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By Peter J. Hilton, Urs Stammbach

We have inserted, during this variation, an additional bankruptcy (Chapter X) entitled "Some purposes and up to date Developments." the 1st portion of this bankruptcy describes how homological algebra arose via abstraction from algebraic topology and the way it has contributed to the data of topology. the opposite 4 sections describe functions of the tools and result of homological algebra to different elements of algebra. many of the fabric provided in those 4 sections was once now not on hand whilst this article used to be first released. evidently, the remedies in those 5 sections are a little bit cursory, the goal being to provide the flavour of the homo­ logical tools instead of the main points of the arguments and effects. we want to specific our appreciation of support obtained in writing bankruptcy X; particularly, to Ross Geoghegan and Peter Kropholler (Section 3), and to Jacques Thevenaz (Sections four and 5). the single different alterations include the correction of small blunders and, in fact, the expansion of the Index. Peter Hilton Binghamton, ny, united states Urs Stammbach Zurich, Switzerland Contents Preface to the second one version vii creation. . I. Modules.

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Definition. If S is a basis of the A-module P, then P is called free on the set S. We shall call P free ifit is free on some subset. 1. Suppose the A-module P is free on the set S. Then A s where As = A as a left module for S E S. Conversely, As P~ EB EB seS is free on the set seS {lA" S E S}. Proof. We define cp: P- EB As seS expressed uniquely in the form a= as follows: Every element a E P is 2: AsS; SES set cp(a) = (As}ses . Conversely, 4. Free and Projective Modules 23 for s E S define ips: As-P by ips (A,) = AsS' By the universal property ofthe direct sum the family {ips}, S E S, gives rise to a map ip = ips> : As- P.

Thus we get an underlying functor U:;:t-> 6. Similarly there are underlying functors from all the examples (a) to U) of categories (in Section 1) to 6. , in which some structure is "forgotten" or "thrown away". (e) The fundamental group may be regarded as a functor n: ;:to->ffi, where ;:to is the category of spaces-with-base-point (see [21J). It may also be regarded as a functor n: ;:t~ -> ffi, where the subscript h indicates that the morphisms are to be regarded as (based) homotopy classes of (based) continuous functions.

Let L Il k h = 0 C. k=1 and let jl

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