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 $b(ER)=F(E{C}_{2+},i_{\ast }ER)$, which we show is related to $E\mathbb{R}$ by a cofiber sequence ${\mathrm{\Sigma }}^{\sigma }b(ER)\to b(ER)\to E\mathbb{R}$. We illustrate our description of ${\pi }_{\star }b(ER)$ with the computation of some $ER$-based Mahowald invariants.