The singular Yoneda category and the stabilization functor

TL;DR

This paper provides an explicit description of the stabilization functor for noetherian rings with a semisimple subring, connecting the singularity category to the homotopy category of acyclic complexes via the singular Yoneda dg category.

Contribution

It introduces an explicit description of the stabilization functor using the $E$-relative singular Yoneda dg category, enhancing understanding of the embedding of singularity categories.

Findings

Explicit description of the stabilization functor for rings with a semisimple subring.

Connection between the singularity category and the homotopy category of acyclic complexes.

Enhanced understanding of the structure of the singularity category via dg categories.

Abstract

For a noetherian ring $\Lambda$, the stabilization functor in the sense of Krause yields an embedding of the singularity category of $\Lambda$ into the homotopy category of acyclic complexes of injective $\Lambda$-modules. When $\Lambda$ contains a semisimple artinian subring $E$, we give an explicit description of the stabilization functor using the Hom complexes in the $E$-relative singular Yoneda dg category of $\Lambda$.