TITLE: Group of units of a ring spectrum vs of its connective cover
QUESTION [5 upvotes]: Let $R$ be a commutative ring spectrum (interpret this as you will;  as an $E_\infty$-ring or as a commutative $S$-algebra etc.) and $\operatorname{GL}_1(R)$ as usual denote its space of units. If $\tilde R$ is the connective cover of $R,$ is there a simple relationship between $\operatorname{GL}_1(\tilde R)$ and $\operatorname{GL}_1(R)$?

REPLY [10 votes]: For symmetric ring spectra $R$ there is also a definition of the graded group of units, $GL_1^J(R)$, which retains information about the negative homotopy groups of $R$. See Sagave-Schlichtkrull, "Diagram spaces and symmetric spectra", Advances in Mathematics (2012).  Here $J = \Sigma^{-1} \Sigma$ is a specific category, and the graded units can be viewed as spaces over $BJ \simeq QS^0$. For instance, $GL_1^J(ku)$ is different from $GL_1^J(KU)$, where $ku$ is the connective cover of the periodic complex $K$-theory spectrum $KU$.