The finitistic dimension conjecture via DG-rings

Given an associative ring $A$, we present a new approach for establishing the finiteness of the big finitistic projective dimension $\mathrm{FPD}(A)$. The idea is to find a sufficiently nice non-positively graded differential graded ring $B$ such that ${\mathrm{H}}^0(B)=A$ and such that $\mathrm{FPD}(B)<\mathrm{\infty }$. We show that one can always find such a $B$ provided that $A$ is noetherian and has a noncommutative dualizing complex. We then use the intimate relation between $\mathrm{D}(B)$ and $\mathrm{D}({\mathrm{H}}^0(B))$ to deduce results about $\mathrm{FPD}(A)$. As an application, we generalize a recent sufficient condition of Rickard, for $\mathrm{FPD}(A)<\mathrm{\infty }$ in terms of generation of $\mathrm{D}(A)$ from finite dimensional algebras over a field to all noetherian rings which admit a dualizing complex.