Remark 1

An element $h/s$ of $S^{-1}H$ with $h \in H, s \in S$, is invertible if and only if the component of degree 0 of $h$ is invertible in $H_0$. Consequently, if the ring $H_0$ is local, the ring $S^{-1}H$ is local and the canonical injection $H_0 \to S^{-1}H$ induces an isomorphism of the residue fields.