Categorical properties of reduction functors over non-positive DG-rings

Given a non-positive DG-ring $A$, associated to it are the reduction and coreduction functors $F(-)={\mathrm{H}}^0(A){\otimes }_A^{\mathrm{L}}-$ and $G(-)=\mathrm{R}{\mathrm{Hom}}_A({\mathrm{H}}^0(A),-)$, considered as functors $\mathrm{D}(A)\to \mathrm{D}({\mathrm{H}}^0(A))$, as well as the forgetful functor $S:\mathrm{D}({\mathrm{H}}^0(A))\to \mathrm{D}(A)$. In this paper we carry a systematic study of the categorical properties of these functors. As an application, a new descent result for vanishing of $\mathrm{Ext}$ and $\mathrm{Tor}$ over ordinary commutative noetherian rings is deduced.