The confusing passage is also here on page 2. May's usage of graded $R$-modules appears to differ from the standard definition, as far as I can tell. He is taking graded $R$-modules to mean the sequence of homology groups. Is this sequence actually a graded module, or is this just an idiosyncratic definition?
2026-02-22 21:33:16.1771795996
Meaning of "graded $R$-modules" in May's "A Concise Course in Algebraic Topology" on Page 89
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Sequences of modules and graded modules (over a non-graded ring) are completely interchangeable. Given any (non-graded) ring $R$ and a sequence $(A_n)$ of $R$-modules, the direct sum $\bigoplus_n A_n$ is a graded $R$-module, with the degree $n$ part just being $A_n$. Conversely, a graded module (up to isomorphism) is determined by the sequence of its degree $n$ parts. So, since the homology modules of a chain complex are a sequence of modules, they can also be thought of as a graded module. (Explicitly, if you define a graded module as a module with a direct sum decomposition, the graded module is the direct sum of all the homology modules.)
(Of course, the direct sum $\bigoplus_n A_n$ does have one major advantage over the bare sequence $(A_n)$, in that you can talk about elements of $\bigoplus A_n$ that are not elements of any single $A_n$. May explicitly mentions that he will not ever need to do this ("we never take the sum of elements in different gradings"), and so there is no loss of convenience in thinking of the graded module as just a sequence.)