I've run across a way of combining the integral cohomology of the real projective space $RP^\infty$ with its cohomology with twisted coefficients, that seems very simple and natural, but which I don't recall every seeing before, so my question is: Has this been noticed before and, if so, where is it published?


Let me describe the result in a relatively simple way, then comment on how I actually came to it. We know that $H^n(RP^\infty;\mathbb{Z})$ is $\mathbb Z$ if $n=0$, $\mathbb Z/2$ if $n>0$ is even, and 0 otherwise. If $\mathbb Z\_-$ denotes the local coefficient system on $RP^\infty$ on which the nontrivial loop acts as $-1$ on $\mathbb Z$, we know that $H^n(RP^\infty;\mathbb Z\_-)$ is $\mathbb Z/2$ if $n>0$ is odd, and 0 otherwise.


Let $R = \mathbb{Z}\times\mathbb{Z}/2$ and write elements of $R$ as $n + \epsilon\gamma$, where $n\in\mathbb Z$ and $\epsilon\in\mathbb Z/2$. Define a group graded on $R$ by
$$
 H^{n+\epsilon\gamma}(RP^\infty) =
 \begin{cases}
 H^n(RP^\infty;\mathbb Z) & \text{if $\epsilon=0$} \\
 H^n(RP^\infty;\mathbb Z\_-) & \text{if $\epsilon=1$.}
 \end{cases}
$$
This is a graded ring, using the pairing of coefficient systems $\mathbb Z\_- \otimes \mathbb Z\_- \cong \mathbb Z$, etc. With this definition,
$$
 H^\*(RP^\infty) \cong \mathbb Z[w]/\langle 2w \rangle
$$
where $w\in H^{1+\gamma}(RP^\infty) = H^1(RP^\infty;\mathbb Z\_-)$. (The class $w$ is the Euler class of the canonical line bundle over $RP^\infty$.) This could also be viewed as a description of the cohomology of the group $\mathbb Z/2$, of course. Similar simple descriptions can be given for the cohomologies of the truncated spaces $RP^k$.


Where this actually came from: Stefan Waner and I recently published a Springer Lecture Notes volume in which we describe ordinary equivariant (co)homology graded on "representations of the fundamental groupoid" of a space. This can be applied in the nonequivariant case as well — the group of representations of the fundamental groupoid of $RP^\infty$ is exactly the $R$ above, and the general theory leads to the calculation of $H^\*(RP^\infty)$ above with this grading.



The integral cohomology of real projective space
This description is given in Lemma 1 of 


* M. Cadek. The cohomology of $BO(n)$ with twisted integer coefficients. J. Math. Kyoto Univ. 39-2 (1999), 277-286.


(and Cadek goes on to give similar descriptions of the cohomology of infinite real Grassmannians)


