The endomorphism ring of a tilting module over a finite-dimensional hereditary algebra (cf. also Algebra; Endomorphism).

Let be a finite-dimensional hereditary -algebra, some field, for example the path-algebra of some finite quiver without oriented cycles. A finite-dimensional -module is called a tilting module if

i) , which always is satisfied in this context;

ii) ; and

iii) there exists a short exact sequence with and in , the category of finite direct sums of direct summands of . Here, is projective dimension. The third condition also says that is maximal with respect to the property . Note further, that a tilting module over a hereditary algebra is uniquely determined by its composition factors. Cf. also Tilting module.

The algebra is called a tilted algebra of type , and becomes an -bimodule (cf. also Bimodule).

In -mod, the category of finite-dimensional -modules, the module defines a torsion pair with torsion class consisting of modules, generated by and torsion-free class . In -mod it defines the torsion pair with torsion class and torsion-free class . The Brenner–Butler theorem says that the functors , respectively , induce equivalences between and , whereas , respectively , induce equivalences between and . In -mod the torsion pair is splitting, that is, any indecomposable -module is either torsion or torsion-free. In this sense, -mod has  "less"  indecomposable modules, and information on the category -mod can be transferred to -mod.

For example, has global dimension at most and any indecomposable -module has projective dimension or injective dimension at most (cf. also Dimension for dimension notions). These condition characterize the more general class of quasi-tilted algebras.

The indecomposable injective -modules are in the torsion class and their images under the tilting functor are contained in one connected component of the Auslander–Reiten quiver of -mod (cf. also Quiver; Riedtmann classification), and they form a complete slice in this component. Moreover, the existence of such a complete slice in a connected component of characterizes tilted algebras. Moreover, the Auslander–Reiten quiver of a tilted algebra always contains pre-projective and pre-injective components.

If is a basic hereditary algebra and is a simple projective module, then , where denotes the Auslander–Reiten translation (cf. Riedtmann classification), is a tilting module, sometimes called APR-tilting module. The induced torsion pair in -mod is splitting and is the unique indecomposable module in . The tilting functor corresponds to the reflection functor introduced by I.N. Bernshtein, I.M. Gel'fand and V.A. Ponomarev for their proof of the Gabriel theorem [a4].

If the hereditary algebra is representation-finite (cf. also Algebra of finite representation type), then any tilted algebra of type also is representation-finite. If is tame (cf. also Representation of an associative algebra), then a tilted algebra of type either is tame and one-parametric, or representation-finite. The latter case is equivalent to the fact that the tilting module contains non-zero pre-projective and pre-injective direct summands simultaneously. If is wild (cf. also Representation of an associative algebra), then a tilted algebra of type may be wild, or tame domestic, or representation-finite.

See also Tilting theory.

References