The Segal conjecture for large objects and semiadditive lifts

We prove a projection formula for homotopy fixed points of trivial actions of finite p-groups. More precisely, for every bounded-below, p-complete spectrum X whose mod-p homology is I-complete, the canonical map \[ ( S_p)BG⊗ X {}{→} XBG \] is an equivalence for every finite p-group G, where I is the augmentation ideal of the Steenrod algebra. Building on earlier joint work with Burklund, our proof combines the Adams--Gunawardena--Miller calculation with an adaptation of Carlsson's induction. As an application, using the work of Yuan, we construct height-one semiadditive lifts of such spectra and show that their cohomology of p-finite spaces depends only on the $1$-type.