Order of element in polynomial ring in Hatcher

Question

So I've been reading Hatcher and I am unsure what they mean when they say things like $H^\ast(\mathbb{R}P^n;\mathbb{Z}_2)\cong \mathbb{Z}_2[\alpha]/(\alpha^{n+1})$ where $|\alpha|=1$. It is this last part I don't understand. What do they mean by $|\alpha|=1$. Aren't the variables in the coefficient ring merely place holders?

Answer 1

Here $|\alpha|$ refers to which dimension cohomology group $\alpha$ is in, i.e. $|\alpha|=1$ means that $\alpha\in H^1(\mathbb{R}P^n;\mathbb{Z}_2)$. Note that the asserted isomorphism is supposed to be not just an isomorphism of rings but an isomorphism of graded rings, so you have to specify a grading on the ring $\mathbb{Z}_2[\alpha]/(\alpha^{n+1})$. Such a grading is uniquely determined by saying that every constant polynomial is in degree $0$ (this is not stated explicitly here, but is a standard assumption) and saying $\alpha$ has degree $1$.