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On the Homotopy Theory of K-Local Spectra at an Odd Prime Author(S): A On the Homotopy Theory of K-Local Spectra at an Odd Prime Author(s): A. K. Bousfield Source: American Journal of Mathematics, Vol. 107, No. 4 (Aug., 1985), pp. 895-932 Published by: The Johns Hopkins University Press Stable URL: http://www.jstor.org/stable/2374361 Accessed: 14/09/2009 17:09 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=jhup. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit organization founded in 1995 to build trusted digital archives for scholarship. We work with the scholarly community to preserve their work and the materials they rely upon, and to build a common research platform that promotes the discovery and use of these resources. For more information about JSTOR, please contact [email protected]. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. http://www.jstor.org ON THE HOMOTOPY THEORY OF K-LOCAL SPECTRA AT AN ODD PRIME By A. K. BOUSFIELD* Introduction. In this paper we investigate the algebraic structure of the stable homotopy category localized with respect to K-homology theory at an odd prime, and we give a purely algebraic classification of all homo- topy types in that localized category. Before describing our results more fully, we must recall the theory of homological localizations of spectra (see [5], [7], [11]). Let HoS denote the homotopy category of CW-spectra (see [5]) or any of the other equivalent versions of Boardman's stable homotopy cateogry. A spectrum E E Ho' determines a homology theory E* with E*X = 7-*E A X for each X E Ho', and a spectrum Y E Ho' is called E*-local if each E*-equivalence A -+ B in Ho' induces an isomorphism [B, Yn* [A, Yn*.For each E E Ho' there is a natural E*-localization which assigns to each spectrum X E Ho' an E*-equivalence X -_ XE in Ho' such that XE is E*-local. It follows that the full subcategory of E*-local spectra in HoS is equivalent to the category of fractions obtained from Ho' by giving formal inverses to the E*-equiva- lences. In this paper, we are interested in the case E = K(p) for an odd prime p where K(P) is the spectrum of nonconnective complex K-theory localized at p. However, we often find it convenient to replace K(P) by its summand E(1) from the well-known splitting K(P) vPi---o2S2iE(1) (see Section 4). Since the K(p)*-equivalencesin Ho5 are the same as the E(1)*- equivalences, it follows that the K(p)*-localizationin Ho5 is the same as the E(1)*-localization. This localization has previously been studied by Ad- ams-Baird (unpublished), Ravenel [11], and the author [7]. We remark that it is the next after the rational localization in Ravenel's hierarchy of localizations which take account of progressivelyhigher sorts of periodicity phenomena in p-local stable homotopy theory. In order to approach the homotopy theory of a K(P)*-local (= E(1)*- local) spectrum X, we first consider the homology groups K(P)*X or Manuscript received April 4, 1983. * Research partially supported by NSF Grant MCS-7900807. 895 896 A. K. BOUSFIELD E(1)*X together with appropriate primary operations. The required infor- mation can be captured by treating K(P)*X or E(1)*X as a comodule over the coalgebraK(p)*K(p) orE(l)*E(1). However, this comodule structure is somewhat awkward to work with, and we find that exactly the same infor- mation is more conveniently captured by the stable Adams operations 1k with k prime to p. To formalize this, we construct graded abelian catego- ries (i(p)* and 63(p)* whose objects have the algebraic properties of the homology groups K( )*X and E(1)*X respectively with their stable Adams operations. We show that (i(p)* and 63(p)* are canonically equivalent to each other and we use 63(p)* in most of our subsequent work. In the last section (Section 10) we demonstrate that (i(p)* and (5(p)* are canonically equivalent to the categories of K(P)*K(p)-comodulesand E(l)*E(1)-com- odules respectively. This is of independent interest and generalizes a some- what similar equivalence obtained by Ravenel [11] for torsion E(1)*E(1)- comodules using a very different approach. In Sections 1, 3 we construct (i(p)* and 63(p)* as graded periodic versions of ungraded abelian catego- ries (i(p) and (5(p) which resemble the abelian category of Adams [1] but differ because we impose rational diagonalizability conditions on our oper- ations and work in a p-local nonfinitely generated context. In Section 5 we give a very simple alternative construction of 63(p) involving a single oper- ation ,r, and we see that a torsion object of 63(p) is merely a p-torsion abelian group with a locally nilpotent operator. Interestingly enough, as noted by Ravenel [11], these same torsion objects have been studied by Iwasawa in connection with cyclotomic fields (see [8], [9], [12]). The struc- ture of 63(p)* is likewise very accesible since that category is equivalent to the product of 2p - 2 copies of 63(p). In Section 7 we obtain detailed results on the groups Ext"j(P)*(M, N) for s = 0, 1, 2 and show that they all vanish for s > 2 so that the objects of 63(p)* have injective dimension c 2. A slightly weaker version of this vanishing result was obtained by Adams- Baird in the equivalent category of K(P)*K(p)-comodules(see [4]). In order to investigate the homotopy theory of E(1)*-local (= K(p*- local) spectra, we need the E(1)*-Adams spectral sequence. In Section 8 we construct a version which has (X Y) Ext3t(p)* (E(l)*X, E(l)*Y) Es,t =z for arbitraryX, Y e Hos and converges strongly to [XE(l), YE(1)] *. We next note that EMt(X, Y) = 0 for s > 2, so that the only possible nontrivial differential is HOMOTOPY THEORY OF K-LOCAL SPECTRA 897 d2: Hom,33(p)*(E(l)*X, E(1)* Y)t -+Ext,33(+)*2tl(E(l)*X,EM1* Y). To determine this d2 algebraically, we introduce the canonical E(1)*-k- invariant kw E Ext 2,1 P)*(E(1)*W, E(1)*W) for each spectrum W E Hos and prove the general formula d2f = k y of + (- )t+ lf ? kx. Our definition of k w involves the theory of E(1)*-Moore spectra. For each object G E 63(p) and each n E Z, we build an E(1)*-local spectrum 9R(G, n) with an isomorphism E(1)n91(G, n) = G in 63(p) and with E(1)iM(G, n) = 0 for i * n mod 2p - 2. We call 91(G, n) an E(1)*- Moore spectrum and show that it is unique up to a canonical equivalence and depends functorially on G E 63(p). Also for each object H E (B(p)* we build a natural E(1)*-local spectrum 9(H) = VML0391(Hn n) with E(1)* 9(H) ~H in (5(p)*, and we call 9(H) a generalized E(l)*-Moore spectrum. For W E Ho', our E(1)*-k-invariant kw measures the obstruc- tion to finding a map W -+ 1(E(1)* W) inducing the identity onE(1)* W. In Section 9 we arrive at our main results on the algebraic classifica- tion of the E(1)*-local (= K(P)*-local) spectra. First we easily see that two spectra X and Y in Ho' have equivalent E(1)*-localizations if and only if the pairs (E(1)*X, kx) and (E(1)* Y, k y) are isomorphic. Then we prove that for each object M E (B(p)* and each element K E Ext2jP) (M, M) there exists a spectrum W E Hos such that (E(1)* W, k w) is isomorphic to (M, K). Indeed, we prove that this W may be chosen to be a finite CW-spectrum if and only if M is of finite type over Z(p). We arrive at the result that the homotopy types of the E(1)*-local spectra correspond to the isomorphism 2,1 classes of the pairs (M, K)with M E (B(p)* and KE Extffl(,,)*(M,M). In fact, we obtain much stronger results on the homotopy category Hos( 1)of E(1)*- local spectra. Using the E(1)*-Adams spectral sequence we algebraically determine the bigraded category obtained from HoSE1)by taking Adams filtration quotients, and also determine the full subcategory of Hos(1)given by the generalized E(1)*-Moore spectra. In particular, for M, N E (p)* we obtain canonical isomorphisms [X(M), 91(N)]n ?) Ext3'(p)*(M, N) such that compositions of homotopy classes correspond to Yoneda prod- ucts.
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