QUILLEN STRATIFICATION FOR THE STEENROD ALGEBRA

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1 QUILLEN STRATIFICATION FOR THE STEENROD ALGEBRA JOHN H. PALMIERI Abstract. LetA be the mod 2 Steenrod algebra, and consider the map ρ: H (A; F 2 ) lim H (E; F 2 ), where the inverse limit is taken over the exterior sub-hopf algebras of A, ordered by inclusion. There is an action of A on this inverse limit, and ρ factors through the invariants under this action, giving a map ( ) A γ : H (A; F 2 ) lim H (E; F 2 ). We show that γ is an F -isomorphism. 1. Results In this paper we give a version of the Quillen stratification theorem [Qui71, 6.2], for the cohomology of the Steenrod algebra. This cohomology ring is the E 2 - term of the Adams spectral sequence converging to the stable homotopy groups of spheres, so any sort of structural information is potentially applicable to stable homotopy theory. Since our result is a version of a powerful group cohomology result, it provides evidence that other structural results about group cohomology (e.g., the work of Benson, Carlson, et al., on varieties for kg-modules) may carry over to results about modules over the Steenrod algebra, and maybe to results about stable homotopy theory. Recall that the mod 2 Steenrod algebra A is a graded connected cocommutative Hopf algebra; Milnor showed in [Mil58] that its dual A is isomorphic, as an algebra, to ( ) A = F2 [ξ 1,ξ 2,ξ 3,...], which is graded by setting ξ n =2 n 1. The product on A is dual to the diagonal map on A, which is given by n : ξ n ξn i 2i ξ i. (We take ξ 0 to be 1.) i=0 Notation 1.1. Given a commutative ring k, a graded augmented k-algebra Γ, and a left Γ-module M, wewriteh (Γ; M)forExt Γ (k, M). Note that this is a bigraded vector space. We write M Γ for Hom Γ (k, M) M; thenm Γ is the set of invariants Date: October 11, Mathematics Subject Classification. 55S10, 55T15, 55Q45, 18G15. Key words and phrases. Steenrod algebra, Ext, stable homotopy, F-isomorphism, Quillen stratification. Research partially supported by National Science Foundation grant DMS

2 2 JOHN H. PALMIERI in M under the Γ-action. It consists of all x in M which support no nontrivial operations by elements of Γ. Note that when Γ is a (graded) cocommutative Hopf algebra over a field k, then H (Γ; k) is a (graded) commutative k-algebra one can use the diagonal map on Γ to induce the product on H (Γ; k), so cocommutativity of Γ implies commutativity of H (Γ; k). This is the case when k = F 2 and Γ = A. Part (a) of this next definition is due to Quillen [Qui71]; part (b) is due to Wilkerson [Wil81]. Definition 1.2. Fix a field k of characteristic p. (a) Given a map f : R S of graded commutative k-algebras, we say that f is an F -isomorphism if (i) every x ker f is nilpotent, and (ii) for all y S, there is an n so that y pn is in the image of f. (b) A Hopf algebra E over k is elementary if E is commutative and cocommutative, and x p =0forallx in IE (where IE = ker(e k) is the augmentation ideal of E). Remark. (a) If one can choose the same n for every y in Definition 1.2(a)(ii), then the F -isomorphism is said to be uniform. We do not expect this to be the case in our results. (b) Note that when char(k) = 2, then a Hopf algebra over k is elementary if and only if it is exterior, as an algebra. The elementary sub-hopf algebras of the Steenrod algebra A were classified by Lin [Lin, 1.1]; they have also been studied in [Wil81] and [Pal96]. One should think of them as the elementary abelian subgroups of A. Given a Hopf algebra Γ over a field k, for every sub-hopf algebra Λ of Γ, there is a restriction map H (Γ; k) H (Λ; k). If we let Q denote the category of elementary sub-hopf algebras of Γ, where the morphisms are inclusions, then we can assemble the restriction maps into ρ: H (Γ; k) lim E QH (E; k). In the case k = F 2 and Γ = A, we show in Section 2 that there is an action of A on lim E Q H (E; F 2 ), making the inverse limit an algebra over A. Here is our main result. Theorem 1.3. The map ρ: H (A; F 2 ) lim E QH (E; F 2 ) factors through ( ) A γ : H (A; F 2 ) lim E Q H (E; F 2 ), and γ is an F -isomorphism. This theorem is based, of course, on a theorem of Quillen [Qui71, 6.2] which identifies the cohomology H (G; k) of a group G, uptof -isomorphism, as the inverse limit lim H (E; k). This inverse limit is taken over the category whose objects are elementary abelian p-subgroups of G (where p =char(k)), and whose morphisms are generated by inclusions and conjugations by elements of G. Using conjugations is technically unpleasant in our setting, so we just use inclusions as our morphisms; this is why we have to consider the invariants under the A-action.

3 QUILLEN STRATIFICATION FOR THE STEENROD ALGEBRA 3 Partial results related to Theorem 1.3 can be found in [Lin], [Wil81], [HP93], and [Pal96]. Lin proves a special case of Theorem 1.3 (actually a version of Theorem 4.1, from which we deduce Theorem 1.3) for certain sub-hopf algebras of A; in his setting, though, one does not have to take invariants. The other papers all focus on the detection of nilpotence (finding a map f that satisfies condition (i) in Definition 1.2), rather than on the identification of all of the non-nilpotent elements. Here is our second result: we can compute the inverse limit in Theorem 1.3, and we have a formula for the A-action. Theorem 1.4. We have an isomorphism of A-algebras lim E Q H (E; F 2 ) = F 2 [h ts s<t] / (h ts h vu u t), where h ts =(1, 2 s (2 t 1)). The action of A is given by the following on the generators: h t 1,s+1 if k = s and t 1 <s+1, Sq 2k (h ts )= h t 1,s if k = s + t 1andt 1 <s, 0 otherwise, and then one extends multiplicatively. The polynomial generator h ts is represented by [ξt 2s ] in the cobar construction (for any elementary E for which this makes sense). By one extends multiplicatively, we mean that one uses the Cartan formula to determine the action of Sq n on products. Alternatively, one can rewrite the action of A as a coaction of the dual A that is, as a map lim H (E; F 2 ) A lim H (E; F 2 ). The coaction is multiplicative, and its value on the polynomial generators is given by the following formula: h ts s+t 1 2 j=0 t j i=j+s+1 ζj 2s ξt i j 2i+j+s h i,j+s. (As usual, ζ j denotes χ(ξ j ), where χ: A A is the conjugation map.) Using this explicit description of the (co)action, one can find elements in the ring of invariants, either by hand or by computer; we give a few examples below. For these examples, we let R denote the bigraded ring F 2 [h ts s<t] / (h ts h vu u t). We give a graphical depiction of the (co)action of A on R in Figure 1-1. Example 1.5. (a) The element h t,t 1 in R 1,2t 1 (2 t 1) is an invariant, for t 1. Indeed, we know that h 10 lifts to H 1,1 (A; F 2 ); also h 4 21 lifts to an element in H 4,24 (A; F 2 ) (the element known as g or κ see [Zac67] for the identification of this element as a lift of a power of h 21 ). We do not know which power of h t,t 1 survives for t 3. (b) We have some families of invariants. Note that h 20 is not invariant it supports a single non-trivial operation: Sq 2 (h 20 )=h 10. Then since h 21 is invariant, the element h 20 h 21 supports at most one possibly nontrivial operation: Sq 2 (h 20 h 21 )=h 10 h 21.

4 4 JOHN H. PALMIERI h 21 h 32 h h 31 h 0 h 10 1 h 20 2 h 30 3 h Figure 1-1. Graphical depiction of the action of A on R. An arrow labeled by k represents an action by Sq 2k A, or equivalently a coaction by ξ1 2k A in other words, a term of the form ξ1 2k (target) in the coaction on the source. But since h 10 h 21 =0inR, thenh 20 h 21 is invariant; indeed, h i 20h j 21 is invariant for all i 0andj 1. It turns out that more of these lift to H (A; F 2 ) than one might expect from Theorem 1.3: the elements in the Mahowald- Tangora wedge [MT68] are lifts of the elements h i 20 h8j 21 for all i 0and j 1. (See [MPT71]; earlier, Zachariou [Zac67] verified this for elements of the form h 2i 20 h2(i+j) 21 for i, j 0.) These elements are distributed over a wedge between lines of slope 1 2 and 1 5 (in the usual Adams spectral sequence picture of H (A; F 2 )). (c) Similarly, while h 30 and h 31 are not invariant, the monomials {h i 30h j 31 hk 32 i 0, j 0, k 1} are invariant elements. Hence some powers of them lift to H (A; F 2 ); we do not know which powers, though. Continuing in this pattern, we find that for n 1, we have sets of invariant elements {h i0 n0 hi1 n1...hin 1 n,n 1 i 0,...,i n 2 0, i n 1 1}. Again, we do not know which powers of these elements lift. In any case, we can see that the family of elements in the Mahowald-Tangora wedge is not a unique phenomenon we have infinitely many such families, and all but the n =1andn = 2 families give more than a lattice of points in H (A; F 2 ). (d) Margolis, Priddy, and Tangora [MPT71, p. 46] have found some other nonnilpotent elements, such as x H 10,63 (A; F 2 )andb 21 H 10,69 (A; F 2 ). These both come from the invariant z = h 2 40 h h2 20 h2 21 h 41 + h 2 30 h 21h h2 20 h3 31 in R. While we do not know which power of z lifts to H (A; F 2 ), we do know that B 21 maps to the product h 3 20h 2 21z, andx maps to h 5 20z. In other words,

5 QUILLEN STRATIFICATION FOR THE STEENROD ALGEBRA 5 we have at least found some elements in the ideal (z) R A which lift to H (A; F 2 ). (e) Computer calculations have led us to a few other such sporadic invariant elements (i.e., invariant element that do not belong to any family any family of which we are aware, anyway): h 8 20h h 8 30h h 11 21h 31 in R 12,80, an element in R 13,104 (a sum of 12 monomials in the variables h i,0 and h i,1,2 i 5), and an element in R 9,104 (a sum of 8 monomials in the same variables). We do not know what powers of these elements lift to H (A; F 2 ), nor are we aware of any elements in the ideal that they generate which are in the image of the restriction map from H (A; F 2 ). Aside from these computational results, Theorem 1.3 is potentially interesting from a structural point of view. In particular, it has led us to several conjectures about the global structure of the category of A-modules. First we give a proposed classification of the thick subcategories of finite A-modules. Conjecture 1.6. The thick subcategories of finite A-modules are in one-to-one correspondence with the radical ideals of lim H (E; F 2 ) which are invariant under the action of A. Given a finite A-module M, weleti(m) be the annihilator ideal of lim H (E; M) in lim H (E; F 2 ), under the Yoneda composition action. The maps in the proposed bijection should go like this: given an invariant radical ideal I lim H (E; F 2 ), then we let D be the full subcategory with objects {M M finite, I(M) I}. This is clearly thick. Given a thick subcategory D, thenweleti be the ideal I(M). M ob D With a little work, one can show that this is an invariant ideal of lim H (E; F 2 ) (and we clearly have I = I). (For the context of this conjecture, one needs to interpret thick subcategories of finite A-modules properly one usually discussses thick subcategories of triangulated categories, and the category of A-modules is, of course, Abelian. The intended interpretation of Conjecture 1.6 is to work in a nice triangulated category in which the category of finite A-modules sits; the best option is the category of cochain complexes of injective A -comodules. See [Pal] for details.) Along similar lines, we have a suggested analogue of the result of Hopkins and Smith [HSb, Theorem 11], in which they identify the center of [X, X] uptof - isomorphism, for any finite spectrum X. Given a ring R, we let Z(R) denote the center of R. Conjecture 1.7. If M is a finite A-module, then there is an F -isomorphism [( ) A Z(Ext A (M,M)) lim H (E; F 2 ) /I(M)].

6 6 JOHN H. PALMIERI Here is the structure of this paper. In Section 2, we discuss elementary and normal sub-hopf algebras of the Steenrod algebra; we describe the action of A on lim H (E; F 2 ), and we prove Theorem 1.4. In Section 3, we prove two results about the spectral sequence associated to a Hopf algebra extension: first, the property of having a vanishing plane with fixed normal vector, with some intercept and at some E r -term, is generic. Second, if we are working with Hopf algebras over the field with 2 elements, given a class y E 0,t 2,theny2i survives to the E 2 i +1-term. In Section 4, we introduce a sub-hopf algebra D of A, and showthat Theorem 1.3 follows if we can prove a similar result Theorem 4.1 about the restriction map H (A; F 2 ) H (D; F 2 ). In Section 5, we prove Theorem 4.1. Remark. All of the main results of this paper Theorems 1.3, 1.4, and 4.1 have obvious analogues if we replace A by any sub-hopf algebra B of A, andq by the category of elementary sub-hopf algebras of B (so the objects will be B E, for E Q). The proofs that we give for the B = A case carry over easily to this slightly more general setting; we leave the details to the interested reader. Convention. Throughout the paper, we have a number of results about the Steenrod algebra, its sub-hopf algebras, and modules over it. We let A denote the mod 2 Steenrod algebra, and we use letters like B, C, D, ande to denote sub-hopf algebras of it. We write B (etc.) for the corresponding quotient Hopf algebra of the dual of A, A. Now, we also have a number of results about general graded cocommutative Hopf algebras over a field k. To set such results apart, we will use capital Greek letters, like Γ and Λ, to denote such Hopf algebras; typically, Λ will denote a sub-hopf algebra of Γ. 2. Sub-Hopf algebras of A In this section we discuss two kinds of sub-hopf algebras of A: elementary ones and normal ones. We also construct the action of A on lim H (E; F 2 ), and we prove Theorem 1.4. First we bring the reader up to speed on elementary sub-hopf algebras of A. We gave a description of the dual of A in Section 1 (equation ( )); we dualize with respect to the monomial basis of A,andweletPt s be the basis element dual to ξt 2s. Proposition 2.1. (a) As an algebra, each elementary sub-hopf algebra E of A is isomorphism to the exterior algebra on the Pt s s that it contains. (b) Hencewehave H (E; F 2 ) = F 2 [h ts Pt s E], where h ts is represented by [ξt 2s] in the cobar complex for E, so h ts = (1, 2 s (2 t 1)). (c) The maximal elementary sub-hopf algebras of A are of the form ) (F 2 [ξ 1,ξ 2,...]/(ξ 1,...,ξ n 1,ξn 2n,ξ2n n+1,ξ2n n+2,...), for n 1. Proof. This is an accumulation of well-known facts about sub-hopf algebras of the Steenrod algebra. Part (b) follows immediately from part (a); parts (a) and (c) can be found in [Mar83, pp ].

7 QUILLEN STRATIFICATION FOR THE STEENROD ALGEBRA 7 We also need to discuss normal sub-hopf algebras of A. Definition 2.2. Suppose that Γ is a cocommutative Hopf algebra over a field k. A sub-hopf algebra Λ of Γ is normal if the left ideal of Γ generated by Λ is equal to the right ideal of Γ generated by Λ (i.e., we have Γ IΛ = IΛ Γ, where IΛ = ker(λ k) is the augmentation ideal of Λ). If Λ is normal, then we have Γ Λ k = k Λ Γ, and this vector space has the structure of a Hopf algebra; we use Γ//Λtodenoteit. In this situation, we say that Λ Γ Γ//ΛisaHopf algebra extension. First we discuss properties of normal sub-hopf algebras of an arbitrary graded cocommutative Hopf algebra Γ, and then we specialize to the case of normal sub- Hopf algebras of the Steenrod algebra A. Proposition 2.3. Let Γ be a graded cocommutative Hopf algebra over a field k. Fix a Γ-module M and a normal sub-hopf algebra Λ of Γ. (a) For each integer t, there is an action of Γ//Λ on H t, (Λ; M). This action is natural in M, and if one views it as an action of Γ on H t, (Λ; M), thenitis natural in Λ as well. This action lowers degrees: given γ Γ//Λ of degree i and x H t,v (Λ; M), thenγx is in H t,v i (Λ; M). (b) There is a spectral sequence, the Lyndon-Hochschild-Serre spectral sequence, with 2 (M) =H s,u (Γ//Λ; H t, (Λ; M)) H s+t,u (Γ; M), and with differentials d r : r E s+r,t r+1,u r. It is natural in M. Proof. This is a simple extension of the corresponding results for group extensions. (We also give a construction of the spectral sequence in the proof of Theorem 3.1.) By the way, we will often be applying Proposition 2.3(a) in the case where Γ = A is the mod 2 Steenrod algebra, yielding an action of A on H (B; F 2 )forb a normal sub-hopf algebra of A. There is another action of A on this cohomology ring, because A acts on the cohomology of any cocommutative Hopf algebra over F 2 see [May70], for example. These two actions are quite different; we will almost always use the action arising as in Proposition 2.3(a), not the other one (the only exception is the proof of Lemma 3.5). Proposition 2.4. Let A be the mod 2 Steenrod algebra. (a) A sub-hopf algebra of A is normal if and only if its dual is of the form F 2 [ξ 1,ξ 2,...]/(ξ 2n 1 1,ξ 2n 2 2,ξ 2n 3 3,...), with n 1 n 2 n 3. (If n i = in this expression, this means that one does not divide out by any power of ξ i.). (b) Hence the maximal elementary sub-hopf algebras of A are normal. Proof. This is well-known; see [Mar83, Theorem 15.6], for example. Now we construct the action of A on lim H (E; F 2 ). We let Q denote the full subcategory of Q consisting of the normal elementary sub-hopf algebras of A. By Proposition 2.4(b), we see that Q is final in Q; hence we have lim Q H (E; F 2 ) = lim Q H (E; F 2 ).

8 8 JOHN H. PALMIERI On the right we have an inverse limit of modules over A; we give this the induced A-module structure. So we define (since taking invariants is an inverse limit) ( ) A ( ) A lim Q H (E; F 2 ) = lim Q H (E; F 2 ) ( = lim Q H (E; F 2 ) A) ( = lim Q H (E; F 2 ) A//E). Theorem 1.4 follows more or less immediately: Proof of Theorem 1.4. We want to compute lim H (E; F 2 ), and we want to compute the action of A on this ring. Proposition 2.1(b) tells us H (E; F 2 )foreach E; since the maps in the category Q are inclusion maps of one exterior algebra into another exterior algebra in which one simply adds more generators, the induced maps on cohomology are surjections of one polynomial ring onto another, each polynomial generator mapping either to zero or to a polynomial generator (of the same name). So assembling the cohomology rings of the maximal elementary sub-hopf algebras of A gives our computation of the inverse limit. To get our formula for the action of A on lim QH (E; F 2 ), we have to compute the A-action on H (E; F 2 )fore A maximal elementary. This is done in [Pal96, Lemmas A.3 and A.5]. (Briefly: for degree reasons, we have αh t 1,s+1 if k = s and t 1 <s+1, Sq 2k (h ts )= βh t 1,s if k = s + t 1andt 1 <s, 0 otherwise, for some α, β F 2. So one only has to check that α = β = 1; this is a reasonably straight-forward computation.) 3. Extensions of Hopf algebras In this section we discuss normal sub-hopf algebras and the Lyndon-Hochschild- Serre spectral sequence that arises from a Hopf algebra extension. In particular, we show in Theorem 3.1 that the presence of a vanishing plane at some term of such a spectral sequence is a generic property (Definition 3.2). We also prove (Lemma 3.5) that if y is in the E 0,t,u 2 -term of this spectral sequence, then y 2i survives to the E 2i +1-term (if we re working over the field F 2 ). (Recall that we defined the notions of normal sub-hopf algebra and Hopf algebra extension in Definition 2.2, and we presented the spectral sequence of a Hopf algebra extension in Proposition 2.3.) Here is our main theorem for this section. It is originally due to Hopkins and Smith [HSa], in the context of the generalized Adams spectral sequence. Since the Lyndon-Hochschild-Serre spectral sequence can be viewed as an Adams spectral sequence in the appropriate context (see [Pal]), it is not surprising that their result applies here. We give a proof below, since there is no published proof of which we are aware. Theorem 3.1. Suppose that we have an extension Λ Γ Γ//Λ of graded cocommutative Hopf algebras over a field k. For any Γ-module M, consider the associated spectral sequence 2 (M) =H s,u (Γ//Λ; H t, (Λ; M)) H s+t,u (Γ; M).

9 QUILLEN STRATIFICATION FOR THE STEENROD ALGEBRA 9 Fix integers a, b, andc with c<0, and consider the following property P of the Γ-module M: P : There are numbers r and d so that Er s,t,u (M) =0whenas + bt + cu > d. Then we have the following results. (a) M satisfies P if and only if ΣM satisfies P. (b) Given a short exact sequence of Γ-modules 0 M M M 0, if two of M, M, andm satisfy property P, then so does the third. (c) If the Γ-module L M satisfies property P,thensodoesL. (d) Every bounded above injective Γ-module satisfies property P. For (a) and (d), recall that Γ is graded; Σ denotes the usual shift functor on the module category, and we say that a Γ-module is bounded above if it is zero in all sufficiently large gradings. Proof. Part (d) is clear: if M is a bounded above injective Γ-module, then the spectral sequence collapses, and 2 = E s,t,u is zero unless s = t =0andu is larger than some number i (where i is the largest nonzero grading of M). Since c<0, then 2 is certainly zero above some plane with normal vector (a, b, c). Parts (a) and (c) are also clear: for part (a), the spectral sequence for ΣM is the same as the spectral sequence for M, but with a shift in dimension; so M and ΣM would have a vanishing plane with the same normal vector at the same E r, possibly with slightly different intercepts. For part (c), the naturality of the spectral sequence ensures that the E r -term for the module L is a retract of the E r -term for L M. So a vanishing plane for the sum induces one on each summand. We are left with (b). We start by giving a construction of the spectral sequence; this is essentially the same construction as in [Ben91, 3.5], for instance, except we use injective rather than projective resolutions. Let I be a fixed injective resolution of k as a Γ//Λ-module; let J be a fixed injective resolution of k as a Γ-module. Then we let 0 =Hom u Γ//Λ (k, Hom Λ (k, I s J t M)) = Hom u Γ(k, I s J t M). (The isomorphism is as in [Ben91, 3.5.1].) The maps in the complexes I and J induce differentials on this, making it a double complex; hence we get two spectral sequences. One can see (again, as in [Ben91]) that one collapses to E 2 = H (Γ; M) and that the other has 2 as in the statement of the theorem. We need to examine the exact couple that produces this spectral sequence: we let For each r, wehaveamap D s,t,u 0 = D s+1,t 1,u r i+j=s+t i s E i,j,u 0. i r D s,t,u r, which is induced by the map for r =0: givend r 1 and i r 1,thenD r is the image of i r 1,andi r is the restriction of i r 1 to this image. Of course, the map i 0 is just the standard inclusion.

10 10 JOHN H. PALMIERI Hence we see that if, for some r and d, the composite i 0 i }{{} 0 : D s+r 1,t r+1,u 0 D s,t,u 0 r 1 is zero whenever as + bt + cu > d, thendr s,t,u =0forallsuch(s, t, u). Conversely, if r then i r : D s+1,t 1,u r D s,t,u r = 0 for all such gradings (s, t, u), so r = 0 whenever as + bt + cu > d, is an isomorphism for all such (s, t, u); indeed, i r+j : D s+1,t 1,u r+j D s,t,u r+j is an isomorphism for all j 0. In other words, the images in D s,t,u 0 of the (r +1+j)-fold composites i 0...i 0 are the same, for all j 0. By the convergence of the spectral sequence, these images must stabilize to zero; hence the (r +1)-fold composite must be zero. So the condition P is equivalent (up to a shift in r) to the condition P : P : There are numbers r and d so that the composite i 0 i }{{} 0 : D s+r 1,t r+1,u 0 D s,t,u 0 r is zero whenever as + bt + cu > d. By our construction of the spectral sequence, D 0 ( ) is functorial; in fact, D 0 (M) = M D 0 (k), so it takes a short exact sequence of modules to a short exact sequence of double complexes. One can easily see that if two of the three complexes satisfy condition P, then so does the third; one may take the intercept d for the third to be the larger of the other two intercepts, and one may take the term r for the third to be the sum of the r s for the other complexes. We borrow the following terminology from the analogous situation in a triangulated category (see [HSb], for example). Definition 3.2. Any property P of Γ-modules which satisfies (a) (d) of Theorem 3.1 is called a generic property. Example 3.3. Suppose that Γ is a cocommutative Hopf algebra over a field k, and let Γ be the dual of Γ, with the standard left Γ-module structure (i.e., if k is the only simple Γ-module, then Γ is the injective hull of k). If M is a bounded above Γ-module which satisfies a generic property P,thenΓ M is a bounded above injective, so the module coker(m Γ M) also satisfies P. We let Ω 1 M denote this kernel. It is standard (and easy to show) that Ext s+1 Γ (Ω 1 L, M) =Ext s Γ(L, M), Ext s Γ (L, Ω 1 M)=Ext s+1 Γ (L, M), for s>0. See [Mar83, 14.8], for example. We point out that if 0 M M M 0 is a short exact sequence, then for some injective modules I and J, soisthefollowing: 0 M I M J Ω 1 M 0.

11 QUILLEN STRATIFICATION FOR THE STEENROD ALGEBRA 11 (See [Mar83, p. 210].) If the modules M, M,andM are bounded above, we may choose I and J to be bounded above, as well. In any case, we say that 0 M M Ω 1 M 0 is exact up to injective summands. We will use Theorem 3.1 in combination with the next result. Proposition 3.4. Fix a graded connected cocommutative Hopf algebra Γ over a field k, and fix a property P which is generic on Γ-modules. Let 0 Σ i N M N 0 be a short exact sequence of Γ-modules, with connecting homomorphism given by h 1 N Ext 1 Γ (N,N) for some h H (Γ; k). Ifh is nilpotent in H (Γ; k), thenm satisfies P if and only if N satisfies P. Proof. By the definition of genericity, if N satisfies P,thensodoesM. For the converse, we assume that h m = 0. Given any ζ in H m (Γ; k), we define a Γ-module L ζ by the following sequence, which is exact up to injective summands: 0 L ζ k Ω m k 0. (So L ζ is only well-defined up to injective summands.) Then the following is also exact up to injective summands: 0 Ω (m 1) k L ζ k 0. We apply this with ζ = h m =0,toseethatk is a summand of L h m, and hence that N is a summand of N L h m. WeclaimthatifM satisfies P,thensodoesN L h i for all i; verifying this claim (and letting i = m) will finish the proof. We prove this by induction on i. We start the induction by noting that M = N L h. So suppose that we know that N L h i 1 satisfies P. Consider the following commutative diagram: Ω (i 2) L h 0 Ω (i 1) L h 0 0 L h i 1 k 0 L h i k h i 1 Ω (i 1) k 0 Ω (i 1) h h i Ω i k The rows and columns of this diagram are exact up to injective summands. Upon tensoring this diagram with N and applying Example 3.3, we see from the righthand column that since M = N L h and N L h i 1 satisfy P, then so does Ω (i 2) N L h, and hence so does N L h i.

12 12 JOHN H. PALMIERI We will also need the following result, about the survival of powers of elements in our spectral sequence. Note that when using trivial coefficients, the Lyndon- Hochschild-Serre spectral sequence is a spectral sequence of commutative algebras: each E r -term is a (graded) commutative algebra, and each differential d r is a derivation. So if we are working with a field k of characteristic p>0, then for any z Er,, z p survives to the E r+1 -term. It turns out that we can greatly improve on this result. Lemma 3.5. Suppose that we have an extension Λ Γ Γ//Λ of graded cocommutative Hopf algebras over the field F 2, and consider the associated spectral sequence Er s,t,u (F 2 ). Fix y E 0,t,u 2. Then for each i, the element y 2i survives to E 0,2i t,2 i u 2 i +1. Proof. This follows from properties of Steenrod operations on this spectral sequence, as discussed by Singer in [Sin73]. Suppose we have a spectral sequence Er s,t which is a spectral sequence of algebras over the Steenrod algebra. Fix z Er s,t,and fix an integer k. Then [Sin73, 1.4] tells us to which term of the spectral sequence Sq k z survives (the result depends on r, s, t, andk). For instance, if z Er 0,t,then Sq t (z) =z 2 survives to E 0,2t 2r 1. There are similar results at odd primes see [Saw82]. By the way, Singer s results [Sin73] are stated in the case of an extension of graded cocommutative Hopf algebras Λ Γ Γ//Λ, where Λ is also commutative. This commutativity condition is not, in fact, necessary, as forthcoming work of Singer will show [Sin]. 4. Reduction of Theorem 1.3 In this section we state a variant of Theorem 1.3 (namely, Theorem 4.1), and we show that Theorem 1.3 follows from it. Let D be the sub-hopf algebra of A generated by the Pt s s with s<t;inother word, D is dual to the following quotient Hopf algebra of A : D = F 2 [ξ 1,ξ 2,ξ 3,...]/(ξ1,ξ 2 2,ξ 4 3,...,ξ 8 n 2n,...). By Proposition 2.1(c), every elementary sub-hopf algebra of A sits inside D. By Proposition 2.4(a), D is a normal sub-hopf algebra of A, so there is an action of A//D on H (D; F 2 ). We have the following result. Theorem 4.1. The restriction map H (A; F 2 ) H (D; F 2 ) factors through and ϕ is an F -isomorphism. ϕ: H (A; F 2 ) H (D; F 2 ) A//D, In this section, we reduce Theorem 1.3 to Theorem 4.1; we prove Theorem 4.1 in Section 5. The first step in the reduction is the following. Proposition 4.2. We have an F -isomorphism f : H (D; F 2 ) lim QH (E; F 2 ).

13 QUILLEN STRATIFICATION FOR THE STEENROD ALGEBRA 13 Proof. We need to show two things: every element in the kernel of f is nilpotent, and we can lift some 2 n th power of any element in the range of f. The first of these statements follows from the main theorem of [HP93] restriction to the elementary sub-hopf algebras of A (and hence of D) detects nilpotence. The second statement follows, almost directly, from a result of Hopkins and Smith [HSb, Theorem 4.12], stated as Theorem 4.3 below, combined with the fact that the Hopf algebra D is an inverse limit of finite-dimensional Hopf algebras. We explain. For each integer r 1, we let B(r) be the sub-hopf algebra of D which is dual to the following quotient Hopf algebra of D : B(r) = D /(ξ 1,ξ 2,...,ξ r ). Then each B(r) is normal in D, D//B(r) is finite-dimensional, and we have D = lim r D//B(r). Note that this inverse limit stabilizes in any given degree. We claim that given any obect E of Q and given any y H (E; F 2 ), some 2 n th power of y is in the image of the restriction map H (D; F 2 ) H (E; F 2 ). We have a normal sub-hopf algebra E(r) =E B(r) ofe, ande//e(r) is a sub- Hopf algebra of D//B(r). Furthermore, E = lim E//E(r), so any y H (E; F 2 )is represented by a class y H (E//E(r); F 2 ), for some r. By Theorem 4.3, then, there is an integer N so that y 2N lifts to H (D//B(r); F 2 ), and hence y 2N lifts to H (D; F 2 ). We have just used the following theorem: Theorem 4.3 (Theorem 4.12 in [HSb]). Suppose that Λ Γ is an inclusion of finite-dimensional graded connected cocommutative Hopf algebras over a field k of characteristic p>0. For any λ H (Λ; k), there is a number N so that λ pn is in the image of the restriction map H (Γ; k) H (Λ; k). Proof of Theorem 1.3, assuming Theorem 4.1. By Proposition 4.2, there is an F - isomorphism f : H (D; F 2 ) lim QH (E; F 2 ). We need to show that this induces an F -isomorphism on the invariants. Then composing with the F -isomorphism in Theorem 4.1 gives the desired result (it is clear that the composite of two F -isomorphisms is again an F -isomorphism). Since each restriction map H (D; F 2 ) H (E; F 2 )isana-algebra map, so is the assembly of these into the map f. So we have an induced map on the invariants: ( ) A f : (H (D; F 2 )) A lim Q H (E; F 2 ). Any x in the kernel of f is also in the kernel of f, and hence is nilpotent. Given any y (lim QH (E; F 2 )) A, we know that y 2n is in im(f) forsomen; by Lemma 4.4 below, we may conclude that y 2n+N im( f) forsomen. (To apply the lemma, we need to know that for every z H (D; F 2 ), Sq 2k (z) is zero for all but finitely many values of k. The action of A decreases degrees, by Proposition 2.3, so this is clear.) This finishes the proof of Theorem 1.3, given Theorem 4.1. We have used the following lemma. We say that an A-module M is locally finite if for every element m M, the submodule generated by m is finite-dimensional.

14 14 JOHN H. PALMIERI Lemma 4.4. Suppose that R and S are commutative A-algebras, with an A-algebra map f : R S that detects nilpotence: every x ker f is nilpotent. Suppose also that R is locally finite as an A-module. Given z R so that f(z) S is invariant under the A-action, then z 2N is also invariant, for some N. Proof. Since R is locally finite, then Sq 2n (z) = 0 for all but finitely many values of n. Since f is an A-module map and f(z) is invariant, then we also know that f(sq 2n (z)) = 0 for all n. Therefore Sq 2n (z) is nilpotent, and therefore, for N large enough, we have Sq 2n (z 2N ) = 0 for all n. 5. Proof of Theorem 4.1 In this section we prove Theorem 4.1. First we show that the restriction map H (A; F 2 ) H (D; F 2 ) factors through the invariants H (D; F 2 ) A,andthenwe show that the resulting map is an F -isomorphism. D is a normal sub-hopf algebra of A, so by Proposition 2.3(b), there is a spectral sequence 2 = H s,u (A//D; H t, (D; F 2 )) H s+t,u (A; F 2 ) It is well-known, and easy to verify from any reasonable construction of the spectral sequence, that the edge homomorphism H (A; F 2 ) E 0, 2 = H (D; F 2 ) A//D factors the restriction map (see [Ben91, p. 114], for instance). This gives us the desired map ϕ: H (A; F 2 ) H (D; F 2 ) A//D = H (D; F 2 ) A. We have to show two things: that every element of ker ϕ is nilpotent, and that for every y H (D; F 2 ) A, there is an integer n so that y 2n im ϕ. (The first of these follows from [Pal96, Theorem 3.1]; see also [HP93, Theorem 1.1]. We provide a proof here, anyway.) These two verifications have many common features, so first we establish some tools that we will use in both parts. For n 1 we define sub-hopf algebras D(n) ofa by D(n) = (F 2 [ξ 1,ξ 2,ξ 3,...]/(ξ1,ξ 2 2,...,ξ 4 n )) 2n. We let D(0) = A. This gives us a diagram of Hopf algebra inclusions: D(2) D(1) D(0) = A. D is the inverse limit of this diagram. Taking cohomology gives H (D(0); F 2 ) H (D(1); F 2 ) H (D(2); F 2 )... By Proposition 2.4, each D(n) is a normal sub-hopf algebra of A; hence there is an action of A on H (D(n); F 2 ). Lemma 5.1. (a) We have H (D; F 2 ) = lim H (D(n); F 2 ). (b) Furthermore, we have H (D; F 2 ) A = lim (H (D(n); F 2 )) A. Proof. Since D is the inverse limit of the D(n) s, and the diagrams D(3) D(2) D(1) D(0), H (D(0); F 2 ) H (D(1); F 2 ) H (D(2); F 2 )...,

15 QUILLEN STRATIFICATION FOR THE STEENROD ALGEBRA 15 stabilize in each degree, then part (a) is clear. Part (b): Since the action of each Sq k A takes H s,t (B; F 2 )toh s,t k (B; F 2 ) (for B = D or B = D(n)), then the action of A on any element of H (D; F 2 )= lim H (D(n); F 2 ) is completely determined at some finite stage of the direct system. Hence the ring of invariants of the direct limit is equal to the direct limit of the invariants. This lemma lets us translate information about H (D; F 2 )toh (D(n); F 2 )for some n 0. We also want to know how to go from D(n) tod(n 1) (and then apply downward induction on n, ending at D(0) = A). There is a Hopf algebra extension D(n) D(n 1) D(n 1)//D(n), and the quotient Hopf algebra is easy to describe: it is dual to F 2 [ξn 2n ] with ξ2n n primitive, so we have an isomorphism of algebras D(n 1)//D(n) = E(Pn n,pn n+1,pn n+2,...). (Recall from Section 2 that Pn n+s is the element of A dual to ξn 2n+s.) Anyway, by Proposition 2.3, for any D(n 1)-module M, we have a spectral sequence 2 (M) =H s,u (D(n 1)//D(n); H t, (D(n); M)) H s+t,u (D(n 1); M). Our main tool for the completion of the proof is the following. Lemma 5.2. Fix an integer N. Forsomer and some c, we have a vanishing plane at the E r -term of the spectral sequence: Er s,t,u (F 2 )=0for Ns+ t u>c. To prove this, we show that for a certain finite module M, we have the appropriate vanishing plane at E 2 (M); then we will use Theorem 3.1 to induce a vanishing plane (with the same normal vector) on E r (F 2 )forsomer. Lemma 5.3. Fix an integer j>n, and let M j be the D(n 1)//D(n)-module dual to E[Pn n,pn+1 n,...,pn j 1 ]. Then we have 2 (M) =0 for 2 j (2 n 1)s + t u>0. (So given an integer N, ifwechoosej so that 2 j (2 n 1) >N, then we will have 2 (M) =0forNs+ t u>0.) Proof. M j is a trivial D(n)-module (by definition), and as algebras, we have D(n 1)//D(n) = Mj E[P n j,pj+1 n,pn j+2,...]. Hence the E 2 -term of the spectral sequence for M j looks like 2 (M j ) = H s,u (D(n 1)//D(n); H t, (D(n); M j )) = H s,u (D(n 1)//D(n); M j H t, (D(n); F 2 )) = H s,u (E[Pn,P j n j+1,...]; H t, (D(n); F 2 )). The Hopf algebra E[Pn,P j n j+1,...]is2 j (2 n 1)-connected, so if L is a module which is zero above degree t, then H s,u (E[P j n,p j+1 n,...]; L) =0

16 16 JOHN H. PALMIERI when u<2 j (2 n 1)s + t. Now note that H t, (D(n); F 2 ) is zero above degree t (remember, the D(n 1)//D(n)-action lowers degrees, so to make H t, (D(n); F 2 ) into a graded module, we put H t,v (D(n); F 2 ) in degree v). We want to apply Theorem 3.1 to get a vanishing plane for the spectral sequence (F 2 ); first we describe how to build M j out of F 2, inductively. E r Lemma 5.4. For j>n, let M j be as in Lemma 5.3. Let M n = F 2. For each j n, there is a short exact sequence of D(n 1)//D(n)-modules 0 Σ 2j (2 n 1) M j M j+1 M j 0. The connecting homomorphism in H (D(n 1); ) is multiplication by h nj 1 Mj, where h nj =[ξn 2j ] H (D(n 1); F 2 ). Proof. This is clear. Proof of Lemma 5.2. Lin has shown [Lin, 3.2] that when j n, then the element h nj H (D(n 1); F 2 ) is nilpotent. Therefore Lemma 5.2 follows from the above lemmas, together with Theorem 3.1 and Proposition 3.4. Now we finish the proof of Theorem 4.1. We have two conditions to verify, and we devote a subsection to each one Every x ker(ϕ) is nilpotent. Fix x H (A; F 2 )sothatϕ(x) = 0; we want to show that x is nilpotent. As mentioned above, this has been proved elsewhere see [Pal96, Theorem 3.1] and [HP93, Theorem 1.1]. We present a proof here anyway, because we have done most of the work already in setting up the other half of the proofoftheorem4.1. Given Hopf algebras C B over a field k, we write res B,C : H (B; k) H (C; k) for the restriction map. We have an element x H (A; F 2 )sothatϕ(x) = 0. Then res A,D (x) =0, so by Lemma 5.1(a), res A,D(n) (x) =0forsomen. So it suffices to show that if res A,D(n) (x) = 0, then res A,D(n 1) (x j )=0forsomej then by downward induction on n, wehaveres A,D(0) (x j ) = 0; but A = D(0), and res A,D(0) is the identity map on H (A; F 2 ). Consider the spectral sequence 2 (F 2 )=H s,u (D(n 1)//D(n); H t, (D(n); F 2 )) H s+t,u (D(n 1); F 2 ) associated to the extension D(n) D(n 1) D(n 1)//D(n). Write z for res A,D(n 1) (x), so that res D(n 1),D(n) (z) =res A,D(n) (x) = 0. Because of this, z must be represented by a class z E p,q,v 2 with p>0. Choose an integer N so that Np+ q v>0. Then for any c, we can find a j so that Npj + qj vj > c c (just choose j> Np+q v ). By Lemma 5.2, for some r and c we have Es,t,u r (F 2 )=0 when Ns+t u >c. As noted above, we can choose j so that z j survives to E r and lies above this vanishing plane, so at the E r -term for which we have the vanishing

17 QUILLEN STRATIFICATION FOR THE STEENROD ALGEBRA 17 plane, z j must be zero. Hence z j =0inH (D(n 1); F 2 ), which is what we wanted to show Some power of every y in range(ϕ) lifts. Fix y H (D; F 2 ) A//D. We want to show that there is an integer m so that y 2m im ϕ. By Lemma 5.1(b), there is an n so that y lifts to H (D(n); F 2 ) A//D(n). Now we show that some power of y lifts to H (D(n 1); F 2 ) A//D(n 1) ;sinced(0) = A, then downward induction on n will finish the proof. Since y is invariant under the A//D(n)-action, then it is also invariant under the action of D(n 1)//D(n) (since the latter is a sub-hopf algebra of the former). So y represents a class at the E 2 -term of the spectral sequence 2 (F 2 )=H s,u (D(n 1)//D(n); H t, (D(n); F 2 )) H s+t,u (D(n 1); F 2 ). Also, y lies in the (t, u)-plane, say y E 0,q,v 2. Choose N large enough so that N + q v 1 is positive. By Lemma 5.2, we know that for some c and r, wehave Er s,t,u =0forc<Ns+ t u. Choose i so that 2 i c N > max(r 2, N + q v 1 ). We claim that y 2i is a permanent cycle. By Lemma 3.5, the possible differentials on y 2i are d j+1 (y 2i ) E j+1,2i q j,2 i v j+1, for j 2 i, so we show that the targets of these differentials lie above the vanishing plane. (Since 2 i was chosen to be at least r 1, then the vanishing plane is present at the E j+1 -term for all j 2 i.) We just have to verify the inequality specified by the vanishing plane: N(j +1)+2 i q j 2 i v =(N 1)j + N +2 i (q v) (N 1)2 i + N +2 i (q v) =2 i (N + q v 1) + N c N > (N + q v 1) + N N + q v 1 = c. Hence all of the differentials on y 2i vanish, so it is a permanent cycle. For degree reasons, it cannot be a boundary; hence it gives a nonzero element of E,and hence a nonzero element of H (D(n 1); F 2 ). Lastly, Lemma 4.4 tells us that some 2 k th power of y 2i is invariant in H (D(n 1); F 2 ). This completes the inductive step, and hence the proof of Theorem 4.1. References [Ben91] D. J. Benson, Representations and cohomology, Volume II, Cambridge studies in advanced mathematics, vol. 31, Cambridge University Press, [HP93] M. J. Hopkins and J. H. Palmieri, A nilpotence theorem for modules over the mod 2 Steenrod algebra, Topology 32 (1993), [HSa] M.J.HopkinsandJ.H.Smith,privatecommunication. [HSb] M. J. Hopkins and J. H. Smith, Nilpotence and stable homotopy theory II, preprint.

18 18 JOHN H. PALMIERI [Lin] W. H. Lin, Cohomology of sub-hopf algebras of the Steenrod algebra I and II, J.Pure Appl. Algebra 10 (1977), , and 11 (1977), [Mar83] H. R. Margolis, Spectra and the Steenrod algebra, North-Holland, [May70] J. P. May, A general algebraic approach to Steenrod operations, The Steenrod Algebra and its Applications (F. P. Peterson, ed.), 1970, Lecture Notes in Mathematics, vol. 168, Springer-Verlag, pp [Mil58] J. W. Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), [MPT71] H. R. Margolis, S. B. Priddy, and M. C. Tangora, Another systematic phenomemon in the cohomology of the Steenrod algebra, Topology 10 (1971), [MT68] M. E. Mahowald and M. C. Tangora, An infinite subalgebra of Ext A (F 2, F 2 ), Trans. Amer. Math. Soc. 132 (1968), [Pal] J. H. Palmieri, Stable homotopy over the Steenrod algebra, in preparation. [Pal96] J. H. Palmieri, Nilpotence for modulex over the mod 2 Steenrod algebra I, Duke Math. J. 82 (1996), [Qui71] D. G. Quillen, The spectrum of an equivariant cohomology ring I, II, Ann. of Math. (2) 94 (1971), , [Saw82] J. Sawka, Odd primary Steenrod operations in first quadrant spectral sequences, Trans. Amer. Math. Soc. 273 (1982), [Sin] W. M. Singer, private communication. [Sin73] W. M. Singer, Steenrod squares in spectral sequences I and II, Trans.Amer.Math.Soc. 175 (1973), , [Wil81] C. Wilkerson, The cohomology algebras of finite dimensional Hopf algebras, Trans. Amer. Math. Soc. 264 (1981), [Zac67] A. Zachariou, A subalgebra of Ext A (F 2, F 2 ), Bull. Amer. Math. Soc. 73 (1967), Room 2-388, Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge (our fair city), MA address: palmieri@math.mit.edu