Tilted algebra

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

[a1]	I. Assem, "Tilting theory - an introduction" N. Balcerzyk (ed.) et al. (ed.) , Topics in Algebra , 26 , Banach Center Publ. (1990) pp. 127–180
[a2]	M. Auslander, M.I. Platzeck, I. Reiten, "Coxeter functors without diagrams" Trans. Amer. Math. Soc. , 250 (1979) pp. 1–46
[a3]	S. Brenner, M. Butler, "Generalizations of the Bernstein–Gelfand–Ponomarev reflection functors" V. Dlab (ed.) P. Gabriel (ed.) , Representation Theory II. Proc. ICRA II , Lecture Notes in Mathematics , 832 , Springer (1980) pp. 103–169
[a4]	I.N. Bernstein, I.M. Gelfand, V.A. Ponomarow, "Coxeter functors and Gabriel's theorem" Russian Math. Surveys , 28 (1973) pp. 17–32
[a5]	K. Bongartz, "Tilted algebras" M. Auslander (ed.) E. Lluis (ed.) , Representations of Algebras. Proc. ICRA III , Lecture Notes in Mathematics , 903 , Springer (1981) pp. 26–38
[a6]	D. Happel, "Triangulated categories in the representation theory of finite dimensional algebras" , London Math. Soc. Lecture Notes , 119 , Cambridge Univ. Press (1988)
[a7]	D. Happel, C.M. Ringel, "Tilted algebras" Trans. Amer. Math. Soc. , 274 (1982) pp. 399–443
[a8]	D. Happel, I. Reiten, S.O. Smalø, "Tilting in abelian categories and quasitilted algebras" Memoirs Amer. Math. Soc. , 575 (1996)
[a9]	O. Kerner, "Tilting wild algebras" J. London Math. Soc. , 39 : 2 (1989) pp. 29–47
[a10]	O. Kerner, "Wild tilted algebras revisited" Colloq. Math. , 73 (1997) pp. 67–81
[a11]	S. Liu, "The connected components of the Auslander–Reiten quiver of a tilted algebra" J. Algebra , 161 (1993) pp. 505–523
[a12]	C.M. Ringel, "The regular components of the Auslander–Reiten Quiver of a tilted algebra" Chinese Ann. Math. Ser. B. , 9 (1988) pp. 1–18
[a13]	C.M. Ringel, "Tame algebras and integral quadratic forms" , Lecture Notes in Mathematics , 1099 , Springer (1984)
[a14]	H. Strauss, "On the perpendicular category of a partial tilting module" J. Algebra , 144 (1991) pp. 43–66