Derived Recollements and Generalised AR Formulas

2016/12/06 by Samuel Dean, Dean, Samuel, Jeremy Russell +1 · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.CT #math.RA #math.RT

paper · pdf · doi:10.48550/arxiv.1612.01651

openalex publication_date 2016/12/06 · arxiv created 2016/12/17 · arxiv updated 2016/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Defect Recollement, Restriction Recollement, Auslander-Gruson-Jensen Recollement, and others, are shown to be instances of a general construction using derived functors and methods from stable module theory. The right derived functors \textsfW_k:=R_k(\hspace0.05cm\underline \hspace0.1cm )^* are computed and it is shown that the functor \textsfW₂:=R₂(\hspace0.05cm\underline \hspace0.1cm )^* is right exact and restricts to a duality \textsfW of the defect zero functors. The duality \textsfW satisfies two identities which we call the Generalised Auslander-Reiten formulas. We show that \textsfW restricts to the generalised Auslander-Bridger transpose and show that the Generalised Auslander-Reiten formulas reduce to the well-known Auslander-Reiten formulas.

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• On the monomorphism category of large modules

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