Recollements for dualizing $k$-varieties and Auslander's formulas
Yasuaki Ogawa
TL;DR
The paper develops a recollement framework for finitely presented functors over a dualizing $k$-variety ${oldsymbol{ m A}}$ relative to a functorially finite subcategory ${oldsymbol{ m B}}$, yielding a recollement $( ext{mod}{oldsymbol{ m B}}, ext{mod}{oldsymbol{ m A}}, ext{mod}({oldsymbol{ m A}}/[{oldsymbol{ m B}}]))$ and a Serre quotient equivalence $ ext{mod}{oldsymbol{ m A}}/ ext{mod}({oldsymbol{ m A}}/[{oldsymbol{ m B}}]) \, ilde{ o}\text{mod}{oldsymbol{ m B}}$. It then connects Auslander–Bridger sequences to right-defining exact sequences in this recollement, providing a unified interpretation of AB theory within the recollement framework. In the higher setting, it develops a higher defect formula for $n$-cluster tilting subcategories, identifies the higher Auslander–Reiten translations $ au_n$ with the associated equivalences $oldsymbol{ au}_n$, and establishes a higher AR duality relating stable and costable categories through Ext functors. Collectively, the results extend Auslander's formulas to dualizing $k$-varieties, offering a cohesive approach to recollements, AB sequences, and higher AR theory with Serre-quotient realizations.
Abstract
Given the pair of a dualizing $k$-variety and its functorially finite subcategory, we show that there exists a recollement consisting of their functor categories of finitely presented objects. We provide several applications for Auslander's formulas: The first one realizes a module category as a Serre quotient of a suitable functor category. The second one shows a close connection between Auslander-Bridger sequences and recollements. The third one gives a new proof of the higher defect formula which includes the higher Auslander-Reiten duality as a special case.