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arXiv · 2209.02068

The finitistic dimension conjecture via DG-rings

Liran Shaul
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Abstract

Given an associative ring $A$, we present a new approach for establishing the finiteness of the big finitistic projective dimension $\operatorname{FPD}(A)$. 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 $\operatorname{FPD}(B) < \infty$. 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 $\operatorname{\mathsf{D}}(B)$ and $\operatorname{\mathsf{D}}(\mathrm{H}^0(B))$ to deduce results about $\operatorname{FPD}(A)$. As an application, we generalize a recent sufficient condition of Rickard, for $\operatorname{FPD}(A) < \infty$ in terms of generation of $\operatorname{\mathsf{D}}(A)$ from finite dimensional algebras over a field to all noetherian rings which admit a dualizing complex.

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Liran Shaul. 2022-09-05. The finitistic dimension conjecture via DG-rings. https://arxiv.org/abs/2209.02068

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