Finitistic dimensions over commutative DG-rings

Abstract: In this paper we study the finitistic dimensions of commutative noetherian non-positive DG-rings with finite amplitude. We prove that any DG-module $M$ of finite flat dimension over such a DG-ring satisfies $\mathrm{projdim}_A(M) \leq \mathrm{dim}(\mathrm{H}^0 (A)) - \inf(M)$. We further provide explicit constructions of DG-modules with prescribed projective dimension and deduce that the big finitistic projective dimension satisfies the bounds $\mathrm{dim}(\mathrm{H}^0 (A)) - \mathrm{amp}(A) \leq \mathsf{FPD}(A) \leq \mathrm{dim}(\mathrm{H}^0(A))$. Moreover, we prove that DG-rings exist which achieve either bound. As a direct application, we prove new vanishing results for the derived Hochschild (co)homology of homologically smooth algebras.