Finitistic dimensions over commutative DG-rings

TL;DR

This paper investigates the finitistic dimensions of commutative noetherian non-positive DG-rings with finite amplitude, establishing bounds for projective dimensions and constructing examples that achieve these bounds, with applications to Hochschild (co)homology.

Contribution

It provides new bounds for finitistic dimensions of such DG-rings, explicit constructions of modules with prescribed projective dimensions, and applications to derived Hochschild (co)homology.

Findings

Bounds for finitistic projective dimension in terms of ring dimension and amplitude.

Existence of DG-rings achieving the established bounds.

Vanishing results for derived Hochschild (co)homology of homologically smooth algebras.

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.