The Real Johnson-Wilson Cohomology of $\mathbb{C}{P}^{\mathrm{\infty }}$

We completely compute the Real Johnson-Wilson cohomology of $\mathbb{C}{P}^{\mathrm{\infty }}$. Applying techniques from equivariant stable homotopy theory to the Bockstein spectral sequence, we produce permanent cycles and solve extension problems to give an explicit description of the ring $ER(n{)}^{\ast }(\mathbb{C}{P}^{\mathrm{\infty }})$