The Real-oriented cohomology of infinite stunted projective spaces

Let $E\mathbb{R}$ be an even-periodic Real Landweber exact $C_2$-spectrum, and $ER$ its spectrum of fixed points. We compute the $ER$-cohomology of the infinite stunted projective spectra $P_j$. These cohomology groups combine to form the $RO(C_2)$-graded coefficient ring of the $C_2$-spectrum $b(ER) = F(EC_{2+},i_\ast ER)$, which we show is related to $E\mathbb{R}$ by a cofiber sequence $\Sigma^\sigma b(ER)\rightarrow b(ER)\rightarrow E\mathbb{R}$. We illustrate our description of $\pi_\star b(ER)$ with the computation of some $ER$-based Mahowald invariants.