Special Tilting Modules for Algebras With Positive Dominant Dimension

Abstract

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 less than or equal to 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.

On Quiver Grassmannians and Orbit Closures for Gen-Finite Modules

“…(2) The 1-shifted tilting module and the 1-coshifted cotilting module of an algebra of positive dominant dimension, studied by the authors in [28] (see also [19,26]) arespecial, where additively generates the category of projective-injective -modules. (3) The characteristic tilting module T of a right ultra-strongly quasihereditary algebra is special cotilting, by a theorem of Conde stated in the introduction to [11].…”

“…Furthermore, it follows from Remark 2.6 that the canonical tilting B Q -module DC Q is the unique B Q e-special tilting module and the canonical cotilting B P -module DT P is the unique D(eB P )-special cotilting module. We note that if E is both a generator and a cogenerator then, in the terminology of [28], the cogenerator-tilted algebra of E is the 1-shifted algebra of , and the generator-cotilted algebra of E is the 1-coshifted algebra of .…”

“…We recall from [28] a description of the images and kernels of some of the functors appearing in the above recollement. This description uses the following notation.…”

Support $τ_2$-tilting and 2-torsion pairs

“…See also [18] for a survey of τ -tilting theory, which has seen much activity in recent years [4], [8], [9], [17], [19], [28], [29] as well as its generalisations such as in silting theory [2], [3], [7]. A natural question to ask is whether similar results are true in the context of higher Auslander-Reiten theory, as introduced by Iyama in [15], [16], and an active area of research [10], [20], [22], [23], [24], [30]. As the name suggests, higher Auslander-Reiten theory has connections to higher-dimensional geometry and topology [11], [12], [25], [27], [32] and is hence a natural generalisation.…”