Homological theory of $k$-idempotent ideals in dualizing varieties

TL;DR

This paper extends the homological theory of $k$-idempotent ideals to dualizing varieties, constructing recollements and exploring conditions for quasi-hereditary algebras with applications to derived categories.

Contribution

It generalizes previous results to dualizing varieties, introduces a canonical recollement for $k$-idempotent ideals, and studies their homological properties and applications.

Findings

Constructed a canonical recollement in dualizing varieties.

Identified conditions for $k$-idempotent ideals to produce quasi-hereditary algebras.

Applied the theory to bounded derived categories.

Abstract

In this work, we develop the theory of $k$-idempotent ideals in the setting of dualizing varieties. Several results given previously in \cite{APG} by M. Auslander, M. I. Platzeck, and G. Todorov are extended to this context. Given an ideal $\mathcal{I}$ (which is the trace of a projective module), we construct a canonical recollement which is the analog to a well-known recollement in categories of modules over artin algebras. Moreover, we study the homological properties of the categories involved in such a recollement. Consequently, we find conditions on the ideal $\mathcal{I}$ to obtain quasi-hereditary algebras in such a recollement. Applications to bounded derived categories are also given.