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Nice homology coalgebras

1970; American Mathematical Society; Volume: 148; Issue: 2 Linguagem: Inglês

10.1090/s0002-9947-1970-0258919-6

1088-6850

A. K. Bousfield,

Homotopy and Cohomology in Algebraic Topology

A. K. BOUSFIELDO) 1. Introduction.Several different unstable versions of the Adams spectral sequence for homotopy groups have recently been constructed ([3], [9], [14]).While these spectral sequences differ for general spaces, it will be shown in [2] that they coincide (at least mod 2) for spaces having nice (2.2) homology coalgrebras, e.g. for spheres and loop spaces but not for S2 v S2.The purpose of this note is to prove two purely algebraic results on nice homology coalgebras which will be needed in [2].The first (4.1)provides a homological characterization of niceness, and the second ( §6) gives the structure of the Hopf algebra Cotorc ik, k) when C is a nice fc-coalgebra.Throughout this note we employ a theory of derived functors for nonadditive functors which is due to André [1], Quillen [13], and others.For convenience, we outline a simplified version of this theory in an appendix ( §7).The definition (2.2) of niceness for a homology coalgebra C comes essentially from Moore-Smith [12, §4] and involves the existence of a certain presentation for C. Our homological characterization of niceness is based on the notion of primitive dimension (3.1).In particular we study ( §3) right derived functors £n£ of the primitive element functor £, and we show (4.1) that a homology coalgebra Cis nice if and only if £n£(C) = 0 for n> 1, i.e.Chas primitive dimension^ 1.For a homology coalgebra C over a field k, £*£(C) is closely related to the Hopf algebra Cotorc ik, k) defined by Eilenberg-Moore [7] (see (5.1)).We construct (5.3) a spectral sequence of Hopf algebras whose £2-term depends only on £*£(C) and which converges to Cotorc ik, k).Using this we determine ( §6) the structure of Cotorc ik, k) when C is nice, and we thereby generalize certain results of [12].It should be noted that when C is of finite type Cotorc ik, k) is isomorphic to Extc.ik, k) and thus §6 gives information on the cohomology of algebras.Finally the author wishes to acknowledge that results similar to 4.1 and 6.1 for commutative rings have been proved by D. G. Quillen [16] using different methods.2. Homology coalgebras and their derived functors.Let k be a fixed field.A homology k-coalgebra is a connected positively graded A>coalgebra with commutative comultiplication [11].We let tfjk denote the category of homology A>coalgebras.Thus H^iX; k) e ^jk for any connected space X.