The space of units $GL_1(R)$ of a commutative ring spectrum $R$ is the infinite loop space of a spectrum $gl_1(R)$. Typically, this spectrum is taken to be connective, meaning it has no nonzero negative homotopy groups. However, there are other interesting deloopings of $GL_1(R)$ which carry important algebraic information about R. One in particular has $\pi_{-1} gl_1(R) = \pi_0$Pic(R), the Picard group of R, and $\pi_{-2} gl_1(R) = \pi_0 Br(R)$, the Brauer group of $R$. If $R$ is connective, there is a spectral sequence for computing the homotopy groups of $Pic(R)$ and $Br(R)$ in terms of the homotopy groups of R and the etale cohomology of the multiplicative group on Spec $\pi_0 R$. If $S$ is a $G$-Galois extension of $R$, one obtains a Galois descent spectral sequence which calculates the Picard and Brauer groups of $R$ relative to $S$.
