Flat and Ghost dimensions in tensor triangulated categories with a compact tilting generator

Abstract

Let $\mathcal{T}$ be a tensor triangulated category with a compact tilting generator $M$ and abelian heart ${\mathcal{H}}_M=M^{{\perp }_{\ne 0}}$. We define a triangulated flat dimension ${\mathrm{flatdim}}_{\mathcal{T}}X$ for $X\in \mathcal{T}$, and give criteria in terms of tensor vanishing and resolutions by the class of pure quotients of $\mathrm{Add}(M)$. We further introduce $M$-ghost length and $M$-ghost dimension, and show that the global dimension of ${\mathcal{H}}_M$ controls the $M$-ghost length of all objects in $\mathcal{T}$, and that the $M$-ghost dimension of $\mathcal{T}$ coincides with the weak global dimension of ${\mathcal{H}}_M$ and the supremum of ${\mathrm{flatdim}}_{\mathcal{T}}X+\inf X$ of bounded objects, and the relative triangulated dimension of ${\mathcal{T}}^c$ with respect to $\mathrm{add}(M)$. This extends the classical Hovey-Lockridge theorem, which is recovered for derived categories of rings.