By Nigel Ray, Grant Walker

J. Frank Adams had a profound impact on algebraic topology, and his paintings maintains to form its improvement. The foreign Symposium on Algebraic Topology held in Manchester in the course of July 1990 used to be devoted to his reminiscence, and nearly the entire world's top specialists took half. This quantity paintings constitutes the court cases of the symposium; the articles contained right here diversity from overviews to experiences of labor nonetheless in growth, in addition to a survey and entire bibliography of Adam's personal paintings. those court cases shape a big compendium of present study in algebraic topology, and person who demonstrates the intensity of Adams' many contributions to the topic. This moment quantity is orientated in the direction of homotopy conception, the Steenrod algebra and the Adams spectral series. within the first quantity the subject is principally risky homotopy concept, homological and specific.

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_1))} is a direct limit of polynomial (exterior) algebras, when r is even (odd) and hence is itself polynomial (exterior). Remark 9. The above also allows us to compute the cohomology ring H*(E(n)2r). Tor of an exterior algebra is a divided power algebra and so the dual Rothenberg-Steenrod spectral sequence {TorH'(E(I')2r-1)(FP, FP)}* ==* H" (E(n)2r) collapses as it is entirely in even dimensions. However, unlike in the case of the MU Hopf ring, the algebra H*(E(n)2r_1) is not of finite type, and Hunton : Detruncating Morava K-theory 40 so we are unable to conclude that the E,,,,,-term of this sequence, and hence H*(E(n);r), is polynomial - it is in fact far larger.

Proof. We begin by proving the second part of the statement. First recall Wilson's splitting theorem: Theorem 6, (W. S. 4)). For r < 2(pn + .. 1)r+2(pi-1) j>n We deduce: Corollary 7. For r sufficiently small the Hopf algebra H*(BP(n)') is bipolynomial if r is even and exterior if r is odd. In particular, for r both even and small, K(n)*(BP(n)r) is a formal power series algebra. Hunton : Detruncating Morava K-theory 39 Proof. Theorem 6 gives a factorisation of the identity map BP(n)', BP;. --* BP(n);.

2 Example. For a KO-module spectrum G and spectrum X, 7r; RTG and KCRTX are Anderson exact. 3 Theorem. For a module Al E CRY, the following are equivalent: (i) M is Anderson exact with M* free abelian; (ii) M is projective in CRT; (iii) M is free in CRT. 4 Corollary. If X is a spectrum with K*X free abelian, then K; RTX K; RTW for some wedge W of suspensions of Q71), S', and C(,q2). 5 Theorem. Y, the following are equivalent: (i) M is Anderson exact; (ii) M has projective dimension < 1 in CRT; (iii) M has injective dimension < 1 in CRT.