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arXiv · 1612.01651

Derived Recollements and Generalised AR Formulas

Samuel Dean·Jeremy Russell
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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 $\textsf{W}_k:=R_k(\hspace{0.05cm}\underline{\ \ }\hspace{0.1cm} )^*$ are computed and it is shown that the functor $\textsf{W}_2:=R_2(\hspace{0.05cm}\underline{\ \ }\hspace{0.1cm} )^*$ is right exact and restricts to a duality $\textsf{W}$ of the defect zero functors. The duality $\textsf{W}$ satisfies two identities which we call the Generalised Auslander-Reiten formulas. We show that $\textsf{W}$ 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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Samuel Dean, Jeremy Russell. 2016-12-06. Derived Recollements and Generalised AR Formulas. https://arxiv.org/abs/1612.01651

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