  We 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 at most that of the original algebra. We characterise minimal
d-Auslander-Gorenstein algebras and d-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.
