@WilliamBalderrama Just want to say thanks for linking to your slides over here -- I've been poring through them, and I'm sure I'll have follow-up questions for you!
@MaximeRamzi I don't know too much about Galois descent, but $KO \rightarrow KU$ should be almost of finite presentation in that it is a base-change of one (some sources, such as Higher Algebra, define almost of finite presentation only in the connective case).
However, you can obtain $KO \rightarrow KU$ as the base-change of the morphism $ko \rightarrow ku$ of connective covers, and the latter is a neotherian context in which almost of finite presentation can be checked at the level of homotopy groups (HA 7.2.4.31 "Hilbert basis theorem")
@PiotrPstrągowski Thanks ! I'm not sure it's good enough in the Galois context though : in the aforementioned paper of Akhil's, he deduces descent for $KO\to KU$ by using his work with Clausen-Naumann-Noel on telescopically localized descent, not by using the étale version of the statement
If I have a morphism of $E_{\infty}$-algebras with vanishing cotangent complex, what can I say about the $E_{k}$-cotangent complex of the same morphism?
