ≥ 0.

2. A long exact sequence

Given two objects A and B in 𝒯, we have a natural map H_{A,B}: 𝒯(A, B) → Hom_{Γ}(H(A), H(B)). In this section we construct a five-term exact sequence which describes the obstruction for this map to be surjective. We start with the following well-known result.

Lemma 2.1. The functor H: 𝒯 ⟶ Mod(Γ) induces an equivalence between Add(Σ^{∗}T) and the category of projective Γ-modules. Moreover, for any X ∈ Add(Σ^{∗}T) and any A ∈ 𝒯 the canonical map H_{X,A}: 𝒯(X, A) ⟶ Hom_{Γ}(H(T), H(A)) is invertible.

The lemma implies that H sends projective presentations in 𝒯 to projective presentations in Mod(Γ). More precisely, if ΩA ⟶ P ⟶ A ⟶ ΣΩA is a projective presentation of A in 𝒯, then the sequence 0 → H(ΩA) → H(P) → H(A) → 0 is a projective presentation of the Γ-module H(A). In particular we have an isomorphism H(ΩA) ≅ ΩH(A) up to projective summands.

The construction of the loop object ΩA is not in general functorial; it depends on the choice of the map P ⟶ A. But it is easy to see that if Ω′A ⟶ P′ ⟶ A ⟶ ΣΩ′A is another projective presentation of A in 𝒯, then the loop objects ΩA and Ω′A are isomorphic up to projective summands: we have an isomorphism ΩA ≅ Ω′A in the stable category 𝒯/𝒫 of 𝒯 modulo the ideal of maps in 𝒯 factorizing through a projective object. Moreover the assignment A ↦ ΩA defines a functor Ω: 𝒯/𝒫 ⟶ 𝒯/𝒫.

Lemma 2.2. The left ideal Ph(ΩA, −) of 𝒯(ΩA, −) is independent of the choice of the loop object ΩA.

Proof. If ΩA ⟶ P ⟶ A ⟶ ΣΩA and Ω′A ⟶ P′ ⟶ A ⟶ ΣΩ′A are projective presentations, then we have the following diagram of morphisms of triangles:

Σ^{−1}A ⟶^{−Σ^{−1}h} ΩA ⟶^{g} P ⟶^{f} A
∥  ^{β}↓  ^{α}↓  ∥
Σ^{−1}A ⟶^{−Σ^{−1}h′} Ω′A ⟶^{g′} P′ ⟶^{f′} A
∥  ^{β′}↓  _{α′}↓  ∥
Σ^{−1}A ⟶^{−Σ^{−1}h} ΩA ⟶^{g} P ⟶^{f} A

Observe that (1_{ΩA} − β′ ^{∘} β)^{∘} Σ^{−1}h = 0. Thus there exists a map ρ: P ⟶ ΩA such that 1_{ΩA} − β′ ^{∘} β = ρ^{∘} g. Similarly there exists a map σ: P′ ⟶ Ω′A such that 1_{Ω′A}−β^{∘} β′ = σ^{∘} g′. Next we define, for any object B ∈ 𝒯, a map β′_{∗}: Ph(ΩA, B) ⟶ Ph(Ω′A, B) by β′_{∗}(ϕ) = ϕ^{∘} β′. Similarly we define β_{∗}: Ph(Ω′A, B) ⟶ Ph(ΩA, B) by β_{∗}(ψ) = ψ^{∘} β. We claim that β′_{∗} is invertible with inverse the map β_{∗}. Indeed for any phantom map ϕ: ΩA ⟶ B we have (β_{∗} ^{∘} β′_{∗})(ϕ) = ϕ^{∘} β′ ^{∘} β. Since 1_{ΩA}−β′ ^{∘} β = ρ^{∘} g, it follows that (β_{∗} ^{∘} β′_{∗})(ϕ) = ϕ − ϕ^{∘} ρ^{∘} g. But ϕ, hence ϕ^{∘} ρ, is a phantom map. Since ϕ^{∘} ρ starts from the projective object P, it follows that ϕ^{∘} ρ = 0. Hence (β_{∗} ^{∘} β′_{∗})(ϕ) = ϕ and we infer that β_{∗} ^{∘} β′_{∗} = 1_{Ph(ΩA,B)}. In a dual way we have β′_{∗} ^{∘} β_{∗} = 1_{Ph(Ω′A,B)} which shows that β′_{∗} is invertible with inverse the map β_{∗}.

Remark 2.3. If α: A′ ⟶ A is a morphism in 𝒯, then we have a morphism of projective presentations

ΩA′ ⟶^{g′} P′ ⟶^{f′} A′ ⟶^{h′} ΣΩA′
Ωα↓  ^{β}↓  ^{α}↓  ^{ΣΩα}↓
ΩA ⟶^{g} P ⟶^{f} A ⟶^{h} ΣΩA

We claim that for any object B in 𝒯 the induced map Ph(Ωα, B): Ph(ΩA, B) ⟶ Ph(ΩA′, B) given by ϕ ↦ ϕ^{∘} Ωα, does not depend on the choices we made for the compatible maps P′ ⟶ P and ΩA′ ⟶ ΩA. Indeed if

ΩA′ ⟶^{g′} P′ ⟶^{f′} A′ ⟶^{h′} ΣΩA′
Ω′α↓  ^{β′}↓  ^{α}↓  _{ΣΩ′α}↓
ΩA ⟶^{g} P ⟶^{f} A ⟶^{h} ΣΩA

is another morphism of projective presentations induced by α, then there exists a morphism σ: P′ ⟶ ΩA such that Ωα − Ω′α = σ^{∘} g′. Then for any phantom map ϕ: ΩA ⟶ B we have ϕ^{∘}(Ωα − Ω′α) = ϕ^{∘} σ^{∘} g′. But ϕ^{∘} σ = 0 as a phantom map out of the projective object P′. Hence ϕ^{∘} Ωα = ϕ^{∘} Ω′α.

Let A be an object in 𝒯 and choose a projective presentation ΩA ⟶^{g} P ⟶^{f} A ⟶^{h} ΣΩA of A in 𝒯. Then for any object B in 𝒯, we define a map

ζ_{A,B}: Ph(ΩA, B) ⟶ Ph^{2}(Σ^{−1}A, B)

by sending a phantom ϕ: ΩA ⟶ B to ζ_{A,B}(ϕ) = ϕ^{∘} Σ^{−1}h. Note that ζ_{A,B}(ϕ) lies in Ph^{2}(Σ^{−1}A, B) since Σ^{−1}h is a phantom map.

Lemma 2.4. The natural map ζ_{A,B}: Ph(ΩA, B) ⟶ Ph^{2}(Σ^{−1}A, B) is independent of the choice of loops.

Proof. Let ζ′_{A,B}: Ph(Ω′A, B) ⟶ Ph^{2}(Σ^{−1}A, B) be the map resulting from another projective presentation Ω′A ⟶ P′ ⟶ A ⟶ ΣΩ′A of A in 𝒯, that is ζ′_{A,B}(ψ) = ψ^{∘} Σ^{−1}h′. We claim that the following diagram commutes:

Ph(ΩA, B) ⟶^{ζ_{A,B}} Ph^{2}(Σ^{−1}A, B)
β′_{∗}↓≅  ∥
Ph(Ω′A, B) ⟶^{ζ′_{A,B}} Ph^{2}(Σ^{−1}A, B)

where β′_{∗} is the isomorphism constructed in Lemma 2.2. Indeed for any phantom map ϕ: ΩA ⟶ B, we have (ζ′_{A,B} ^{∘} β′_{∗})(ϕ) = ζ′_{A,B}(ϕ^{∘} β′) = ϕ^{∘} β′ ^{∘} Σ^{−1}h′ = ϕ^{∘} Σ^{−1}h = ζ_{A,B}(ϕ). Hence the above diagram commutes and this shows that the map ζ_{A,B} does not depend on the choices of the loop objects.

Remark 2.5. As in Remark 2.3 one can prove that any morphism α: A′ ⟶ A in 𝒯 induces a commutative square

Ph(ΩA, B) ⟶^{ζ_{A,B}} Ph^{2}(Σ^{−1}A, B)
Ph(Ωα,B)↓  ^{Ph^{2}(Σ^{−1}α,B)}↓
Ph(ΩA′, B) ⟶^{ζ_{A′,B}} Ph^{2}(Σ^{−1}A′, B)

The main result of this section is the following.

Proposition 2.6. For any objects A, B in 𝒯, there exists a natural map

ϑ_{A,B}: Hom_{Γ}(H(A), H(B)) ⟶ Ph(ΩA, B)

which induces a functorial exact sequence

0 ⟶ Ph(A, B) ⟶ 𝒯(A, B) ⟶^{H_{A,B}}Hom_{Γ}(H(A), H(B)) ⟶^{ϑ_{A,B}}  (2.1)

Ph(ΩA, B) ⟶^{ζ_{A,B}} Ph^{2}(Σ^{−1}A, B) ⟶ 0 .

Proof. First we need to define the map ϑ_{A,B}. To this end choose a projective

presentation ΩA ⟶^{g} P ⟶^{f} A ⟶^{h} ΣΩA of A. Then we have a short exact se-

quence 0 ⟶ H(ΩA) ⟶^{H(g)} H(P) ⟶^{H(f)} H(A) ⟶ 0 in Mod(Γ). For any morphism α̃: H(A) ⟶ H(B) in Mod(Γ), there exists a unique morphism α′: P ⟶ B such that H(α′) = α̃^{∘} H(f). Then the morphism α′ ^{∘} g: ΩA ⟶ B is phantom, since H(α′ ^{∘} g) = α̃^{∘} H(f)^{∘} H(g) = 0. We define ϑ_{A,B}(α̃) = α′ ^{∘} g, and it is easy to see that this gives a well-defined homomorphism ϑ_{A,B}: Hom_{Γ}(H(A), H(B)) ⟶ Ph(ΩA, B).

Let α̃: H(A) ⟶ H(B) be in Ker ϑ_{A,B}, i.e. α′ ^{∘} g = 0. Then there exists a morphism ρ: A ⟶ B such that α′ = ρ^{∘} f. Then H(α′) = H(ρ)^{∘} H(f) = α̃^{∘} H(f), hence α̃ = H(ρ). The morphism ρ is uniquely determined up to phantom maps. Indeed if α′ = σ^{∘} f, then the morphism ρ − σ factors through the phantom map h, hence it is phantom. It follows that the map 𝒯(A, B)/Ph(A, B) ⟶ Hom_{Γ}(H(A), H(B)) induced by H_{A,B} is the kernel of ϑ_{A,B}, and this proves the exactness of the first part of the sequence (2.1).

We now show that ζ_{A,B} is surjective. So let β be in Ph^{2}(Σ^{−1}A, B). Then

there exists a factorization β = β_{2} ^{∘} β_{1} : Σ^{−1}A ⟶^{β_{1}} X ⟶^{β_{2}} B where the β_{i} are phantoms. Since β_{1} is phantom, the composition β_{1} ^{∘} Σ^{−1}f is zero. Hence there exists a morphism γ: ΩA ⟶ X such that β_{1} = γ^{∘} Σ^{−1}h. Consider the phantom map δ = β_{2} ^{∘} γ: ΩA ⟶ B. Then by construction we have ζ_{A,B}(δ) = δ^{∘} Σ^{−1}h = β. We infer that ζ_{A,B} is surjective.

It remains to prove that Ker ζ_{A,B} = Im ϑ_{A,B}. First let α: ΩA ⟶ B be a phantom map such that ζ_{A,B}(α) = α^{∘} Σ^{−1}h = 0. Then there exists a morphism β: P ⟶ B such that α = β^{∘} g. Since α is phantom we have 0 = H(α) = H(β)^{∘} H(g). Hence there exists a unique morphism γ: H(A) ⟶ H(B) such that γ^{∘} H(f) = H(β). Then by construction ϑ_{A,B}(γ) = β^{∘} g = α. Hence α lies in Im ϑ_{A,B}. Finally let α: ΩA ⟶ B be in the image of ϑ_{A,B}, that is α = α′ ^{∘} g where α′: P ⟶ B is the unique morphism such that H(α′) = α̃^{∘} H(f) for some morphism α̃: H(A) ⟶ H(B). Then ζ_{A,B}(α) = α^{∘} Σ^{−1}h = α′ ^{∘} g^{∘} Σ^{−1}h = 0. Hence ϑ_{A,B}(α) ∈ Ker ζ_{A,B}. We infer that Ker ζ_{A,B} = Im ϑ_{A,B}, hence the sequence (2.1) is exact. The naturality of (2.1) follows from the preceding discussion in this section

The above result suggest the following definition.

Definition 2.7. Let A, B be objects of 𝒯. The obstruction group 𝒪_{A,B} of A and

B is the cokernel of the map 𝒯(A, B) ⟶^{H_{A,B}} Hom_{Γ}(H(A), H(B)).

Thus we have an isomorphism 𝒪_{A,B} ≅ Ker ζ_{A,B} = Im ϑ_{A,B}. Next we construct a sequence which computes Ext_{Γ}^{n+1}(H(A), H(B)).

Proposition 2.8. For any objects A, B in 𝒯 and n ≥ 0, there exist a natural map

φ_{Ω^{n}A,B}: Ph(Σ^{−1}Ω^{n}A, B) ⟶ Ext_{Γ}^{n+1}(H(A), H(B))

which induces a functorial exact sequence

0 ⟶ Ph^{2}(Σ^{−1}Ω^{n}A, B) ⟶Ph(Σ^{−1}Ω^{n}A, B) ⟶^{φ_{Ω^{n}A,B}}  (2.2)
Ext_{Γ}^{n+1}(H(A), H(B)) ⟶ 𝒪_{Ω^{n+1}A,B} ⟶ 0 .

Proof. The exact triangle ΩA ⟶^{g} P ⟶^{f} A ⟶^{h} ΣΩA induces the following exact commutative diagram

⋯ ⟶  𝒯(A, B)  ⟶^{𝒯(f,B)}  𝒯(P, B)  ⟶^{𝒯(g,B)}
H_{A,B} ↓  ^{H_{P,B}} ↓≅
0 ⟶ Hom_{Γ}(H(A), H(B)) \underrightarrow{(H(f),H(B))} Hom_{Γ}(H(P), H(B)) \underrightarrow{(H(g),H(B))}
𝒯(ΩA, B)  ⟶^{ε} Im𝒯(Σ^{−1}h, B) ⟶ 0
H_{ΩA,B} ↓  ^{φ}↓
Hom_{Γ}(H(ΩA), H(B)) ⟶^{δ} Ext_{Γ}^{1}(H(A), H(B)) ⟶ 0

in which the morphism H_{P,B} is invertible. Therefore we have an exact sequence 0 ⟶ 𝒪_{A,B} ⟶ Ker ε ⟶ Ker δ ⟶ 0. Next observe that Im𝒯(Σ^{−1}h, B) = Ph(Σ^{−1}A, B). Using the property of ζ_{A,B} from Proposition 2.6, the right hand square induces the following exact commutative diagram

0  0  0
↓  ↓  ↓
0 ⟶ 𝒪_{A,B} ⟶  Ph(ΩA, B)  ⟶^{ζ_{A,B}} Ph^{2}(Σ^{−1}A, B) ⟶ 0
↓  ↓  ↓
0 ⟶ Ker ε ⟶  𝒯(ΩA, B)  ⟶ Ph(Σ^{−1}A, B) ⟶ 0
↓  ↓  ^{φ_{A,B}} ↓
0 ⟶ Ker δ ⟶ Hom_{Γ}(H(ΩA), H(B)) ⟶ Ext_{Γ}^{1}(H(A), H(B)) ⟶ 0
↓  ↓  ↓
0  𝒪_{ΩA,B}  ⟶^{=}  𝒪_{ΩA,B}
↓  ↓
0  0

The right hand vertical exact sequences is the desired sequences for n = 0. The general case is obtained by replacing A with Ω^{n}A for n ≥ 1.

Remark 2.9. The map φ_{A,B}: Ph(Σ^{−1}A, B) → Ext_{Γ}^{1}(H(A), H(B)) has an explicit description which we include for later reference. Let α: Σ^{−1}A → B be a phantom map. We complete this to a triangle Σ^{−1}A → B → C → A and apply H to obtain an exact sequence ε: 0 → H(B) → H(C) → H(A) → 0. We have φ_{A,B}(α) = ε.

Combining the exact sequences (2.1) and (2.2) we have the following.

Corollary 2.10. For any objects A, B in 𝒯 and any n ≥ 1, there exists an exact sequence

0 ⟶ Ph^{2}(Σ^{−1}Ω^{n−1}A, B) ⟶Ph(Σ^{−1}Ω^{n−1}A, B) ⟶^{φ_{Ω^{n−1}A,B}}
Ext_{Γ}^{n}(H(A), H(B)) ⟶Ph(Ω^{n+1}A, B) ⟶^{ζ_{Ω^{n}A,B}} Ph^{2}(Σ^{−1}Ω^{n}A, B) ⟶ 0 .

We end this section with a remark which is useful for the computation of the ideal Ph(−, −).

Remark 2.11. Let A ⟶^{α} B ⟶^{β} C ⟶^{γ} ΣA be an exact triangle. Then for any X ∈ 𝒯, the following sequences of phantom ideals are exact

Ph(C, X) ⟶^{β_{∗}} Ph(B, X) ⟶^{α_{∗}} Ph(A, X)

Ph(X, A) ⟶^{α^{∗}} Ph(X, B) ⟶^{β^{∗}} Ph(X, C) .

Indeed, let ρ: B ⟶ X be a phantom map such that α_{∗}(ρ) = ρ^{∘} α = 0. Then there exists a morphism σ: C ⟶ X such that σ^{∘} β = ρ. Hence β_{∗}(σ) = ρ and it suffices to show that σ is phantom. But H(σ)^{∘} H(β) = H(ρ) = 0. Hence H(σ) = 0 since H(β) is an epimorphism. It follows that σ is phantom. The exactness of the second sequence is proved similarly.

3. Realizability of morphisms

We say that a morphism α̃: H(A) ⟶ H(B) in Mod(Γ) is realizable if there exists a morphism α: A ⟶ B in 𝒯 such that α̃ = H(α). The exact sequence

𝒯(A, B) ⟶^{H_{A,B}} Hom_{Γ}(H(A), H(B)) ⟶ 𝒪_{A,B} ⟶ 0

shows that the map α̃: H(A) ⟶ H(B) in Mod(Γ) is realizable if and only if its image in the obstruction group 𝒪_{A,B} vanishes. Hence all maps between H(A) and H(B) are realizable if and only if 𝒪_{A,B} = 0. We denote by 𝒪_{A,−}: 𝒯 ⟶ 𝒜b the functor defined by 𝒪_{A,−}(B) = 𝒪_{A,B}. The functor 𝒪_{−,B} is defined dually. Note that 𝒪_{A,−} preserves all products, whereas 𝒪_{−,B} preserves all coproducts.

Our aim in this section is a characterization of the objects A satisfying 𝒪_{A,−} = 0. We start with some preparations. The Jacobson radical of a ring Λ is denoted by Rad Λ.

Lemma 3.1. Let A be an object in 𝒯. Then the following are equivalent. (i) Ph^{2}(A, A) = 0 and 𝒪_{A,−} = 0. (ii) Ph(A, A) ⊆ Rad 𝒯(A, A) and 𝒪_{A,−} = 0. (iii) pd A ≤ 1.

Proof. (i) ⇒ (ii): Clear, since Rad 𝒯(A, A) contains any nilpotent ideal of 𝒯(A, A).

(ii) ⇒ (iii): Consider the following morphism of triangles which is induced by the maps from a projective resolution of A

P_{1} ⟶^{δ_{1}} P_{0} ⟶^{ε} B ⟶ ΣP_{1}
f_{1}↓  ∥  ^{β}↓  ^{Σf_{1}}↓
ΩA ⟶^{g_{0}} P_{0} ⟶^{f_{0}} A ⟶^{h_{0}} ΣΩA

Applying to the above diagram the functor H we see directly that there exists a morphism α̃: H(A) ⟶ H(B) such that α̃^{∘} H(f_{0}) = H(ε). Since 𝒪_{A,B} = 0, there exists a map α: A ⟶ B such that H(α) = α̃. Moreover, α^{∘} f_{0} = ε by Lemma 2.1. We obtain therefore a second morphism of triangles

ΩA ⟶^{g_{0}} P_{0} ⟶^{f_{0}} A ⟶^{h_{0}} ΣΩA
τ↓  ∥  ^{α}↓  ^{Στ}↓
P_{1} ⟶^{δ_{1}} P_{0} ⟶^{ε} B ⟶ ΣP_{1}

The equality ε^{∘}(β^{∘} α) = ε implies H(β^{∘} α) = 1_{H(A)}. Therefore 1_{A} − β^{∘} α belongs to Ph(A, A). Using the hypothesis Ph(A, A) ⊆ Rad 𝒯(A, A), we infer that β^{∘} α is invertible. Thus f_{1} ^{∘} τ is invertible, and therefore ΩA is projective. We conclude that pd A ≤ 1.

(iii) ⇒ (i): Since ΩA is projective, the exact sequence (2.1) shows that 𝒪_{A,−} = 0 and Ph^{2}(A, −) = Ph^{2}(Σ^{−1}A, −) = 0.

We denote by 𝒯′ the localizing subcategory of 𝒯 which is generated by the compact object T representing H. That is, 𝒯′ is the smallest full triangulated subcategory containing T which is closed under taking arbitrary coproducts. This category is compactly generated, and therefore the inclusion functor 𝒯′ → 𝒯 has a right adjoint R: 𝒯 → 𝒯′; see [9]. For each object A ∈ 𝒯, we denote by ε_{A}: R(A) → A the corresponding counit.

Lemma 3.2. Let A be an object in 𝒯′.

(i) Ph(A, A) ⊆ Rad 𝒯(A, A). (ii) pd A = pd_{Γ} H(A).

Proof. (i) This follows from the fact that H reflects isomorphisms between objects in 𝒯′.

(ii) We have always pd A ≥ pd H(A). Now suppose A belongs to 𝒯′. It is not difficult to see that A is projective if and only H(A) is projective. The equality pd A = pd H(A) then follows by induction.

The following result characterizes the realizability of morphisms, using homological conditions in the module category Mod(Γ).

Theorem 3.3. Let A be an object of 𝒯. Then the following are equivalent. (i) For any object B in 𝒯, all maps H(A) ⟶ H(B) are realizable. (ii) 𝒪_{A,−} = 0. (iii) pd_{Γ} H(A) ≤ 1 and the counit ε_{A}: R(A) → A is a section. (iv) pd_{Γ} H(A) ≤ 1 and there is a decomposition A = A′ ⊕ A′′ such that A′ belongs

to the localizing subcategory which is generated by the object representing H, and H(A′′) = 0.

Proof. (i) ⇔ (ii): This follows from the definitions.

(ii) ⇒ (iii): The counit ε_{A}: R(A) → A induces an isomorphism H(R(A)) ⟶^{≅} H(A). Therefore 𝒪_{R(A),−} = 0. Combining Lemma 3.1 and 3.2 we obtain

pd_{Γ} H(A) = pd_{Γ} H(R(A)) = pd R(A) ≤ 1.

The inverse for the counit ε_{A}: R(A) → A is obtained by realizing the inverse of the isomorphism H(ε_{R(A)}).

(iii) ⇒ (iv): Take A′ = R(A) and let A′′ be the cofibre of the counit ε_{A}. This gives a decomposition A = A′ ⊕ A′′.

(iv) ⇒ (ii): We apply again Lemma 3.1 and 3.2. We have pd_{Γ} H(A′) = pd_{Γ} H(A) ≤ 1 and therefore 𝒪_{A′,−} = 0. Using the canonical map A → A′, we see that 𝒪_{A,−} = 0.

The following example shows that we cannot expect any homomological criterion on H(A) which decides the realizability of maps H(A) → H(B).

Example 3.4. Let Λ be the ring of upper triangular 2 × 2 matrices over a field k, and let 𝒯 = 𝒟(Mod(Λ)) be the derived category of unbounded complexes over Λ. We number the indecomposable projective Λ-modules P_{1} and P_{2} such that Hom_{Λ}(P_{1}, P_{2}) ≅ k and view them as complexes concentrated in degree 0. Let H = 𝒯(P_{1}, −)_{∗}. Then Γ = 𝒯(P_{1}, P_{1})_{∗} ≅ k and the non-zero map α: P_{1} → P_{2} induces an isomorphism H(P_{1}) ⟶^{≅} H(P_{2}). Clearly, H(P_{2}) ≅ k is a projective Γmodule. However, the map H(α)^{−1} is not realizable since 𝒯(P_{2}, P_{1}) = 0.

4. A dual result

In this section we discuss the dual statement of Theorem 3.3. Thus we characterize the fact that for a fixed object B in 𝒯 all maps H(A) → H(B) are realizable. This requires an extra assumption on the triangulated category 𝒯. Throughout this section we assume that 𝒯 satisfies Brown representability for cohomology theories, that is, for every exact contravariant functor F: 𝒯 → 𝒜b which sends coproducts in 𝒯 to products, there exists an object X in 𝒯 such that F ≅ 𝒯(−, X). For example, a compactly generated triangulated category satisfies Brown representability for cohomology theories [9]. In particular, all examples from Section 1 have this property.

An immediate consequence of our assumption on 𝒯 is the fact that every object in 𝒯 has an injective presentation. In addition, we have the following lemma.

Lemma 4.1. There exists a localization functor L: 𝒯 → 𝒯 such that L(A) = 0 if and only if H(A) = 0 for every object A in 𝒯.

Proof. Recall from Section 3 that the inclusion 𝒯′ → 𝒯 for the localizing subcategory 𝒯′ generated by T has a right adjoint R: 𝒯 → 𝒯′. This right adjoint preserves coproducts. Thus R has a right adjoint R′: 𝒯′ → 𝒯 since 𝒯 satisfies Brown representability. Now put L = R′ ^{∘} R.

We are now in a position to state the analogue of Theorem 3.3. To this end we say that an object A in 𝒯 is H-local if the natural map A → L(A) is an isomorphism.

Theorem 4.2. Let B be an object of 𝒯. Then the following are equivalent. (i) For any object A in 𝒯, all maps H(A) ⟶ H(B) are realizable. (ii) 𝒪_{−,B} = 0. (iii) id_{Γ} H(B) ≤ 1 and the natural map B → L(B) is a retraction. (iv) id_{Γ} H(B) ≤ 1 and there is a decomposition B = B′ ⊕ B′′ such that B′ is

H-local and H(B′′) = 0.

The proof of this result is similar to that of Theorem 3.3. It is therefore omitted.

5. Realizing extensions

Now we turn our attention to the problem of realizing short exact sequences.

We call an exact sequence 0 ⟶ H(B) \underrightarrow{β̃} E \underrightarrow{α̃} H(A) ⟶ 0 in Mod(Γ) realizable if there exists a triangle B ⟶^{β} C ⟶^{α} A ⟶ ΣB in 𝒯 and a commutative diagram

0 ⟶ H(B) ⟶^{H(β)} H(C) ⟶^{H(α)} H(A) ⟶ 0
∥  ≅↓  ∥
0 ⟶ H(B) \underrightarrow{β̃} E \underrightarrow{α̃} H(A) ⟶ 0

Our analysis of realizable sequences is based on the long exact sequence

0 ⟶ Ph^{2}(Σ^{−1}A, B) ⟶ Ph(Σ^{−1}A, B) ⟶^{φ_{A,B}}  (5.1)
Ext_{Γ}^{1}(H(A), H(B)) ⟶ 𝒪_{ΩA,B} ⟶ 0

which is obtained from the sequence (2.2). Using the description of the map φ_{A,B}, we get the following lemma.

Lemma 5.1. A short exact sequence ε: 0 → H(B) → E → H(A) → 0 is realizable if and only if ε belongs to the image of the map φ_{A,B}. Moreover, the map A → ΣB in a triangle B → C → A → ΣB realizing ε is unique up to a map in Ph^{2}(A, ΣB).

Proof. Combine the description of φ_{A,B} from Remark 2.9 and the long exact sequence (5.1).

We apply again the exactness of the sequence (5.1) and obtain the following criterion for realizability.

Corollary 5.2. Let A, B be objects in 𝒯. Then every exact sequence 0 → H(B) → E → H(A) → 0 in Mod(Γ) is realizable if and only if 𝒪_{ΩA,B} = 0.

6. Applications

6.1. Tate cohomology. We fix a finite group G and a field k. Let 𝒯 = \underline{Mod}(kG) be the stable module category over the group algebra kG and let H = Ĥ^{∗}(G, −) be the functor which assigns to each kG-module its Tate cohomology. Note that H = \underline{Hom}_{kG}(k, −)_{∗}. Theorem 1 tells us when all maps starting in a fixed module Ĥ^{∗}(G, A) over the Tate cohomology ring are realizable. It is clear that Theorem 1 is the reformulation of our general Theorem 3.3 in this setting. Moreover, Theorem 1 has an analogue for maps ending in a fixed module Ĥ^{∗}(G, B) which is a consequence of Theorem 4.2.

Next we apply Corollary 5.2 and consider extensions over the Tate cohomology ring. We assume for simplicity that G is a p-group where p denotes the characteristic of the field k.

Corollary 6.1. For a kG-module A the following are equivalent:

(i) Each exact sequence 0 → Ĥ^{∗}(G, B) → E → Ĥ^{∗}(G, A) → 0 of graded Ĥ^{∗}(G, k)-modules is realizable (for every kG-module B).

(ii) Each exact sequence 0 → Ĥ^{∗}(G, B) → E → Ĥ^{∗}(G, A) → 0 of graded Ĥ^{∗}(G, k)-modules is of the form Ĥ^{∗}(G, ε) for some exact sequence ε: 0 → B → C → A → 0 of kG-modules (for every kG-module B).

(iii) pd_{Ĥ^{∗}(G,k)} Ĥ^{∗}(G, A) ≤ 2.

Proof. We combine Theorem 3.3 and Corollary 5.2. The condition 𝒪_{ΩA,−} = 0 is equivalent to Ĥ^{∗}(G, ΩA) ≅ ΩĤ^{∗}(G, A) being of projective dimension at most 1, since k generates the stable module category \underline{Mod}(kG) by our assumption on G. In addition one observes that triangles B → C → A → ΣB in \underline{Mod}(kG) correspond to exact sequences 0 → B → C′ → A → 0 in Mod(kG) via the isomorphism \underline{Hom}_{kG}(A, ΣB) ≅ Ext_{kG}^{1}(A, B).

Remark 6.2. In Theorem 1 and all its applications, the question arises when a kGmodule belongs to the subcategory which is generated by the trivial representation k. It is known that k generates the stable category of the principal block B_{0}(kG) if and only if the centralizer of each element of order p in G is p-nilpotent [2]. In particular, k generates all kG-modules if G is a p-group. If k generates the principal block, we have for any module A in the stable category of the principal block that all maps out of Ĥ^{∗}(G, A) are realizible if and only if the projective dimension of Ĥ^{∗}(G, A) over the Tate cohomology ring is at most 1.

6.2. Derived categories. We consider the derived category 𝒟(Mod(Λ)) of unbounded complexes of modules over some ring Λ. Le H: 𝒟(Mod(Λ)) → Mod(Λ) be the functor which takes the cohomology H^{0}A in degree 0 of a complex A. The following result is another application of Theorem 3.3.

Corollary 6.3. For any complex A in 𝒟(Mod(Λ)) the following are equivalent: (i) For any complex B in 𝒟(Mod(Λ)), all maps H^{0}A → H^{0}B are realizable. (ii) 𝒪_{A,−} = 0. (iii) pd_{Λ} H^{0}A ≤ 1. In particular, Λ is right hereditary if and only if 𝒪_{−,−} vanishes.

6.3. An extended Milnor sequence. We consider again the five-term exact sequence (2.1)

0 ⟶ Ph(A, B) ⟶ 𝒯(A, B) ⟶^{H_{A,B}}Hom_{Γ}(H(A), H(B)) ⟶^{ϑ_{A,B}}

Ph(ΩA, B) ⟶^{ζ_{A,B}} Ph^{2}(Σ^{−1}A, B) ⟶ 0 .

This sequence generalizes and extends some sequences which appear in the literature. Whenever there is an appropriate model for the triangulated category 𝒯, the sequence (2.1) is induced from the Roos spectral sequence

E_{2}^{p,q} = \underleftarrow{lim}^{p}Ext^{q}(A_{α}, B) ⟹ Ext^{n}(\underrightarrow{lim} A_{α}, B)

where {A_{α}} is a filtered system of compact objects having A as its colimit. This spectral sequence has been studied by Christensen (see [6]) in relation to purity and phantom maps in the derived category of a ring. Here, we follow Benson and Gnacadja (see [4]) and work in the stable category 𝒯 = \underline{Mod}(kG) over a group algebra kG. We choose for H: \underline{Mod}(kG) → Mod(Γ) the functor sending a module A to \underline{Hom}_{kG}(−, A)|_{\underline{mod}(kG)}. It turns out that our map

ϑ_{A,B}: Hom_{Γ}(H(A), H(B)) ⟶ Ph(ΩA, B)

can be identified with the transgression

d_{2}^{0,1} : \underleftarrow{lim} \underline{Hom}_{kG}(A_{α}, B) ⟶ PExt_{kG}^{2}(A, Σ^{−1}B) .

appearing in [4].

Let us be more precise. We denote for kG-modules A and B by PExt_{kG}^{1}(A, B) the space of pure exact sequences 0 → B → E → A → 0 in the category of kG-modules. The isomorphism Ext_{kG}^{1}(A, B) ⟶^{≅} \underline{Hom}_{kG}(A, ΣB) induces an isomorphism

PExt_{kG}^{1}(A, B) ⟶^{≅} \underline{Ph}_{kG}(A, ΣB)

onto the space of phantom maps A → ΣB in \underline{Mod}(kG). Let ΩA be the kernel of a pure epimorphism P → A where P is pure-projective. Then we obtain an isomorphism

PExt_{kG}^{2}(A, B) ⟶^{≅} PExt_{kG}^{1}(ΩA, B) ⟶^{≅} \underline{Ph}_{kG}(ΩA, ΣB).

Now write A = \underrightarrow{lim} A_{α} as a filtered colimit of finitely presented kG-modules. Then

we have an isomorphism \underrightarrow{lim} H(A_{α}) ⟶^{≅} H(A) which induces an isomorphism

Hom_{Γ}(H(A), H(B)) ⟶^{≅} \underleftarrow{lim} \underline{Hom}_{kG}(A_{α}, B).

Having made these identifications, we obtain from the five-term exact sequence (2.1) the extended Milnor sequence which appears in Theorem 1.2 of [4].

Corollary 6.4. For kG-modules A = \underrightarrow{lim} A_{α} and B, there is an exact sequence

0 ⟶ \underline{Ph}_{kG}(A, B) ⟶ \underline{Hom}_{kG}(A, B) ⟶ \underleftarrow{lim} \underline{Hom}_{kG}(A_{α}, B)
⟶^{d_{2}^{0,1}} PExt_{kG}^{2}(A, Σ^{−1}B) ⟶ \underline{Ph}_{kG}^{2}(Σ^{−1}A, B) ⟶ 0 .

References

[1] A. Beligiannis, Relative homological algebra and purity in triangulated categories, J. Algebra 227 (2000), 268–361.

[2] D.J. Benson, Cohomology of modules in the principal block of a finite group, New York J. Math. 1 (1999), 196–205.

[3] D.J. Benson and G.Ph. Gnacadja, Phantom maps and purity in modular representation theory, I, Fund. Math. 161 (1999), 37–91.

[4] D.J. Benson and G.Ph. Gnacadja, Phantom maps and purity in modular representation theory, II, Algebr. Represent. Theory 4 (2001), 395–404.

[5] D.J. Benson, H. Krause and S. Shwede, Realizability of modules over Tate cohomology rings, preprint (2001).

[6] J.D. Christensen, Ideals in triangulated categories: phantoms, ghosts and skeleta, Adv. Math. 136 (1998), 284–339.

[7] J.D. Christensen, B. Keller and A. Neeman, Failure of Brown representability in derived categories, Topology 40 (2001), 1331–1361.

[8] H. Krause, Brown representability and flat covers, J. Pure Appl. Algebra 157 (2001), 81–86.

[9] A. Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), 205–236.

Department of Mathematics, University of the Aegean, 83200 Karlovassi, Samos, Greece

  E-mail address: abeligia@mail.otenet.gr, abeligia@aegean.gr

Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany

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