Cotilting duality for Artinian rings
Francesca Mantese, Lorenzo Martini
TL;DR
This paper studies cotilting bimodules as a generalisation of Morita bimodules, and proves that cotilting dualities act on bounded derived categories in a manner mirroring Azumaya–Morita dualities for artinian rings. It provides a characterization: a right artinian ring $R$ admits a cotilting duality iff there exists a finitely generated product-complete cotilting module $U_R$. It also shows representability: under suitable finiteness and acyclicity hypotheses, the induced duality on $\mathcal{D}^b$ is realized by a faithfully balanced cotilting bimodule ${}_S U_R$ with $R$ right artinian and $S$ left artinian, and the $\mathcal{D}$-reflexive modules correspond to those with finitely generated cohomology. Moreover, for cotilting dualities over artinian rings, the $\mathcal{D}$-reflexive right $R$-modules coincide with the finitely generated ones, paralleling Azumaya’s results.
Abstract
A classical result due to Morita and Azumaya establishes that given two arbitrary rings, any duality between their finitely generated modules is representable by a faithfully balanced bimodule which is a finitely generated injective cogenerator of both rings and, equivalently, these latter are one-sided artinian. We extend this well-known result to the case of a cotilting bimodule, by analysing the duality it represents in the bounded derived categories of the given rings.