Openness with respect to levels in triangulated categories
Souvik Dey, Jian Liu, Liran Shaul
TL;DR
The paper establishes a general openness principle for level loci in $R$-linear, compactly generated triangulated categories, extending Letz's results to a broad, model-friendly setting that includes compact subcategories of the compact objects. It leverages the converse ghost lemma and the Benson–Iyengar–Krause framework to show that for any $n$, a fixed object $X$ and subcategory $\\\mathcal{G}$, the locus of primes where $X$ can be built within $n$ steps from $\\\mathcal{G}$ is open, with consequences for the finiteness of level and local-global principles. The paper then translates these ideas to DG rings, proving a local-global principle for $\\mathsf{D}^f_b(A)$ and a DG version of Gabber’s injective-dimension results, leading to openness of the finite injective dimension locus and openness of Gorenstein and related loci in the DG context. Applications span derived categories of commutative Noetherian rings and DG rings, singularity categories, and modular representation theory, including cases arising from group cohomology. Overall, the work unifies locality phenomena in homological algebra with modern triangulated-category techniques to yield robust, widely applicable openness results.
Abstract
Given a compactly generated triangulated category $\mathcal{T}$ equipped with an action of a graded-commutative Noetherian ring $R$, generalizing results of Letz, we prove a general result concerning the openness with respect to levels of compact objects in $\mathcal{T}$. Applications are given to derived categories of commutative Noetherian rings, derived categories of commutative Noetherian DG rings and singularity categories.