Kitchloo and Wilson have used the homotopy fixed points spectrum ER(2) of the classical complex-oriented Johnson-Wilson spectrum E(2) to deduce certain non-immmersion results for real projective spaces. ER(n) is a 2 n+2 (2 n − 1)-periodic spectrum. The key result to use is the existence of a stable cofibration Σ λ(n) ER(n) → ER(n) → E(n) connecting the real Johnson-Wilson spectrum with the classical one. The value of λ(n) is 2 2n+1 − 2 n+2 + 1. We extend Kitchloo-Wilson's results on non-immersions of real projective spaces by computing the second real Johnson-Wilson cohomology ER(2) of the odd-dimensional real projective spaces RP 16K+9 . This enables us to solve certain non-immersion problems of projective spaces using obstructions in ER(2)-cohomology.For b = 2n and c = 2k Don Davis shows in [2] that there is no such map when n = m + α(m) − 1 and k = 2m − α(m), where α(m) is the number of ones in the binary expression of m by finding an obstruction to James's map (2) in E(2)-cohomology. Kitchloo and Wilson get new non-immersion results by computing obstructions in ER(2)-cohomology. In this paper we extend Kitchloo-Wilson's results by computing the ER(2)-cohomology of the odd projective space RP 16K+9 . This will give us newer non-immersion results. The main results are the following.Theorem 1.1. A 2-adic basis of ER(2) 8 * (RP 16K+9 , * ) is given by the elementsTheorem 1.2. Let α(m) be the number of ones in the binary expansion of m. If (m, α(m)) ≡ (6,2) or (1,0) mod 8, RP 2(m+α(m)−1) does not immerse in R 2(2m−α(m))+1 . This shall give us new non-immersions that are often new and different from those of [7] and [8]. Using Davis's table [1] the first new result is RP 2 13 −2 does not immerse in R 2 14 −59 .

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Abstract. We prove analogs of faithfully flat descent and Galois descent for categories of modules over E∞-ring spectra using the ∞-categorical Barr-Beck theorem proved by Lurie. In particular, faithful G-Galois extensions are shown to be of effective descent for modules. Using this we study the category of ER(n)-modules, where ER(n) is the Z/2-fixed points under complex conjugation of a generalized Johnson-Wilson spectrum E(n). In particular, we show that ER(n)-modules is equivalent to Z/2-equivariant E(n)-modules as stable ∞-categories.

We develop an obstruction theory for lifting compact objects to the stable ∞ category of quasi-coherent modules over a derived geometric stack X from the category of modules over its underlying classical stack X cl . The obstructions live in Andre-Quillen cohomology.

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