Eventually homological isomorphisms in recollements of derived categories

TL;DR

This paper studies conditions under which certain functors in recollements of derived categories become eventually homological isomorphisms, linking algebraic properties like Gorensteinness and Hochschild cohomology.

Contribution

It characterizes when the functor $j^*$ is an eventually homological isomorphism in recollements, connecting algebraic invariants of $A$ and $C$.

Findings

Identifies conditions for $j^*$ to be an eventually homological isomorphism.

Relates Gorensteinness and singularity categories of $A$ and $C$.

Applies results to stratifying ideals, triangular matrix algebras, and derived discrete algebras.

Abstract

For a recollement $(\mathcal{D}B,\mathcal{D}A,\mathcal{D}C)$ of derived categories of algebras, we investigate when the functor $j^*:\mathcal{D}A\rightarrow\mathcal{D}C$ is an eventually homological isomorphism. In this context, we compare the algebras $A$ and $C$ with respect to Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology. The results are applied to stratifying ideals, triangular matrix algebras and derived discrete algebras.