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We present a very short way of calculating additively the stable (co)homology of Eilenberg–MacLane spaces K(Z/p, n). Our method depends only on homological algebra in appropriate categories of functors. Introduction. In the present paper we compute the stable homology of the Eilenberg–MacLane space K(Z/p, n). Of course the answer has been known for years and not the answer but the method is important here. We start our calculation from the point where the problem was approached classically—from the homotopical equality K(Z, n) = SP∞Sn and its mod p version. This leads to the interpretation of the stable homology of K(Z/p, n) in terms of stable derived functors in the sense of Dold–Puppe of symmetric power functors (also very classical). But then we use a more modern interpretation of these stable derived functors as Tor groups in the abelian category of functors from sets to Z/p-vector spaces. Then surprisingly the main result is derived from the properties of Koszul and de Rham sequences relating symmetric and exterior power functors. There are several reasons for writing this note. First of all we present a very short “stable” calculation of the stable homology of K(Z/p, n). Classically calculations followed the following route: first one had to do them unstably for all n and then to analyse the stabilization process to check which classes survive. It was always lengthy and complicated. Here we skip the unstable part. It seems to us that this point of view on the Steenrod algebra can be fruitful and we are going to explore it in further papers. Secondly we show that the “shape” of the Steenrod algebra (dimensions of 2000 Mathematics Subject Classification: Primary 55P20; Secondary 55S10, 55U99, 18G15. The author was partially supported by the Polish Scientific Grant (KBN) 2 P03A 002 18.

Stable derived functors, the Steenrod algebra and homological algebra in the category of functors