A Presheaf Interpretation of the Generalized Freyd Conjecture

In algebraic topology, the generating hypothesis, or Freyd conjecture, is a longstanding conjecture about the structure of the stable homotopy category. It was initially formulated in 1965 [5] and remains open to this day. Because the original conjecture has proved difficult to analyze, recent work has turned to studying similar conjectures in categories that share many properties with the stable homotopy category in hopes of further understanding the types of categories in which such a conclusion holds. In this note, we state a version of the generating hypothesis for an arbitrary triangulated category and analyze conditions under which this hypothesis holds. We emphasize that we impose no additional conditions on our triangulated categories, so that our results show which formal properties of a category imply the Freyd conjecture. Therefore our results give conceptual insight into the kind of category in which the generating hypothesis can be expected to hold. Let T be a triangulated category and write [X, Y ] for the abelian group of maps X → Y in T . Let B be a (small) full subcategory of T closed under its translation (or shift) functor Σ and let C be the thick full subcategory of T that is generated by B; write ι : B → C for the inclusion. For emphasis, we often write B(X, Y ) = [X, Y ] when X, Y ∈ B and C (X, Y ) = [X, Y ] when X, Y ∈ C . The category B is pre-additive (enriched over Ab), and C is additive (has biproducts). Let PB and PC denote the categories of abelian presheaves defined on B and C . They consist of the additive functors from B op or C op to Ab and the additive natural transformations between them. Conjecture 1 (The generalized Freyd conjecture). The functor F : C → PB is faithful. Equivalently, Ff = 0 if and only if f = 0. We then say that the Freyd conjecture holds for the pair (C , B). We hasten to add that the conjecture is false in this generality. Additional hypotheses are needed, but none will be relevant to our formal analysis. For example, we might as well assume that a map f : X → Y in C is an isomorphism if and only f * : FX → FY is an isomorphism, as holds in the stable homotopy category. This condition is necessary but not sufficient for the Freyd conjecture to hold. Indeed, if Z is the third term in an exact triangle with one map f , then f is an isomorphism if and only if Z = 0 and Ff is an isomorphism if and only if FZ = 0. If FZ = 0 and the Freyd conjecture is true, then Z = 0 since its identity map is the zero map. Example 2. Take T = HoS to be the stable homotopy category. Let B consist of the sphere spectra S n for integers n. Then C is the homotopy category of finite CW spectra. Freyd [5] conjectured that a map f in C is zero if and only if it induces the zero homomorphism f * : π * (X) → π * (Y ). By the following observation, this is a special case of our Conjecture 1. Lemma 1. In Example 2, the category PB is isomorphic to the category M of right modules over the ring π * of stable homotopy groups of spheres. Under this isomorphism, the Freyd functor F coincides with the stable homotopy group functor π * : The additive contravariant functor T gives homomorphisms By adjunction, these give homomorphisms T n ⊗π m-n → T m . The functoriality gives the formula (tx)y = t(xy). No signs appear since we are taking right modules, as is dictated categorically by contravariance. Conversely, given a right π * -module M , define T (S n ) = M n and, for t ∈ M n and x ∈ π m-n ∼ = B(S m , S n ) define T (x)(t) = tx ∈ M m . The module axioms ensure that T is a functor. This specifies the isomorphism of categories, and the consistency of F and π * is clear. The following example originally prompted us to take a presheaf perspective on the Freyd conjecture. Example 3. Let G be a compact Lie group and take T = HoGS to be the equivariant stable homotopy category. Let GB consist of the orbit G-spectra for integers n and closed subgroups H of G. The thick subcategory GC generated by GB is the category of retracts of finite G-CW spectra. The equivariant version of the Freyd conjecture asserts that a map f in GC is zero if and only if it induces the zero homomorphism Again, this is a special case of our Conjecture 1. This example is the focus of [2]. Composition is induced by suspension and composition in GB in the evident fashion. Define a right Gπ * -module M to be a graded presheaf, that is, a contravariant functor Gπ * → Ab * , where Ab * is the category of graded abelian groups. Lemma 2. In Example 3, the category PGB is isomorphic to the category GM of right Gπ * -modules. Under this isomorphism, the Freyd functor F coincides with the equivariant stable homotopy group functor π (-) * : HoGS → GM . The proof is the same as that of Example 2. In this case, the presheaf formulation of the Freyd conjecture appears more natural to us than the equivalent homotopy group reformulation. Example 6. In the rational equivariant stable homotopy category, the Freyd conjecture is true if G is finite [6] and is false if Here is another example where the presheaf formulation, and the use of many objects and their shifts rather than a single object and its shifts in B, may be more natural than the formulation of the Freyd conjecture studied so far. Example 7. In several recent papers, Benson, Carlson, Chebolu, Christensen and Mináč [1,3,4] study the Freyd conjecture in the stable module category of a finite group G over a field k whose characteristic divides the order of G. This is a triangulated category StMod kG obtained from the category of kG-modules by modding out by maps that factor through a projective module. They restrict to the thick subcategory generated by the trivial representation k and ask whether the Tate cohomology functor is faithful on this subcategory. Since the Tate cohomology of a module M is given by maps out of k in the stable module category, Ĥi (G, M ) = Stmod kG (Ω i k, M ), their formulation is equivalent to our Conjecture 1 with B = {Ω i k}. In this context, they prove that the Freyd conjecture holds in the stable module category of kGmodules if and only if the p-Sylow subgroup of G is either C 2 or C 3 , where p is the characteristic of k [3]. Their proof in fact shows that this variant of the Freyd conjecture holds if and only if the thick subcategory generated by k consists of finite direct sums of suspensions of k. For p-groups, the trivial module is the only irreducible module over kG, and the thick subcategory generated by k is the subcategory of compact objects in the stable module category. For non-p-groups, other irreducible modules exist. It is thus natural to generalize the Freyd conjecture to our presheaf context by letting B be the category whose objects are the suspensions of the irreducible kG-modules. The thick subcategory C generated by this B is then the subcategory of compact objects in the stable module category and one can ask when Conjecture 1 holds. This presheaf version of the generating hypothesis in StMod kG takes into account all the generating objects in the stable module category. The presheaf perspective suggests a method of attack on the generalized Freyd conjecture. We have the forgetful functor given by restricting presheaves defined on C op to the full subcategory B op . The functor U has a left adjoint prolongation functor For T ∈ PB and K ∈ C , PT (K) is the categorical tensor product Observe that the Freyd functor F : T → PB is the composite UY. This leads to the following observation. Proposition 1. The Freyd conjecture holds for (C , B) if Proof. Since the unit of the adjunction (P, U) is an Uε an isomorphism by one of the triangle identities. Thus F ∼ = UPF. Therefore, for a map f : X → Y in C , Ff = 0 if and only if PFf = 0. By the Yoneda lemma, f = 0 if and only if Yf = 0. If ε is an isomorphism, then PFf = 0 if and only if Yf = 0. In fact, less is needed. Consider YX(X) = [X, X]. We will shortly prove the following result. Proposition 2. The Freyd conjecture holds for (C , B) if the identity map of X is in the image of ε for all X ∈ C . We have the following starting point towards verification of the hypothesis. Lemma 3. The map ε : PFX(J) → YX(J) is an isomorphism for all X ∈ T and J ∈ B. Proof. We have seen that Uε is an isomorphism, and by definition UT (J) = T (J) for any J ∈ B and any T ∈ PC . K / / L / / M / / ΣK in T , where K, L, and M are in C . We have the commutative diagram Since YX(K) = [K, X], the lower row is exact. By definition, C is the smallest subcategory of T that contains B, is closed under retracts, and has the property that if two terms of an exact triangle are in C then so is the third. By an easy retract argument and the five lemma, this gives the following conclusion. 2) is exact for every exact triangle (1) and every X ∈ C , then ε : PFX → YX is an isomorphism for every X ∈ C and the Freyd conjecture holds for (C , B). There is a reinterpretation of the exactness hypothesis that makes it reminiscent of the standard result that the adjoint (if it exists) of an exact functor between triangulated categories is exact. For K and X in C , the abelian group PFX(K) is the coequalizer in Ab displayed in the diagram where the parallel arrows are given by composition in C . We use this to interpolate the proof of Proposition 2. The composition maps and the following diagram commutes: Take K = X and suppose that ε(z) = id X (for the top map ε). But in this dual reformulation, the exactness hypothesis on K for fixed X is now a levelwise exactness statement about the composite functor takes exact triangles in the variable K to exact sequences for each fixed J. Thus a more general question to ask is whether or not P d : is exact for all X ∈ C ? Observe that we have not yet used any hypothesis on B, other than that it generates the thick subcategory C of the triangulated category T . Thus all that we have done is to give a purely formal reduction of the general problem. Our framework for understanding the Freyd conjecture leads to a transparent proof of the result of Lockridge [10, 3.9] that the generalized Freyd conjecture holds in the derived category of a von Neumann regular ring. We simply observe that the hypotheses of Proposition 3 hold in this case by one of the equivalent definitions of a von Neumann regular ring. However, our methods do not prove the converse, which is proven in [7,10]. Using right R-modules for definiteness, let D(R) be the derived category of a ring R and let B be the full subcategory of D(R) whose objects are the shifts Σ i R of the chain complex that is R concentrated in degree 0. Then C is the category of perfect chain complexes, namely those isomorphic in D(R) to bounded chain complexes of finitely generated projective R-modules. The Freyd functor assigns the homology groups B(Σ i R, X) = H i X to a chain complex X, and Conjecture 1 is the assertion that a map between perfect chain complexes is 0 in D(R) if it induces the zero map on homology. Defining H i (K) = C (K, Σ i R), as usual, we have the following observation in this case. Lemma 4. For perfect chain complexes K and X, PFX(K) is isomorphic to the abelian group There are no maps Σ specify the right action of R on H i (X). Therefore our coequalizer diagram can be rewritten as The conclusion follows. Proposition 4. If the homology R-modules H i (X) of any perfect chain complex X are R-flat, then the Freyd conjecture holds for the pair (C , B) in D(R). Proof. Let K → L → M → ΣK be an exact triangle in D(R). Since the functor C (-, Σ i R) takes exact triangles to exact sequences, the sequence is exact. By our flatness hypothesis, this sequence remains exact on tensoring with each H i (X). By Lemma 4, when we take the direct sum over i of these sequences, we obtain the exact sequence The conclusion follows from Proposition 2.6. By one definition, the ring R is von Neumann regular if every R-module is flat. Corollary 1. The Freyd conjecture holds for the derived category D(R) of a von Neumann regular ring R. We have the following more general analogous result. Corollary 2. The Freyd conjecture holds for the derived category D(R) if R has weak dimension at most one and all finitely generated submodules of finitely generated projective R-modules are FP-injective. The ring R has weak dimension at most one if all submodules of flat modules are flat. A module M is FP-injective if Ext 1 (F, M ) = 0 for all finitely presented modules F . By [8,Theorem 4.89], M is FP-injective if and only if any short exact sequence of right modules 0 → M → L → N → 0 is pure exact, meaning that it remains exact on tensoring with any left module. Proof. Let R satisfy the stated hypotheses. We show that the homology modules H i (X) of a perfect complex X are flat. Proposition 4 then applies. Consider the short exact sequence 0 → B i (X) → Z i (X) → H i (X) → 0. (5) The module Z i (X) is flat since it is a submodule of the projective, hence flat, module X i . By [8,Corollary 4.86], it follows that H i (X) is flat if and only if the sequence ( 5) is pure exact. The module B i (X) is finitely generated since it is a quotient of the finitely generated module X i+1 , and it is a submodule of the finitely generated projective R-module X i . By assumption, this implies that B i (X) is FP-injective. Therefore ( 5) is pure exact and H i (X) is flat. This elementary result comes close to one direction of the best possible result about D(R), which is proven in [7, Theorem 2.1] and states that the Freyd conjecture holds for D(R) if and only if R has weak dimension at most one and all finitely presented R-modules are FP-injective.