A theorem in homological algebra and stable homotopy of projective spaces

Abstract

Introduction. The paper exhibits a general change of rings theorem in homological algebra and shows how it enables to systematize the computation of the stable homotopy of projective spaces. Chapter I considers the following situation: R and S are rings with unit, h: R -+ S is a ring homomorphism, M is a left S-module. If an S-free resolution of M and an R-free resolution of S are given, Theorem I.1. shows how to construct an R-free resolution of M. Chapter II is devoted to computing the initial stable homotopy groups of projective spaces. Here the results of Chapter I are applied to the homomorphism a: A-+ A of the Steenrod algebra over Z2 (see 1.3). The main tool in computing stable homotopy is the Adams spectral sequence [1]. Let RP?, CP', HP' be the real, complex, and quaternionic infinite-dimensional projective spaces, respectively. If X is a space, let n s(X) denote the mth stable homotopy group of X [1]. Part of the results of Chapter II can be presented as follows: