Use that there is an injection \(0\to \operatorname{Pic}(\pi_* R) \to \pi_0 {\operatorname{Pic}}(R)\) when \(R\) is connective or \(R\) is weakly even periodic and \(\pi_0 R\) is regular Noetherian. - This is \(\operatorname{Pic}\) over graded rings - But it’s much more complicated to have anything like this for the Brauer group. - Theorem: the functors \(\operatorname{Pic}\) and \(\mathop{\mathrm{Br}}\), \(\mathsf{CAlg}({\mathsf{Sp}}) \to {\Omega}^\infty{\mathsf{Top}}\) satisfy etale descent and Galois descent respectively - \(R\to S\) a map of ring spectra if \(\pi_0 R\to \pi_0 S\) is etale as a map of rings (smooth of dimension zero, or flat + unramified) and there is an equivalence \(\pi_k R \otimes_{\pi_0 R} \pi_0 S \xrightarrow{\sim} \pi_k S\). - \({\operatorname{KO}}\) has no interesting etale extension - \(R\to S^{?}\) is \(G{\hbox{-}}\)Galois if - \(R \xrightarrow{\sim}S^{hG}\), mapping to homotopy fixed points is an equivalence - \(S\otimes_R S \xrightarrow{\sim} \prod_G S\) - \(\pi_* {\operatorname{ku}}= {\mathbf{Z}}[\beta ^{\pm 1}]\) and \(\operatorname{Pic}(\pi_* {\operatorname{ku}}) = {\mathbf{Z}}/2\) where \(\beta\) is the Bott class. In fact \(\operatorname{Pic}({\operatorname{KU}}) = {\mathbf{Z}}/2\), and descent yields \(\operatorname{Pic}({\operatorname{KO}}) = \operatorname{Pic}({\operatorname{KU}})^{hC_2}\) - See descent spectral sequence? - Descent is like a local to global principle.