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We study certain special tilting and cotilting modules for an algebra withpositive dominant dimension, each of which is generated or cogenerated (andusually both) by projective-injectives. These modules have various interestingproperties, for example that their endomorphism algebras always have globaldimension at most that of the original algebra. We characterise minimald-Auslander-Gorenstein algebras and d-Auslander algebras via the property thatthese special tilting and cotilting modules coincide. By the Morita-Tachikawacorrespondence, any algebra of dominant dimension at least 2 may be expressed(essentially uniquely) as the endomorphism algebra of a generator-cogeneratorfor another algebra, and we also study our special tilting and cotiltingmodules from this point of view, via the theory of recollements andintermediate extension functors.

Special tilting modules for algebras with positive dominant dimension