The finitistic dimension conjecture via DG-rings

Abstract

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 $\mathrm{FPD}(B)<\infty$. 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 $D(B)$ and $D({\mathrm{H}}^0(B))$ to deduce results about $\mathrm{FPD}(A)$. As an application, we generalize a recent sufficient condition of Rickard, for $\mathrm{FPD}(A)<\infty$ in terms of generation of $D(A)$ from finite dimensional algebras over a field to all noetherian rings which admit a dualizing complex.

14 pages, comments are welcome! v3: additional improvement of the main result. Now stated in terms of generation of injectives