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 ${\mathrm{projdim}}_A(M)\le \dim ({\mathrm{H}}^0(A))-\inf (M)$. We further provide explicit constructions of DG-modules with prescribed projective dimension and deduce that the big finitistic projective dimension satisfies the bounds $\dim ({\mathrm{H}}^0(A))-\mathrm{amp}(A)\le \mathsf{FPD}(A)\le \dim ({\mathrm{H}}^0(A))$. 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.

v4: 25pp, final version, to appear in Math. Z