The Bockstein and the Adams Spectral Sequences

PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 83, Number 1, September 1981 THE BOCKSTEIN AND THE ADAMS SPECTRAL SEQUENCES J. P. MAY AND R. J. MILGRAM Abstract. We show that, above the appropriate "vanishing line", the Adams spectral sequence of a connective spectrum can be read off from its Bockstein spectral sequence. In this short note, we prove a basic folklore theorem which relates the modp homology Bockstein spectral sequence of X to the Adams spectral sequence {ErX} converging from E2X = ExtA(H*X, Zp) to tr^X where A' is a bounded below spectrum with integral homology of finite type. As usual, we grade {ErX} so that ES2'X = Ex?A\H*X, Zp), with <L: Es/X -> Ers+r-'+r~xX, the total degree being t - s. We have a natural homomorphism 7s2'*Ar —»HmX which factors the modp Hurewicz homomor- phism, and we shall sometimes identify elements of 77, X with their inverse images in E°2*X. We have a pairing of spectral sequences ErS ® ErX -» ErX, where S is the sphere spectrum. Finally, we have an infinite cycle a0 E E2X,XSsuch that if x E E^X and if y E F\_SX projects to x, ihenpy E Fs+xir,_sX and py projects to a0x. Our main theorem will be a consequence of the following vanishing theorem, which is due to Adams [1] when p = 2 and to Liulevicius [4] when p > 2. Let A0= E{ß] 0. Theorem c A and recall that an .40-module M is free if and only if H(M;ß) 1. Let M be an (m + \)-connected A0-free A-module. = Then ExfA'(M, Zp) = 0for s > 1 and t - s < m + fis), where fis) = 2(p - \)s if p > 2 and, if p = 2, f(4k) = 8/c + 1, f(4k + 1) = 8/c + 2, f(4k + 2) = 8k + 3, and f(4k + 3) = 8Ä:+ 5. Definition 2. Let M be an A -module. We say that x E ExtA(M, Zp) generates a spike if x is not of the form a0x' and if a¿x f* 0 for all i. The set of spikes in ExtA(M, Zp) has its evident meaning. The same language will be applied to each ErX. Let K(R, n) denote the «th Eilenberg-Mac Lane spectrum of R and abbreviate 777Î = K(R, 0). Let y denote the canonical generator of HQ(HZp,) and let ßr denote the /-th mod/? Bockstein (in homology or cohomology according to context); let y = ßrZ- Received by the editors September 23, 1980. 1980 Mathematics Subject Classification. Primary 55T15; Secondary 55P42. © American Mathematical Society 0002-9939/81/0000-0429/$01.75 128 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 129 BOCKSTEIN AND ADAMS SPECTRAL SEQUENCES Lemma 3. (i) E2HZp = E^HZp is Zp in bidegree (0, 0). (ii) For r > 2, E2HZp, is the sum of a spike generated by y E E2'°HZ', and a spike generated by z E E20xHZpr; {ay\o < i < /•}. moreover, dr(a¿z) = a¿+ry and ExHZp, has basis (iii) E2HZ = EXHZ is a spike generated by y E E2fiHZ. Proof. H*(HZp) = A ■i, H*(HZp,) = (A/Aß) ■i © (A/Aß) ■ßri for r > 2, and H*(HZ) = (A/Aß) ■t, where t denotes the fundamental class. The calculation of the specified E2 terms is immediate by change of rings [2, VT.4.13], and the differentials in (ii) follow (up to a nonzero constant) by convergence. That the constant is 1 can be checked by a comparison of the constructions of the Adams and the Bockstein spectral sequences. We also record the following triviality. Lemma 4. Er(X V Y) = ErX © ErY, with dr = dr® dr. Here, now, is the main result. Its proof derives from a discussion of the edge theorem one of us had with Mark Mahowald many years ago. Theorem 5. Let X be an (m — \)-connected spectrum with integral homology of finite type. Let Cr, r > I, be a basis for the rth term ErX of the modp homology Bockstein spectral sequence of X and assume the Cr chosen so that Cr = Dr u ßrDr u Cr+X, where Dr, ßrDr, and Cr+ x are disjoint linearly independent subsets of ErX such that ßrDr = {ßrd\d E Dr) and Cr+X is a set of cycles under ßr which projects chosen basis for Er+xX. to the (i) The set of spikes in ErX, 2 < r and r = oo, is in one-to-one correspondence with Cr; if'c E C, has degree q and y E E*''X generates the corresponding spike, then fis) + m < q = / —s. (ii) If d E Dr and if 8 E Es/X and e E E"'VX, v-u = t-s-\, generate the spikes corresponding to d and to ßrd, then dr{a08) = a0+'+°-ut provided that m + fii + s) > t — s. Proof. Modulo torsion prime top, Hm(X;Z) is the direct sum of cyclic groups of order pr whose generators reduce modp to the elements of ßrDr and of infinite cyclic groups whose generators reduce modp to the elements of Cx. By exploiting the universal coefficients theorem and the representability of integral and modpr cohomology, we can use this decomposition to construct maps +,iX-*K{H/(X,Z\i) which induce isomorphisms (modulo torsion prime to p) on integral homology in degree i. The map * - 2 +,'.X-* V K(Ht(X;Z), i) = Y License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 130 J. P. MAY AND R. J. MILGRAM induces a monomorphism short exact sequence (•) on modp' homology for all r. In particular, we have a 0^77,(;r;Z/))^77,(r;Z/,)^A/,^0, and closer inspection of the construction of the <£,shows that M*" Since H(A/Aß;ß) 2 q>i + 2 H4{K{HiXtZ),i);Z,). = 0, we find (from the proof of Lemma 3) that the dual M of M^ is yl0-free and (m + l)-connected. exact sequence -► The exact sequence (*) gives rise to a long Extr ''(A/, Zp)-» E?X-* JÇ«y->Ext^M, Z,)-> • • • . By Theorem 1, E2''X -» E2y'Y is an epimorphism if s > I and t — s < m + fis) and is an isomorphism if s > 2 and / — j < m + fis — 1). The conclusions follow directly from Lemmas 3 and 4, by naturality. Remark 6. Spikes of E2X can be generated by elements lying in lower filtration degree than the range of isomorphism. Such generators can have nontrivial differentials earlier than predicted by the theorem (hitting classes annihilated by appropriate powers of a0); in particular, such differentials can occur on the bottoms of spikes the top parts of which survive to E^X. When X is a ring spectrum, such anomalous behavior is sometimes prevented by the relationship between the algebra structure of HtX and its Bockstein spectral sequence. We shall apply Theorem 5 to the study of the Adams spectral sequence converging to ir^MS Top in [3]. As will be illustrated there, the result can be a powerful tool for the computation of differentials in the Adams spectral sequence. Bibliography 1. J. F. Adams, Stable homotopy theory, Lecture Notes in Math., vol 3, Springer-Verlag, Berlin and New York, 1966. 2. H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N.J., 1956. 3. H. Ligaard, B. Mann, J. P. May and R. J. Milgram, The odd primary torsion of the topological cobordism ring (to appear). 4. A. Liulevicius, Zeroes of the cohomology of the Steenrod algebra, Proc. Amer. Math. Soc. 14 (1963), 972-976. Department of Mathematics, University of Chicago, Chicago, Illinois 60637 Department of Mathematics, Stanford University, Stanford, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use California 94305