Generation of singularity categories and infinite injective dimension locus via annihilation of cohomologies
Souvik Dey, Jian Liu, Yuki Mifune, Yuya Otake
TL;DR
The paper develops a coherent framework connecting the generation of singularity categories with annihilator ideals derived from Ext-functors. It introduces and analyzes cohomological annihilators $\mathop{\mathrm{ca}}_{R}(M)$ and their ring-level version $\mathop{\mathrm{ca}}(R)$, establishing precise links to strong generators, the singular locus, and the infinite injective-dimension locus via new co-cohomological annihilators $\mathrm{coca}_{R}(M)$. A central theme is that generation properties of $\mathsf{D}_{\mathsf{sg}}(R)$ and the module category $\mathop{\mathrm{mod}}(R)$ reflect and control the structure of $\operatorname{Spec}R$, especially in low Krull dimension scenarios, with explicit equivalences for domains of dimension at most one and for rings with isolated singularities. The work ties together open/closed loci of homological dimensions, extension- vs. resolving-generators, and finiteness of Krull dimension, providing a unified account of when singularity categories admit strong generators and how this interacts with annihilator/conductor ideals. Overall, the results illuminate the interplay between homological generation, annihilator theory, and geometric properties of the underlying ring.
Abstract
Let R be a commutative Noetherian ring. We establish a close relationship between the strong generation of the singularity category of R and the nonvanishing of the annihilator of the singularity category of R. As an application, we prove that the singularity category of R has a strong generator if and only if the annihilator of the singularity category of R is nonzero when R is a Noetherian domain with Krull dimension at most one. We introduce the notion of the co-cohomological annihilator of modules. If the category of finitely generated R-modules has a strong generator, we show that the infinite injective dimension locus of a finitely generated R-module M is closed, with the defining ideal given by the co-cohomological annihilator of M. Finally, we provide a connection between the existence of an extension generator of the category of finitely generated R-modules and the finiteness of the Krull dimension of R.