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Additional info for Algebra, Topology, and Category Theory. A Collection of Papers in Honor of Samuel Eilenberg

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Sometimes the results are valid for arbitrary coefficient rings in 33 34 CHARLES C H I N G - A N C H E N G A N D BARRY MITCHELL place of Ζ, and this leads to questions of when projective modules over ringoids of the form KC (K a field) are free, or in other words, coproducts of representables. Some instances where this is the case are mentioned at the end, but the reader is advised that the conjecture which this gives rise to is to be accepted with a grain of salt. 1. Nonadditive Flatness c P € Let C be a small category, M e S e t s ° , and i V e S e t s .

The g r o u p V is s free in the variety of nilpotent groups of class η and exponent p . The action of t on K, induced by any lifting of π, defines an action on V a r ee and a subsequent action on V/y2(V). VhiiV) is f Z p*-module and is also a Z p S[Z(f)]-module. Further reduction of V/y2(V) by its subgroup of pth-powers yields HX(K\ Z p ) ; the induced action by t yields the previous Z p [Z(i)]-module structure on H^K; Z p) . Zp) = F2g_2 © Zp(y). Let x l 5 x 2 g- 2 be By the proposition, Ηγ{Κ\ M a Z p[Z(i)]-basis for F2g_2.

Since j is onto, it follows from the theorem of W. 4 of [15] that j is an isomorphism if H^; Z ) is an isomorphism. Hence it suffices to show that Ηγ(]\ Zp) is an isomorphism for all primes p. ) W e know already that HJO'; Zp) is onto. By the propositon of the previous section we have and H1(K;Zp)*F2g-2®Zp(b) f Hx(K \Zp)^ F'2g_2®Zp(b') as Z p( Z ( f ) ] - m o d u l e s , where F2g_2 and F'2g_2 are each free on 2g — 2 generators and Zp(b) and Zp{b') are Z p- v e c t o r spaces of dimension one on which t operates trivially.

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