RESEARCH STATEMENT. 1. Introduction

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1 RESEARCH STATEMENT EVA BELMONT. Introduction One of the most fundamental problems in stable homotopy theory is calculating the stable homotopy groups of spheres, πns s = lim n π n+k S n. The simplest theorem about πns s is the calculation of its rationalization, which goes back to Serre [Ser53] inverting p in the spectral sequence for the p-local sphere. Adams [Ada66] proved that the rationalization of the Adams spectral sequence E 2 = Ext A (F p, F p ) = π S p s collapses at E 2, and E r = q 0 E r (where q 0 = [τ 0 ] Ext A (F p, F p ) converges to (S p S) π 0 S) above a line of slope 2p 2 for all r 2. Chromatic homotopy theory situates Adams theorem as the n = 0 case of a more general phenomenon. After fixing a prime p, there is a filtration of the category of p-local finite spectra F H (p) = F p,0 F p, F p,2... { } where F p,n is the full subcategory of all K(n )-acyclics [HS98]. If X is in F p,n and not F p,n+ we say it has type n. This means that X has a non-nilpotent self-map Σ k X X (for some k depending on X) that gives rise to a non-nilpotent operator v i n acting on the E page of the Adams spectral sequence for π X E 2 = Ext A (F p, H X) = π (X p). So we can consider the v n -localization, the colimit of repeated applications of v i n. Then: Theorem. (Hopkins-Palmieri-Smith [HPS99]). If X has type n, then its Adams E page has a vanishing line of slope 2p n 2 = v, and agrees with its v n n-localization above a line of slope 2p n+ 2 = v n+. From this point forward, we will work localized at an odd prime p (which will eventually be specialized to 3) and write k = F p. A longstanding program in chromatic homotopy theory is to compute chromatic localizations: Serre s computation of v0 π S discussed above was the first, followed by Miller s computation [Mil78, Corollary 3.6] of v π (S/p) at odd primes; here S/p is the mod-p Moore spectrum, which has type. These localizations give information about π S: given an element x π (S/p) we can form an infinite family S S/p vk S/p x S (where the first map is inclusion of the bottom cell), and similarly one studies infinite v n -periodic families in π S for higher n. Date: October 23, 207.

2 Chromatic homotopy theory determines vanishing lines on the E page of the Adams spectral sequence, but there are vanishing lines for Adams E 2 pages that are not given by powers of v n. Miller and Wilkerson [MW8] classify the possible E 2 page vanishing lines in terms of operators determined by the elements ξ ps t (for s < t) and τ n (for n 0) in A. Mahowald and Shick [MS87] clarify the analogy with chromatic vanishing lines: there are operators b ts for s < t (corresponding to ξ ps t ) and q n (corresponding to τ n, with q n = v n ), ordered by degree, which act parallel to vanishing lines in Ext A (k, M) for A-comodules M. If M = H X for a type n spectrum X, then the E 2 page might have the same vanishing line as the E page, corresponding to an action of q n, or it might have a higher slope, corresponding to an action of b ij with q n < b ij < q n. For example, at p > 3 Smith s complex V () has a v 2 -self map (and hence a vanishing line in its E page determined by powers of v 2 ), but its Adams E 2 page has the higher vanishing line determined by powers of b 0. Palmieri [Pal0] shows that these E 2 vanishing lines arise as part of a more general framework, gotten by mimicking the entire machinery of chromatic homotopy theory in the category Stable(A) of unbounded cochain complexes of A-comodules up to chain homotopy. Working over Stable(P ), where P = k[ξ, ξ 2,... ] A is the reduced powers, instead of Stable(A) restricts to just the algebraic periodicity operators b ij ; my aim is to compute the first algebraic chromatic localization in this setting. Goal.2. Compute 0 Ext P (k, M) for any A-comodule M. So far, my work has focused on the case M = k, but the most interesting application is to the case M = Ext E (k, k) =: Q (where E is the exterior algebra quotient of A generated by the τ i s): since the Cartan-Eilenberg spectral sequence E 2 = Ext P (k, Q) = Ext A (k, k) collapses for an odd prime, computing 0 Ext P (k, Q) gives a computation of a localization of the Adams E 2 page at p = 3. In other words, this computes the Adams E 2 page in the b 0 -periodic region; unlike the classical chromatic case, this is not the region above a line in the standard Adams grading, but rather the region above a vanishing plane defined using a third grading of Ext A (k, k). Proposition.3. At p > 2, if s > p(p 2 ) (t s ) + c (for a constant c) then Ext s,t P (k, Qs ) =,t 0 Exts P (k, Qs ). (Here Q i denotes the homological dimension i part of Q, Adams filtration is s = s + s, and t denotes internal topological dimension as usual.) In particular, at p = 3, 0 Ext P (k, k) = Ext P (k, k) above a line of slope 23. This region doesn t neatly fit in the hierarchy of chromatic height n wedges described by the vanishing line theorems: there are b 0 -periodic elements that correspond to height > (such as b 0 itself, which converges to β, the first element of coker J), but there are also elements in im J (such as an infinite family along the vanishing line) which are not b 0 -periodic. Much of my work thus far has been in studying 0 Ext P (k, k) at p = 3 via an analogue of an Adams spectral sequence constructed in the category Stable(P ). 2

3 Theorem.4. At p = 3, there is a convergent spectral sequence E 2 = k[h 0, b ± 0 ]/h2 0 k[w 2, w 3,... ] = 0 Ext P (k, k) for polynomial generators in Adams dimension (2, 2(3 n + ) 2). In section 4 we present evidence for and strategies for proving the following conjecture, which is equivalent to a conjecture about the remaining differentials and extensions in the spectral sequence. Conjecture.5. Let D = k[ξ ]/ξ 3 as a comodule-algebra. There is an isomorphism 0 Ext P (k, k) = 0 Ext D(k, W ) where W = k[w 2, w 3,... ] and the D-coaction is given by ψ(w n ) = w n + ξ w2 2w3 n. The element b 0 Ext 2 A (k, k) survives the Adams spectral sequence and converges to β π s. While β is nilpotent in classical homotopy, it is non-nilpotent in Ext BP BP (BP, BP ), as well as in the homotopy of the p-complete C-motivic sphere (S mot ) p. So studying its localization gives topological, as opposed to purely algebraic, information in the context of motivic homotopy theory. In particular, there is an element τ π 0, ((S mot ) p) in the homotopy of the p-completed motivic sphere such that the realization map from motivic homotopy theory to classical homotopy theory corresponds to inverting τ. That is, the τ-periodic part of π ((S mot ) p) corresponds to classical homotopy theory, and so recent work on understanding the unique properties of motivic homotopy theory centers around studying Cτ, the cofiber of multiplication by τ. Gheorghe, Wang, and Xu [GWX] show that π Cτ = Ext BP BP (BP, BP ), and the motivic Adams spectral sequence for Cτ coincides with the algebraic Novikov spectral sequence E 2 = Ext P (k, Q) = Ext BP BP (BP, BP ). The element b 0 Ext P (k, Q) converges to β in Ext BP BP (BP, BP ), which acts parallel to the vanishing line. One of my motivating goals is to understand the β -localization of the Adams spectral sequence for Cτ and compute β Ext BP BP (BP, BP ) = β π Cτ at p = 3. This would be the p = 3 analogue to Andrews and Miller s computation [AM4] of α Ext BP BP (BP, BP ). Then I could compute the τ-bockstein spectral sequence to compute β π ((S mot ) p). This would be a p = 3 analogue of Andrews and Miller s computation [AM4] of η π ((S mot ) 2 ). 2. The Margolis-Palmieri Adams spectral sequence We will use the following notation: k = F p D = k[ξ ]/ξ p D[x] = k[x]/x p and E[x] = k[x]/x 2 P = k[ξ, ξ 2,... ] (the algebra of reduced powers) P = k[ξ p, ξ 2, ξ 3,... ] = P Dk 3

4 Given an extension of Hopf algebras B A C and an A-comodule M, one can define the Cartan-Eilenberg spectral sequence E 2 = Ext B (k, Ext C (k, M)) = Ext A (k, M). This is the spectral sequence of the double complex C B (k) C A (k) where C B (k) denotes the cobar complex B. This can only be defined when B is a coalgebra, but if all we have is an A-comodule algebra structure on B, we can define a variant C B (k) of the cobar complex where the tensor products carry the diagonal A-coaction and the differentials are. We can describe this more functorially as follows. The forgetful functor A-comod C-comod has a left adjoint A C. Given an A-comodule M, the cosimplicial object associated to the free-forgetful monad is M A CM A C(A CM)... If B = A Dk, using the change of rings theorem we can write this as M B M B B M... where all tensor products carry the diagonal A-coaction. Taking normalized chain complexes, we have a quasi-isomorphism M B + M and C A (M) = C A (B + M). This gives rise to a double-complex spectral sequence (2.) E = Ext t A(k, B B s M) = Ext t C(k, B s M) = Ext A (k, M). where the isomorphism is from the change of rings theorem. Due to its construction as the spectral sequence associated to a bi-cosimplicial algebra, this has power operations [Sin06]. This spectral sequence is more general than the Cartan-Eilenberg spectral sequence it can be defined using just a map A C of coalgebras. When A C is a normal map of Hopf algebras, then this is isomorphic to the Cartan-Eilenberg spectral sequence (using repeated applications of the shear isomorphism to turn tensor products with diagonal coactions into left coactions as in the cobar complex). In our case, we use the map P D of coalgebras with M = k, and take the b 0 -localization of the entire spectral sequence. If we write P := P Dk, this gives a spectral sequence (2.2) E s,t = 0 Extt D(k, P s ) = which has been my main focus. 0 Exts+t P (k, k) The construction (2.) coincides with a spectral sequence introduced by Margolis [Mar83] and developed by Palmieri [Pal0], obtained by constructing the Adams spectral sequence in the category Stable(A) of unbounded chain complexes of A-comodules up to chain homotopy. This is a stable homotopy category in the sense of Hovey-Palmieri-Strickland [HPS97]. There is a functor A-comod Stable(A) sending a comodule M to its injective resolution. If we also let M denote its image in Stable(A), we have Hom Stable(A) (M, N) = Ext A (M, N). The analogue of the classical functor π ( ) = [S, ]: Ho(Top ) Ho(Top ) is Hom Stable(A) (k, ); we will also call this π ( ). If M is a A-comodule then π (M) = Ext A (k, M), and in particular, the analogue of π S is π (k) = Ext A(k, k). Margolis [Mar83] and Palmieri [Pal0] study the Adams spectral sequence in Stable(A), defined by mimicking the construction of the classical Adams spectral sequence: given objects 4

5 X, E Stable(A) satisfying conditions analogous to those required to construct the classical Adams spectral sequence, there is a spectral sequence [Pal0, Theorem.4.2] (2.3) E2 = Ext s,t,u E (E E, E X) abuts = π X which converges when X is connective and E-complete. We will refer to this as the Margolis- Palmieri Adams spectral sequence (MPASS). If X = C is an A-comodule, under favorable conditions Adams flatness is satisfied and this recovers the Cartan-Eilenberg spectral sequence for the extension A Ck A C. The spectral sequence (2.2) coincides with the MPASS, but working in Stable(P ) instead of Stable(A), where X = k and E = K(h 0 ) := 0 (P Dk). At p = 3 (but not for higher primes), flatness conditions are satisfied for K(h 0 ), and hence the spectral sequence (2.2) has the E 2 term given by (2.3). This fits into a larger program to compute chromatic localizations in this algebraic category. There are objects of Stable(A) which function like Morava K-theories: define k(h ts ) = A D[ξ p s t ] k for s < t, which has π (h ts ) = Ext D[ξ p s t ] (k, k) = k[b ts, h ts ]/h 2 ts by change of rings, and k(q i ) = A E[τi ]k, which has π k(q i ) = k[q i ]. As in the classical case, these have nonconnective versions K(h ts ) = ts k(h ts) and K(q i ) = qi k(q i ), where q i -periodicity corresponds to v i -periodicity in ordinary chromatic homotopy theory. Working over with Stable(P ) instead of Stable(A), as we choose to do, has the effect of restricting to just the K(h ts ) s, the Morava K-theories that do not correspond to homotopy-theoretic periodicity operators v n. Margolis [Mar83] and Palmieri [Pal94] studied the following spectral sequence (a special case of the MPASS above with E = vn K(q n )) to compute localized Ext groups: if a comodule M has a q n -self map then E 2 = Ext K(qn) K(q n)(k(q n ), K(q n ) M) k[v ± n ] = v n Ext A (k, M) converges. Palmieri computed K(q 0 ) K(q 0 ) explicitly, which showed that the MPASS converging to q0 Ext A (k, k) collapses at E 2, recovering Adams result on the rationalization of the Adams spectral sequence for the sphere. Palmieri also gave an explicit computation of K(q ) K(q ) and showed that the MPASS collapses at E 2 for odd primes, recovering Miller s computation of the v -inverted Adams E 2 page for the Moore space S/p. The next chromatic localization in Stable(A) would be at K(h 0 ); localizing at K(h 0 ) is also the first chromatic localization in Stable(P ), which is the subject of our study. 3. E and E 2 pages of MPASS Recall the E 2 page of the MPASS is Ext K(h0 ) K(h 0 )(K(h 0 ), K(h 0 ) ), where K(h 0 ) = 0 (P Dk) = 0 P and K(h 0 ) = k[b ± 0, h 0]/h 2 0. By the change of rings theorem, we have K(h 0 ) K(h 0 ) = 0 Ext P (k, P 2 ) = 0 Ext D(k, P ). We compute this cooperation algebra by explicitly determining the D-comodule structure of P modulo D-free summands, which do not contribute to the b 0 -localized cohomology. At any prime, every D-comodule is a sum of comodules of the form M(c) = k[x]/x c+, where x is primitive. 5

6 Proposition 3.. Let p > 2. Then we have a decomposition of D-comodules: P Dk = M(c i )ξ c i k i D-free summands. monomials ξ c k...ξ cn kn c i <p where M(c)ξk c denotes the copy of M(c) generated by ξc k. At p = 3, this is M()ξ ki D-free summands. ξ k...ξ kn i i One of the reasons that the analogous computation at p = 2 is much easier is that there is a Künneth formula for Ext E (k, ). No such formula holds for odd primes: Renaud [Ren79] gives a formula for the decomposition of M(n) M(m) in terms of other M(i) s, but it quickly becomes unwieldy. At p = 3, one can make significant simplifications over the situation at an arbitrary odd prime. There are only three D-comodules, namely k, M(), and M(2) = D, and moreover in Stable(D) we have M() = Σ k (where Σ indicates chain complex shift in Stable(D)), which implies 0 Ext D(k, M()) = Σ 0 Ext D(k, k). Thus, at p = 3, 0 Ext D(k, P ) = ξ k...ξ kn 0 Ext +n D (k, k ξ k...ξ kn ). Theorem 3.2. Let p = 3. Then 0 Ext D (k, P ) is generated as an algebra by classes e n = 2[ξ ξ n ] + [ξ 2 ξ 3 n ] 0 Ext D(k, k ξn ) and moreover these classes are primitive. More precisely, if K(h 0 ) = 0 P then the associated Hopf algebroid (K(h 0 ), K(h 0 ) K(h 0 )) is a Hopf algebra with coefficient ring K(h 0 ) = 0 Ext D (k, k) = k[h 0, b ± 0 ]/h2 0 and K(h 0 ) K(h 0 ) = K(h 0 ) [e 2, e 3,... ]/(e 2 2, e 2 3,... ). Corollary 3.3. Let w n = [e n ] Ext K(h0 ) K(h 0 )(k, k). Then the E 2 term of the K(h 0 )-based MPASS computing π k is E 2 = Ext K(h0 ) K(h 0 )(K(h 0 ), K(h 0 ) ) = K(h 0 ) [w 2, w 3,... ] where w n has MPASS filtration, internal topological degree 2(3 n +), and internal homological degree. In particular, K(h 0 ) K(h 0 ) is flat over K(h 0 ). The classes w n have representatives 2[ξ ξ n ] + [ξ 2 ξ3 n ] P P in the cobar complex for Ext P (k, k). For example, w 2 = g 0 = h 0, h 0, h is a permanent cycle, as is w2 2 = b 0k 0 = b 0 h 0, h, h and w2 3 = b2 0 b. But the rest of the w n s are not cocycles in the P -cobar complex they support MPASS differentials. Most of what made this calculation possible was the very simple Hopf algebra structure of 0 Ext D(k, P ) at p = 3. At higher primes, the situation is significantly more complicated: K(h 0 ) K(h 0 ) is not flat over K(h 0 ), and does not have a natural Hopf algebroid structure, though we can still study the E page and try to compute the differentials in the double complex spectral sequence described in section 2. 6

7 4. Higher differentials at p = 3 For degree reasons, the only nontrivial differentials in (2.2) are d r with r 4 (mod 9) and r 8 (mod 9). Moreover, d 4+9k is only (possibly) nonzero on the non-h-divisible part of E 4+9k, and d 8+9k is only (possibly) nonzero on the h-divisible part of E 8+9k. Conjecture 4.. () For n 3, d 4 (w n ) = b 4 0 h 0w 2 2 w3 n. (2) If d 4 (x) = h 0 y and d 4 (y) = h 0 z, then d 8 (h 0 x) = b 0 z. (3) For r 9, d r = 0. This suggests a stronger conjecture, which is equivalent to the above plus information about multiplicative extensions: Conjecture 4.2. Let W = k[w 2, w 3,... ]. Then 0 Ext P (k, k) = 0 Ext D(k, W ) where the D-coaction on W is given by (4.)(). One can verify this conjecture for n = 3, 4, 5 by comparing against a computer-generated chart for Ext P (k, k). 4.. Spectral sequence comparison. Let D = k[ξ, ξ 2,... ]/(ξ 3, ξ9 2, ξ9 3,... ) and let B = P D k = k[ξ3, ξ9 2, ξ9 3,... ]. There is an inclusion B P Dk, and (after localizing at b 0 ) we study the resulting map of Margolis-Palmieri Adams spectral sequences. The 0 B -based MPASS for π ( 0 k) has E 2 term Ext (b 0 B ) ( 0 B ) ((b 0 B ), ( 0 B ) ) where ( 0 B ) = 0 Ext P (k, B ) = 0 Ext D (k, k) (by the change of rings theorem) and ( 0 B ) ( 0 B ) = 0 Ext P (k, B B ) = 0 Ext D (k, B ) = 0 Ext D (k, k) B since B is a trivial D -comodule. To calculate Ext D (k, k), we use a Cartan-Eilenberg spectral sequence for the extension D D k[ξ 2, ξ 3,... ]/(ξ 9 i ). We calculate the E 2 page and verify that all of the classes in the E 2 page are permanent cycles. Proposition Ext D (k, k) = k[b± 0, h 0, h 20, w n, b n,0 ] n 3 /(h 2 0, h2 20 ) Using a minimal resolution for k over D to compute the coaction H D B H D, we obtain the following. Proposition 4.4. In the 0 B -based MPASS, there are differentials d (w n ) = b 4 0 h 0w 2 2 w3 n. 7

8 There is an inclusion map B P Dk that induces a ring map and hence a map of spectral sequences. Note that the differentials in Proposition 4.4 increase filtration from 0 to, whereas the desired differentials in the 0 (P Dk)-based MPASS increase filtration from to 5. In topology, given a map of ring spectra one can try to use Miller s square of spectral sequences [Mil8, Theorem 6.] to deduce differentials in the second Adams spectral sequence from differentials in the first. Though the theorem is formal enough to have an analogue in Stable(P ), it does not apply in this case, as the spectral sequence collapse hypothesis is not satisfied here A more explicit form for the conjectured answer. We have a partial computation of 0 Ext D(k, W ) as a vector space, which we conjecture above to be isomorphic to 0 Ext P (k, k). As an intermediate step, we consider the cohomology of W := k[w3 3, w 4, w 5,... ] equipped with the D-comodule structure given by x ξ x where : w n wn 3 for n 4. Proposition 4.5. We have an explicit list of generators of 0 Ext D(k, W ) in terms of a D-comodule decomposition W = W (k) W (M()) W (D) where W (M) is a direct sum of copies of the D-comodule M. For example, for every vector space basis element x W (k) there are generators w2 i (2[ξ w 3 x] + [ξ 2 w5 2x]) for 0 i 4 and w2 i x for 0 i 4 in b 0 Ext D(k, W ). Moreover, we have an explicit formula for a basis of W (k) and W (M()). However W (D) is much more complicated, though we can describe the degrees in which generators live. I plan to attack this problem by focusing on algebra structure, which might provide a simpler description of 0 Ext D(k, W ). 5. Future work 5.. Generalization to 0 Ext P (k, M). My immediate goal is to finish and extend the work outlined in this document: proving Conjecture 4.2 about the structure of 0 Ext P (k, k) and extending it to arbitrary coefficients 0 Ext P (k, M). In addition to the methods outlined above, I am also working on using iterated Cartan-Eilenberg spectral sequences to take advantage of power operations in the non-localized Ext over various subalgebras of P Non-nilpotence in β Ext BP BP (BP, BP ). Completing the work above would result in a b 0 -localization of the E 2 page of the 3-primary algebraic Novikov spectral sequence E 2 = Ext P (k, Ext E (k, k)) = Ext BP BP (BP, BP ). The element b 0 converges to β in Ext BP BP (BP, BP ), so computing the rest of the localized spectral sequence would result in a calculation of β Ext BP BP (BP, BP ) (note that β acts parallel to the vanishing line in the Adams-Novikov E 2 page). This would be the p = 3 analogue of a calculation of Andrews and Miller [AM4] at the prime 2. In addition to being of independent interest, this is especially relevant to motivic homotopy theory: there is an element τ π 0, ((S mot ) p) in the homotopy of the p-complete motivic sphere, 8

9 and Gheorghe, Wang, and Xu [GWX] have proved that π (Cτ) = Ext BP BP (BP, BP ). So understanding the β -localization of the algebraic Novikov spectral sequence would lead to a computation of β π (Cτ). Moreover, at this point I also plan to compute the β -localization of the τ-bockstein spectral sequence, leading to a computation of β π ((S mot ) p). This would be the first chromatic localization of the odd-primary motivic sphere, and would be the analogue of a computation by Andrews and Miller [AM4] of η π ((S mot ) 2 ) Nilpotence of b at higher primes. The differential d 8 (h 0 w3 2) = b 7 0 w0 2 = b 7 0 w 2b 3 shows that w2 3b3 = b4 = 0 in b 0 Ext P (k, k). Since b n is in the b 0-periodic part of the Adams E 2 page for sufficiently large n, this shows that b is nilpotent in Ext A (k, k). This was already known by Nakamura [Nak75], who showed using explicit cobar computations that b 2 is divisible by h. However, the question of whether b is nilpotent at p > 3 is still open, and Palmieri conjectures that it is nilpotent. Since the slope of b -multiplication is higher than the slope of the line delimiting the b 0 -periodic region, it suffices to check whether b is nilpotent after inverting b 0. While the E 2 page of the MPASS at higher primes is probably quite complicated, in order to verify this conjecture it would simply suffice to compute the MPASS for the first few generators. References [Ada66] J.F. Adams. A periodicity theorem in homological algebra. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 62, pages Cambridge Univ Press, 966. [AM4] Michael Andrews and Haynes Miller. Inverting the Hopf map in the Adams-Novikov spectral sequence Preprint: [GWX] Bogdan Gheorghe, Guozhen Wang, and Zhouli Xu. BP BP -comodules and cofiber of tau modules. [HPS97] Mark Hovey, John Palmieri, and Neil Strickland. Axiomatic stable homotopy theory, volume 60. American Mathematical Soc., 997. [HPS99] M J Hopkins, J H Palmieri, and J H Smith. Vanishing lines in generalized Adams spectral sequences are generic. Geometry & Topology, 3():55 65, jul 999. [HS98] Michael J. Hopkins and Jeffrey H. Smith. Nilpotence and stable homotopy theory II. The Annals of Mathematics, 48():, jul 998. [Mar83] Harvey Margolis. Spectra and the Steenrod algebra. North-Holland Publishing Co., Amsterdam, 983. [Mil78] Haynes Miller. A localization theorem in homological algebra. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 84, pages Cambridge Univ Press, 978. [Mil8] Haynes Miller. On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore space. Journal of Pure and Applied Algebra, 20(3):287 32, 98. [MS87] Mark Mahowald and Paul Shick. Periodic phenomena in the classical adams spectral sequence. Transactions of the American Mathematical Society, 300():9 9, jan 987. [MW8] Haynes Miller and Clarence Wilkerson. Vanishing lines for modules over the Steenrod algebra. Journal of Pure and Applied Algebra, 22(3): , 98. [Nak75] Osamu Nakamura. Some differentials in the mod 3 Adams spectral sequence. Bull. Sci. Engrg. Div. Univ. Ryukyus Math. Natur. Sci, (9): 25, 975. [Pal94] John Palmieri. The chromatic filtration and the Steenrod algebra. Contemporary Mathematics, 58:87 87, 994. [Pal99] John Palmieri. Quillen stratification for the Steenrod algebra. Annals of mathematics, 49:42 449, 999. [Pal0] John H. Palmieri. Stable homotopy over the Steenrod algebra, volume 5. American Mathematical Society (AMS), 200. [Ren79] J-C Renaud. The decomposition of products in the modular representation ring of a cyclic group of prime power order. Journal of Algebra, 58():, 979. [Ser53] Jean-Pierre Serre. Groupes d homotopie et classes de groupes abéliens. Annals of Mathematics, pages , 953. [Sin06] William Singer. Steenrod Squares in Spectral Sequences. American Mathematical Society, aug