Graded comodule categories with enough projectives

It is well-known that the category of comodules over a flat Hopf algebroid is abelian but typically fails to have enough projectives, and more generally, the category of graded comodules over a graded flat Hopf algebroid is abelian but typically fails to have enough projectives. In this short paper we prove that the category of connective graded comodules over a connective, graded, flat, finite-type Hopf algebroid has enough projectives. Applications to algebraic topology are given: the Hopf algebroids of stable co-operations in complex bordism, Brown-Peterson homology, and classical mod $p$ homology all have the property that their categories of connective graded comodules have enough projectives. We also prove that categories of connective graded comodules over appropriate Hopf algebras fail to be equivalent to categories of graded connective modules over a ring.


Introduction.
Let (A, Γ) be a graded Hopf algebroid (that is, a cogroupoid object in the category of graded-commutative rings) such that Γ is flat over A. Then the category of graded Γ-comodules is abelian, and homological algebra in this category is of central importance in algebraic topology, since the input for generalized Adams spectral sequences is a (relative) Ext functor in a category of graded Γ-comodules; see chapters 2 and 3 of [14] for a textbook account of this material. Appendix 1 of [14] is the standard reference for Hopf algebroids and homological algebra in their comodule categories.
Some homological constructions in comodule categories are made problematic, however, by the lack of enough projectives. It is well-known that the category of comodules over a Hopf algebroid typically fails to have enough projectives; even when A is a field and Γ a Hopf algebra over A, the category of Γ-comodules has enough projectives if and only if Γ is semiperfect, i.e., every simple comodule has an injective hull which is finite-dimensional as an A-vector space. (This result is attributed by B. I. Lin, in [10], to unpublished work of Larson, Sweedler, and Sullivan; the generalization of this result which replaces Hopf algebras with coalgebras is a result of Lin's, from the same paper.) Here is an example: in the paper [7] (see the Remark preceding Proposition 1.2.3), M. Hovey shows that the category of comodules over the Hopf algebra Q[x], with x primitive, has the property that infinite products fail to be exact. That is, Grothendieck's axiom AB4 * fails in this category of comodules. It is standard that a complete abelian category which has enough projectives also satisfies axiom AB4 * (see e.g. Lemma A.3.15 of [13]), so this category of comodules cannot have enough projectives. Hovey's example also works in the graded case (although, crucially, not the connective graded case, if x is in a positive grading degree).
The purpose of this short paper is to prove that, under some reasonable assumptions (which are satisfied in cases of topological interest), appropriate categories of graded comodules over graded Hopf algebroids do have enough projectives. The essential point is to work with connective graded comodules, that is, graded comodules which are trivial in all negative grading degrees; the category of connective graded comodules over the Hopf algebra Q[x] of Hovey's example does have enough projectives, and much more generally, our main result is Theorem 3.8: Theorem. Let (A, Γ) be a connective finite-type flat graded Hopf algebroid. Then the category of connective graded Γ-comodules is a Grothendieck category with a projective generator. Consequently, the category of connective graded Γ-comodules has enough projectives and enough injectives, and satisfies Grothendieck's axiom AB4 * (that is, infinite products exist and are exact).
However, if A is not the zero ring, then this category of connective graded Γcomodules fails to have a compact projective generator, so it is not equivalent to the category of (ungraded) modules over any ring. 1 This is proven in Proposition 3.9.
Of course it is then natural to ask whether the category of Γ-comodules might be equivalent to the category of connective graded modules over a connective graded ring. This takes a bit more work: in Theorem 4.7 we show, under some reasonable hypotheses on Γ, that the category of graded connective Γ-comodules cannot be equivalent by a suspension-preserving equivalence to the category of connective graded modules over a ring.
An amusing consequence is Corollary 4.8: the category of connective graded comodules over the mod p dual Steenrod algebra is not equivalent, via a suspensionpreserving functor, to the category of connective graded modules over a ring. Nevertheless, that comodule category does have enough projectives, by Theorem 3.8. Some terminology used above may not be immediately familiar. The relevant definitions are as follows: • a graded Hopf algebroid (A, Γ) is flat if Γ is flat over A, • connective if A and Γ are concentrated in nonnegative grading degrees (i.e., (A, Γ) is N-graded, not just Z-graded), 1 As a peculiar but elementary special case, which must certainly already be well-known: if A = Γ with trivial grading, the category of connective graded A-modules-which is, of course, isomorphic to a countable infinite product of copies of the category Mod(A)-is not equivalent to the category of modules over a ring.
• and finite-type if there exists an exact sequence of graded A-modules for some sequences of integers (. . . , a −1 , a 0 , a 1 , a 2 , . . . ) and (. . (Of course, if Γ is also connective, then a i and b i each must vanish for sufficiently small i.) Following the usual convention in topology, we write Σ for the suspension operator, i.e., ΣA is A with all grading degrees increased by one.
Special cases of Theorem 3.8 include some of the most important Hopf algebroids for topological applications, as we see in Corollary 3.10: Corollary. The categories of connective graded comodules over the Hopf algebroids (M U * , M U * M U ), (BP * , BP * BP ), and ((HF p ) * , (HF p ) * HF p ) all have enough projectives.
These Hopf algebroids are very well-known in algebraic topology: (M U * , M U * M U ) is the Hopf algebroid of stable natural co-operations of the complex bordism functor M U * , (BP * , BP * BP ) is the Hopf algebroid of stable natural co-operations of the p-local Brown-Peterson homology functor BP * , and ((HF p ) * , (HF p ) * HF p ) is the mod p dual Steenrod algebra, i.e., the Hopf algebra of stable natural co-operations of the mod p classical homology functor (HF p ) * . These are the Hopf algebroids whose comodule categories have the most important homological invariants: appropriate relative Ext groups over these three Hopf algebroids recover the E 2 -terms of the global Adams-Novikov, p-local Adams-Novikov, and classical p-primary Adams spectral sequences, respectively. See chapters 2, 3, and 4 of [14] for this material.
I am grateful to G. Valenzuela for useful conversations relating to this material, and to A. Baker and an anonymous referee for their patience with how long I took to make revisions on this paper.

2.
When does tensor product of modules commute with infinite products?
Conventions 2.1. In this paper, all gradings will be assumed to be Z-gradings. When a graded object is trivial in all negative grading degrees, we will say that the object is connective. We write N for the set of nonnegative integers.
is surjective. • For every set I, the canonical graded A-module map • As a graded A-module, Γ has finite-type generators. Proof.
• If the first condition is satisfied, then letting M i = A for all i ∈ I, we immediately get that the second condition is satisfied.
• Suppose that the second condition is satisfied. Choose an integer n, and let I be the set of homogeneous elements of Γ of grading degree exactly n. We will write i∈I Γ{e i } for the product i∈I Γ, using e i as formal symbols to index the factors in the product. Let x n ∈ i∈I Γ{e i } be the element x n = i∈I i · e i . Since the map (2.3) is grading-preserving and surjective, there exists some element mn j=1 c j,n ⊗ i∈I a i,j,n e i ∈ Γ ⊗ A i∈I A{e i } which is sent by the map (2.3) to x n , in which each c j,n is a homogeneous element of Γ and in which each a i,j,n is a homogeneous element of A. The grading degrees of these elements satisfy |c j,n | + |a i,j,n | = n, and consequently |c j,n | ≤ n.
Consequently we have the formula mn j=1 i∈I c j,n a i,j,n e i = i∈I i · e i , and consequently mn j=1 c j,n a i,j,n = i. Consequently the set of elements S = {c j,n : n ∈ Z, 1 ≤ j ≤ m n } is a set of homogeneous A-module generators for Γ. Let S n be the set {c j,n : 1 ≤ j ≤ m n } ⊆ Γ, so that S = n∈Z S n . Then each S n is finite, and, given an element of S in grading degree N , that element must be contained in S n for some n ≤ N , of which there are only finitely many, since Γ is connective. So, for each integer N , there are only finitely many elements of S of grading degree ≤ N . Hence there are only finitely many elements of S in each grading degree. Hence Γ has finite-type generators.
• Now suppose that Γ has finite-type generators, and that {M i } i∈I is a set of graded left A-modules. We need to show that map (2.2) is surjective. Choose a set of homogeneous A-module generators {c j } j∈J for Γ, with at most finitely many c j in each grading degree. Let D : J → Z be the function that sends j to the grading degree of c j . For each integer n, let Γ ≤n be the graded sub-A-module of Γ generated by all the elements c j such that D(j) ≤ n. Since A is connective and all M i are connective, the natural map Γ ≤n ֒→ Γ of graded A-modules is bijective in grading degrees ≤ n.
Write J n for the set of elements j ∈ J such that D(j) ≤ n. Now we have an exact sequence of A-modules where s(e j ) = c j ; here the elements e j are formal symbols indexing the coproduct summands. The map s now fits into the commutative square of graded A-modules where the vertical maps are the canonical comparison maps, as in map (2.2). The map s ⊗ id is a surjection, since each s ⊗ id is a surjection and since infinite direct products are exact in the category of graded A-modules. The left-hand vertical map in diagram (2.4) is an isomorphism, since J n is finite. Hence the right-hand vertical map in diagram (2.4) is also surjective. The square of graded A-modules commutes, and the horizontal maps are isomorphisms in grading degrees ≤ n, so surjectivity of the right-hand vertical map in diagram (2.4), i.e., the left-hand vertical map in diagram (2.5), tells us that the right-hand vertical map in diagram (2.5), i.e., the map (2.2), is surjective in grading degree n. But this holds for all integers n; so the map (2.2) is surjective.

Lemma 2.4. Let
A be an connective graded ring and let Γ be a connective graded left A-module. The following conditions are equivalent: is an isomorphism. • For every set I, the canonical graded A-module map Proof.
• If the first condition is satisfied, then letting M i = A for all i ∈ I, we immediately get that the second condition is satisfied.
• Suppose that the second condition is satisfied. We will write i∈I Γ{e i } for the product i∈I Γ, using e i as formal symbols to index the factors in the product. By Lemma 2.3, we know that Γ has finite-type generators. Choose an exact sequence of graded A-modules with F 0 finite-type and free. We can arrange maps as in (2.7) into a commutative diagram with exact rows 3. An easy diagram chase shows that the vertical map ( i∈I A) ⊗ A K → i∈I K is then also surjective. By Lemma 2.3, K then has finite-type generators, hence we can choose a finite-type and free graded A-module F 1 and a surjective graded A-module map F 1 → K, and consequently is an exact sequence of graded A-modules with F 1 , F 0 finite-type and free. So Γ is finite-type. • Now suppose that Γ is finite-type. First, suppose that Γ is finite-type and free. Choose a set of homogeneous A-module generators S for Γ with at most finitely many elements of S in each grading degree, and then let Γ ≤n be the graded sub-A-module of Γ generated by the elements of S of degree ≤ n. Since A and Γ and all M i are connective, the horizontal maps in the commutative square are isomorphisms in grading degrees ≤ n, and the left-hand vertical map is an isomorphism in grading degrees ≤ n, since Γ ≤n is a direct sum of finitely many copies of A (up to suspension), and finite direct sums coincide with finite products in module categories, including graded module categories. Consequently the right-hand vertical map in square (2.9) is also an isomorphism in grading degrees ≤ n. Since this is true for all n, the canonical map (2.6) is an isomorphism when Γ is finite-type and free. Now lift the assumption that Γ is finite-type and free, and assume it is only finite-type. Choose an exact sequence of graded A-modules with F 1 , F 0 finite-type and free. We can fit maps as in (2.6) into the commutative diagram of graded A-modules with exact rows Similarly, we will say that a comodule is connective if it is connective as an graded A-module. We will say that the Hopf algebroid (A, Γ) is connective if A and Γ are both connective as graded A-modules.
For another class of examples: (k, A) is flat, connective, and finite-type for any commutative graded connected finite-type Hopf algebra A over a field k (as studied in [12]). Proof. I am grateful to the anonymous referee for pointing out that this lemma follows from Lemma 2.4, above, together with a general result about limits in the category of coalgebras over a comonad, which one can find as (the dual to) Proposition 4.3.2 in [2]. I include a self-contained proof here as well, because the proof is short, direct, and (in my opinion) illuminating. Write gr ≥0 C for the connective graded objects in an abelian category C . Write G : gr ≥0 Comod(Γ) → gr ≥0 Mod(A) for the forgetful functor and E : gr ≥0 Mod(A) → gr ≥0 Comod(Γ) for its right adjoint, the extended comodule functor given by For each i ∈ I, we have the exact sequence of graded Γ-comodules, where δ 0 is the difference of the two unit maps arising from the adjunction G ⊣ E. (This is well-known; it is the reason that the cobar resolution of a comodule is indeed a resolution, as in Appendix 1 of [14]. The reader who prefers a self-contained, categorical argument may be satisfied with the observation that, for any adjunction f ⊣ g, the cofork (3.11) X / / gf X / / / / gf gf X splits after applying f ; see section VI.6 of [11]. But in our setting, f = G, the left adjoint functor f reflects isomorphisms, so the canonical map X → ker δ 0 being an isomorphism after applying f , due to the splitting of the cofork, implies that X → ker δ 0 is already an isomorphism.) Now the fact that products preserve kernels tells us that we have the commutative diagram with exact rows The maps indicated as isomorphisms are isomorphisms due to E being a right adjoint, hence preserving products. The vertical composites GE We now give a sequence of lemmas which refer to generators, cogenerators, and compactness. Recall that, given an abelian category C , an object M of C is said to be compact if the functor hom gr ≥0 Comod(Γ) (M, −) : gr ≥0 Comod(Γ) → Ab commutes with filtered colimits; and M is said to be a generator if the functor hom gr ≥0 Comod(Γ) (M, −) is faithful. "Cogenerator" is defined dually to "generator." Proof. Let gr ≥0 Comod(Γ) denote the category of connective graded Γ-comodules, let gr Comod(Γ) denote the category of graded Γ-comodules, and let gr ≥0 Mod(A) denote the category of connective graded A-modules. It is standard that gr Comod(Γ) is abelian as long as Γ is flat over A; see Theorem 1.1.3 of [14], for example. Since gr ≥0 Comod(Γ) is a full additive subcategory of gr Comod(Γ) which is closed under finite biproducts and kernels and cokernels computed in gr Comod(Γ), the category gr ≥0 Comod(Γ) is abelian as well; see Theorem 3.41 of [4], for example. Now let E : gr ≥0 Mod(A) → gr ≥0 Comod(Γ) be the extended comodule functor. The idea here is to apply E to a cogenerator in the category of graded A-modules, but if A is not concentrated in a single grading degree, then a cogenerator for the category of graded A-modules will typically fail to be connective, so applying Γ⊗ A − to such a cogenerator does not yield a connective graded comodule.
Instead, we will apply E to an injective cogenerator I in the category gr ≥0 Mod(A) of connective graded A-modules-but we must show that I exists. Since kernels and colimits in gr ≥0 Mod(A) are computed in the underlying category of graded A-modules, and since graded A-modules form an AB5 abelian category, the category gr ≥0 Mod(A) is also AB5. The coproduct n≥0 Σ n A is a generator for gr ≥0 Mod(A), so gr ≥0 Mod(A) is Grothendieck, so by Grothendieck's famous theorem in [6] (that every Grothendieck category has an injective cogenerator), gr ≥0 Mod(A) has an injective cogenerator. So I exists.
Now the functor E is right adjoint to the forgetful functor G : gr ≥0 Comod(Γ) → gr ≥0 Mod(A), and G preserves monomorphisms since kernels of comodule maps are computed in the underlying module category; it is an elementary exercise to show that a functor sends injectives to injectives if it has a monomorphism-preserving left adjoint. So E(I) is an injective object in connective graded Γ-comodules. We claim that E(I) is also a cogenerator. Let f : X → Y be a morphism in gr ≥0 Comod(Γ) whose induced map hom gr ≥0 Comod(Γ) (Y, E(I)) → hom gr ≥0 Comod(Γ) (X, E(I)) is zero. Then the adjunction G ⊣ E tells us that the map hom gr ≥0 Mod(A) (G(Y ), I) → hom gr ≥0 Mod(A) (G(X), I) is zero, and hence that G(f ) : G(X) → G(Y ) is zero, since I is a cogenerator in gr ≥0 Mod(A). Since G is faithful and additive, this then tells us that f = 0. So E(I) is an injective cogenerator in gr ≥0 Comod(Γ). Proof. Let A be a small category and let H : A → gr ≥0 Mod(A) be a functor. We continue to write G for the forgetful functor gr ≥0 Comod(Γ) → gr ≥0 Mod(A) which is left adjoint to E. The composite GE is Γ ⊗ A −, hence preserves colimits in gr ≥0 Mod(A). So the composite natural map is an isomorphism, so the comparison map colim d∈D EH(d) → E colim d∈D H(d) is an isomorphism after applying G, hence is already an isomorphism since G reflects isomorphisms.  is an isomorphism, and consequently that the identity map on M factors through some M i , i.e., M is a summand in a finitely generated graded A-module, so M is itself a finitely generated graded A-module.  [15] for examples, due to Gabber and Roos, of Grothendieck categories satisfying axiom AB4 * but having no nonzero projectives at all!) More precisely, Lemma 3.3 shows that the forgetful functor G : gr ≥0 Comod(Γ) → gr ≥0 Mod(A) preserves products. In this paragraph and the next two, we show that G preserving products is the key result which causes gr ≥0 Comod(Γ) to have enough projectives. The functor G is also easily seen to preserve kernels (see e.g. Appendix 1 of [14] for the usual construction of kernels in graded Γ-comodules; the salient point is that they are computed in the underlying category of graded A-modules), so G preserves all limits. Now gr ≥0 Comod(Γ) is certainly "well-powered," that is, every connective graded Γ-comodule has only a set (not a proper class) of subcomodules; and by Lemma 3.4, gr ≥0 Comod(Γ) has a cogenerator. So by Freyd's Special Adjoint Functor Theorem (standard; see e.g. Theorem V.8.2 of [11], or for a statement closer to our application here, section 3.M of [4]), G has a left adjoint. Call this left adjoint F . Since F has a right adjoint (namely, G) which preserves epimorphisms, F sends projectives to projectives. So F ( n≥0 Σ n A) is a projective object of gr ≥0 Comod(Γ), since n≥0 Σ n A is projective in gr ≥0 Mod(A).
We claim that F ( n≥0 Σ n A) is also a generator in gr ≥0 Comod(Γ). The proof is as follows: if V is a generator of gr ≥0 Mod(A) and f : X → Y a map in gr ≥0 Comod(Γ) whose induced map is zero, then the adjunction F ⊣ G gives us that the induced map is zero and hence that Gf : GX → GY is zero. Since G is faithful and additive, f = 0. So F V is a generator in gr ≥0 Comod(Γ).
Consequently gr ≥0 Comod(Γ) is a cocomplete abelian category with a projective generator. It is standard that this now implies that gr ≥0 Comod(Γ) has enough projectives: if C is a cocomplete abelian category with projective generator P , then for any object X of C , the object f ∈homC (P,X) P is projective, and the evaluation map f ∈homC (P,X) P → X is epic.
Since gr ≥0 Mod(A) satisfies Grothendieck's axiom AB5 (see the proof of Lemma 3.4 for this), and since G is faithful, additive, has both a left and a right adjoint and hence is exact and preserves all colimits, gr ≥0 Comod(Γ) also satisfies Grothendieck's axiom AB5. So gr ≥0 Comod(Γ) satisfies AB5 and has a generator, hence gr ≥0 Comod(Γ) is Grothendieck.
There is a classical "recognition principle" for the category of modules over a ring (see Corollary V.1 of [5]): an abelian category is equivalent to the category of modules over a ring if and only if that abelian category is cocomplete and has a compact projective generator. Theorem 3.8 tells us that the category gr ≥0 Comod(Γ) of connective comodules over a connective finite-type graded flat Hopf algebroid is co-complete and has a projective generator. This seems, at a glance, like it is awfully close to saying that gr ≥0 Comod(Γ) is equivalent to the category of modules over a ring. However, the projective generator for gr ≥0 Comod(Γ) that we construct in the proof of Theorem 3.8 is infinitely generated, hence far from being compact. One might ask if it is possible to find a smaller projective generator, one which is compact. We now give the simple argument for why, except in trivial cases, this is impossible: Proposition 3.9. Let (A, Γ) be as in Theorem 3.8. If A is not the zero ring, then gr ≥0 Comod(Γ) is not equivalent to the category of modules over a ring.
Proof. Suppose that M is a compact generator for gr ≥0 Comod(Γ). If we assume that the underlying graded A-module of M admits a set of homogeneous generators concentrated in finitely many grading degrees, then we get a contradiction as follows: let S denote a minimal set of homogeneous generators for the underlying Amodule of M , and let n be an upper bound for the grading degrees of the elements of S. Every map of graded Γ-comodules M → Σ n+1 A must send all A-module generators of M to zero, so the functor hom gr ≥0 Comod(Γ) (M, −) fails to distinguish between the zero map Σ n+1 A → Σ n+1 A and the identity map on Σ n+1 A, contradicting faithfulness of hom gr ≥0 Comod(Γ) (M, −). (The previous sentence is where we have used the assumption A = 0.) So, if we choose a set of homogeneous generators {m i } i∈I for the underlying A-module of M , there must be elements m i in arbitrarily high grading degrees. In particular, the underlying A-module of M is not finitely generated, consequently not compact by Lemma 3.7. But applying Lemmas 3.5, 3.6, and 3.7, G(M ) is a finitely generated graded A-module, a contradiction. So M must not exist. Perhaps the statement of Proposition 3.9 sounds a bit strange: after all, if Γ = A, then the category of graded Γ-comodules is simply the category of graded A-modules. The reason that Proposition 3.9 works is that the category of connective graded modules over a connective graded ring A is not equivalent to the category of ungraded modules over a ring. So what makes Proposition 3.9 work is not really specific to comodules at all: it is essentially the same phenomenon which is responsible for graded module categories being inequivalent to ungraded module categories.
Consequently, we ought to show that gr ≥0 Comod(Γ) is not equivalent to the category of connective graded modules over a graded ring. This takes a bit more work. The purpose of the next section is to do this in the simplest and most classical nontrivial case, when Γ is a graded commutative Hopf algebra over a field. (This is of course a very commonly-occurring case: it occurs when Γ is the dual Steenrod algebra at any prime, for example.) 4. The case of a Hopf algebra over a field.
Throughout this section, we restrict our attention to the situation where the Hopf algebroid (A, Γ) is a connective graded commutative Hopf algebra over a field. Consequently A will be a field in this section, and to reinforce this running assumption, we change notation slightly, and write k in place of A.

4.1.
Calculation of the k-vector space underlying a generator for the category of connective graded Γ-comodules. Recall from Theorem 3.8 that we constructed a left adjoint F : gr ≥0 Mod(k) → gr ≥0 Comod(Γ) to the forgetful functor G : gr ≥0 Comod(Γ) → gr ≥0 Mod(k). The effect of F on suspensions Σ n k of the ground ring was especially important in the rest of the proof of Theorem 3.8, since we showed that n≥0 F (Σ n k) is a projective generator for the category gr ≥0 Comod(Γ). Our first task in this section is to give a more concrete identification of the Γ-comodule F Σ n k. Proposition 4.1 identifies the underlying graded k-vector space of F Σ n k: Proposition 4.1. Let Γ be a connective finite-type graded commutative Hopf algebra over a field k. Let n be a nonnegative integer, and let m be an integer. Then the k-linear dual vector space ((F Σ n k) m ) * of the degree m summand (F Σ n k) m of F Σ n k is isomorphic to the degree n − m summand Γ n−m of Γ. That is, ((F Σ n k) m ) * ∼ = Γ n−m as k-vector spaces.
Proof. Using the adjunctions F ⊣ G ⊣ E, we have the isomorphisms of k-vector spaces Corollary 4.2. Let Γ, n be as in Proposition 4.1. Then the projective connective graded Γ-comodule F Σ n k is trivial in degrees > n.
If we furthermore assume that Γ is not concentrated in a single grading degree, then there exist positive integers n such that F Σ n k fails to be isomorphic to Σ n F k. That is, the free functor F : gr ≥0 Mod(k) → gr ≥0 Comod(Γ) fails to commute with suspension.
Example 4.3. Suppose that Γ is the mod 2 dual Steenrod algebra, and k = F 2 . Then Proposition 4.1 gives us that F Σ 0 k ∼ = k as a k-vector space, and hence also as a k-comodule. That is, F Σ 0 k ∼ = F 2 . Meanwhile, as a graded k-vector space, F Σ 1 k is isomorphic to k in degree 0, isomorphic to k in degree 1, and trivial in all other degrees. So F Σk fails to be isomorphic to ΣF k. Remark 4.4. I hope the reader will forgive me for offering this warning about an easy way to make mistakes when reasoning about the free functor F and the projective connective graded Γ-comodules F Σ n k. We adopt the following convenient notation: if M and N are connective graded k-vector spaces, we write hom gr ≥0 Mod(k) (M, N ) for the graded hom-group whose homogeneous degree n summand is trivial for n < 0, and if n ≥ 0, it is the set of homomorphisms M → N which increase degree by n, i.e., hom C (M, N ) n = hom C (Σ n M, N ). This is the natural choice of self-enrichment (in the sense of [8]) of the category of connective graded k-vector spaces, so that we have the isomorphism of connective graded k-modules for all connective graded k-vector spaces M, N, Q.
That is, G • Σ n ≃ Σ n • G and E • Σ n ≃ Σ n • E for all nonnegative integers n. It is easy to imagine that the isomorphism of k-vector spaces That is, the natural map GF Σ j M → Σ j GF M of graded k-vector spaces induces an isomorphism on k-linear duals in each grading degree. Hence GF Σ j M → Σ j GF M is an isomorphism. Since G reflects isomorphisms and commutes with suspension, F Σ j M → Σ j F M is an isomorphism, contradicting the failure of F to commute with suspension, demonstrated in Corollary 4.2. Hence we cannot have an adjunction of the form (4.13).
Another corollary of Proposition 4.1 is an identification of the k-vector space underlying the generator n F Σ n k for the category of connective graded Γ-comodules: Corollary 4.5. Let Γ, n be as in Proposition 4.1. Recall that we have the projective generator F ( n Σ n k) for the category gr ≥0 Comod(Γ) of connective graded Γcomodules constructed in Theorem 3.8. Then the degree m summand of F ( n Σ n k) has k-linear dual vector space isomorphic to the k-vector space product n≥0 Γ n−m .

4.2.
Failure of the category of connective graded comodules to be equivalent to the category of connective graded modules over a ring. Corollary 4.5 identified the generator n F Σ n k for the category of connective graded Γcomodules, but only as a k-vector space. In order to prove our main result in this section, Theorem 4.7, we will need slightly more information about the structure of n F Σ n k as a Γ-comodule.
The needed information will be expressed in terms of the covariant embedding of Γ-comodules into Γ * -modules. This construction is classical: I do not know its historically earliest appearance in the literature, but see [1] for a discussion from a topological perspective, or [3] for a discussion from a purely algebraic perspective. The construction goes as follows: given a graded Γ-comodule M with coaction map ψ : M → Γ ⊗ k M , the action map M × Γ * → M sends a pair (m, f ) ∈ M × Γ * to image of m under the composite This action of Γ * on M is called the adjoint action. This construction yields a covariant, exact, faithful, full functor Cov : gr Comod(Γ) → gr Mod(Γ * ) which admits a right adjoint Rat : gr Mod(Γ * ) → gr Comod(Γ). Given a graded Γ *module, the graded Γ * -module Cov(Rat(M )) is called the rational submodule of M , and it is indeed a graded Γ * -submodule of M , via the counit map Cov(Rat(M )) → M of the adjunction Cov ⊣ Rat. See section 4 of [3] for a presentation of these well-known results. These results rely on Γ being projective as a k-module, which of course is automatic from our assumption that k is a field. To avoid potential confusion, it is important to fix our grading convention for the k-linear dual of a graded k-vector space. Given a graded k-vector space V , we grade its k-linear dual V * as follows: the degree n summand of V * is the k-linear dual of the degree −n summand of V −n . That is, (V * ) n = (V −n ) * . Consequently, if Γ is a connective graded Hopf algebra over k, then its k-linear dual Γ * is a co-connective graded Hopf algebra over k. With this convention, the covariant embedding Cov : gr Comod(Γ) → gr Mod(Γ * ) preserves the gradings. Proposition 4.6. Let Γ be a connective finite-type graded commutative Hopf algebra over a field k. Let n be a nonnegative integer. Let I n denote the two-sided ideal of the co-connective dual Hopf algebra Γ * generated by all homogeneous elements of degree < −n. Then, for every connective graded Γ-comodule M , we have an isomorphism of k-vector spaces (4.15) hom  The isomorphism of (4.16) with (4.17) is induced by the natural map Σ n Γ * /I n → Cov F Σ n k which picks out the copy of k in degree n of F Σ n k, so the chain of isomorphisms from (4.16) to (4.17) yields naturality of (4.15) in the variable M .
Theorem 4.7. Let Γ be a connected 2 finite-type graded commutative Hopf algebra over a field k. Suppose that Γ is nontrivial in infinitely many degrees. Then the category gr ≥0 Comod(Γ) of connective graded Γ-comodules is not equivalent to the category of connective graded modules over a graded ring by a suspension-preserving equivalence of categories.
Proof. We argue by contrapositive. The category of connective graded modules over a graded ring does not have a compact projective generator, but what it does have is a compact projective object C such that the coproduct n≥0 Σ n C is a generator. (Namely, C = R, as a free R-module generated in degree zero.) So suppose that gr ≥0 Comod(Γ) has a compact projective object C such that the coproduct n≥0 Σ n C is a generator for gr ≥0 Comod(Γ). In the proof of Theorem 3.8, we showed that n≥0 F Σ n k is a generator for gr ≥0 Comod(Γ). Consequently there exists an epimorphism ǫ : n≥0 F Σ n k ⊕κ → C in gr ≥0 Comod(Γ) for some cardinal number κ. Since C is projective, the epimorphism ǫ splits. Choose a section σ : C → n≥0 F Σ n k ⊕κ of ǫ in the category gr ≥0 Comod(Γ). We claim that the image of σ contains nonzero elements of at most finitely many of the summands F Σ n k of n≥0 F Σ n k ⊕κ . This is easily seen: since C is compact, the natural morphism is an isomorphism, and elements of the direct sum in the domain of (4.18) are zero except in finitely many of the summands.
Since im σ nontrivially intersects only finitely many of the summands F Σ n k, there exists some largest integer N such that im σ nontrivially intersects summands of the form F Σ N k. Consequently the integer N and the connective graded Γcomodule C have the following properties: (1) C is a coproduct of retracts of copies of F Σ n k for n ≤ N .
(2) For every connective graded Γ-comodule M and every homogeneous element m ∈ M , there exists a coproductC of suspensions of copies of C and a graded Γ-comodule morphismC → M whose image contains m.
As a consequence of 1 and 2, we have the following: (3) For every connective graded Γ-comodule M and every homogeneous element m ∈ M , there exists a coproductC of suspensions of copies of F Σ n k, for various integers n ≤ N , and a graded Γ-comodule morphismC → M whose image contains m.
Bringing Proposition 4.6 to bear now yields: (4) For every connective graded Γ-comodule M and every homogeneous element m ∈ M , there exists a coproduct C of suspensions of copies of Σ n Γ * /I n , for various integers n ≤ N , and a graded Γ * -module morphism C → Cov(M ) whose image contains m, and consequently, (5) For every connective graded Γ-comodule M and every homogeneous element m ∈ M , the element m ∈ M is annihilated by the ideal I N of Γ * Recall our assumption that Γ is nontrivial in infinitely many degrees. Consequently there must be some nonzero homogeneous element of I 0 which is in a degree d with d < −N . Choose such an element in Γ * in degree d, and call it γ * . Choose also a homogeneous element γ ∈ Γ in degree −d such that γ * , when evaluated on γ, is equal to 1 ∈ k. By counitality and connectedness of Γ, we have where J 0 is the augmentation ideal in Γ.(To avoid possible confusion, we remind the reader that the notation I 0 already is reserved for the augmentation ideal in Γ * .) Then, following the recipe for the adjoint action of Γ * on Γ described above in (4.14), we have that γ * · γ = 1. Consequently we have an element γ in the graded Γ * -module Cov(Γ) such that γ is not γ * -torsion. But since γ * is in degree d < −N , γ * is in the ideal I N of Γ * generated by all elements of grading degree < −N . Hence Cov(Γ) contains a homogeneous element which is not I N -torsion. This contradicts claim (5) above, whose truth we already established. So C must not exist.