Abstract
AbstractWe study certain special tilting and cotilting modules for an algebra with positive dominant dimension, each of which is generated or cogenerated (and usually both) by projective-injectives. These modules have various interesting properties, for example, that their endomorphism algebras always have global dimension less than or equal to that of the original algebra. We characterise minimald-Auslander–Gorenstein algebras andd-Auslander algebras via the property that these special tilting and cotilting modules coincide. By the Morita–Tachikawa correspondence, any algebra of dominant dimension at least 2 may be expressed (essentially uniquely) as the endomorphism algebra of a generator-cogenerator for another algebra, and we also study our special tilting and cotilting modules from this point of view, via the theory of recollements and intermediate extension functors.
Highlights
In [8], Crawley-Boevey and the second author associated with each Auslander algebra a distinguished tilting-cotilting module T, with the property that it is both generated and cogenerated by a projective-injective module
We study more general instances of tilting modules generated by projective-injectives and cotilting modules cogenerated by projective-injectives
For each 0 ≤ k ≤ d + 1, consider the shifted module Tk with endomorphism algebra Bk and let ck : -mod → Bk-mod be the intermediate extension functor from the recollement in (4.2)
Summary
In [8], Crawley-Boevey and the second author associated with each Auslander algebra a distinguished tilting-cotilting module T, with the property that it is both generated and cogenerated by a projective-injective module. In contrast to the case of Auslander algebras, we consider here tilting and cotilting modules of arbitrary finite projective or injective dimension.
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