q: for R K(n)-local commutative, the map Σ gl_1(R) -> pic(R) is a K(n)-equivalence, and thus rezk's logarithm gl_1(R) -> R can be extended uniquely to a map pic(R) -> Σ R. in particular, each invertible R-module is assigned an invariant in pi_{-1}(R). is there a description of this invariant?

@TylerLawson I sent you an email about this

I've never figured out a "description" for this invariant, i.e., a theoretical description for it other than via its definition. You can compute it pretty easily at height 1, where it induces a surjective map $\pi_0 \mathrm{pic}(S_{K(1)}) \to \pi_{-1} S_{K(1)}$ at all primes. At higher heights I don't know much, except I think it sends the Tate sphere (aka the determinant sphere) to a torsion element.

Does anyone know of any papers which deal with adjunctions between categories of Left/Right modules over Koszul dual algebras? (in the sense of HA)

In particular how to set up appropriately monoisal adjunctions between these categories