Auslander's Formula: Variations and Applications

TL;DR

This paper explores various versions of Auslander's formula, including derived and unbounded cases, and applies these to recollements of triangulated categories, enhancing understanding of abelian and derived categories.

Contribution

It provides a detailed analysis of different forms of Auslander's formula and extends their applications to recollements in triangulated categories.

Findings

Derived versions of Auslander's formula are developed.

Applications to recollements of triangulated categories are demonstrated.

Enhanced understanding of homological properties of abelian categories.

Abstract

According to the Auslander's formula one way of studying an abelian category ${\mathcal{C}}$ is to study ${\rm mod}\mbox{-}{\mathcal{C}}$, that has nicer homological properties than ${\mathcal{C}}$, and then translate the results back to ${\mathcal{C}}$. Recently Krause gave a derived version of this formula and thus renewed the subject. This paper contains a detailed study of various versions of Auslander formula including the versions for all modules and for unbounded derived categories. We apply them to include some results concerning recollements of triangulated categories.