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 $\operatorname{GL}\_1(R)$?



Group of units of a ring spectrum vs of its connective cover
By definition, the space $GL\_1(R)$ is the subspace of $\Omega^\infty R$ consisting of those elements whose path component $\alpha \in \pi\_0(\Omega^\infty R) = \pi\_0(R)$ is a unit in $\pi\_0(R)$.


If $\tilde R \to R$ is a connective cover, then the map of spaces $\Omega^\infty \tilde R \to \Omega^\infty R$ is an isomorphism on all homotopy groups, and hence we get a weak equivalence $GL\_1(\tilde R) \to GL\_1(R)$.


