Tilting module

A (classical) tilting module over a finite-dimensional $k$-algebra $A$ (cf. also Algebra), where $k$ is a field, is a (finitely-generated) $A$-module $T$ satisfying:

i) the projective dimension of $T$ is at most one;

ii) $\operatorname { Ext } _ { A } ^ { 1 } ( T , T ) = 0$; and

iii) the number of non-isomorphic indecomposable summands of $T$ equals the number of simple $A$-modules. The fundamental work by S. Brenner and M.C.R. Butler, and D. Happel and C.M. Ringel, on tilting theory have established the relations between the module categories $\mod A$ and $\operatorname { mod} B$, where $B = \operatorname { End } _ { A } ( T )$, through the tilting functors $\text{Hom}_A( T , - )$ and $\operatorname { Ext } _ { A } ^ { 1 } ( T , - )$ (cf. also Tilting functor). The particular case where $A$ is a hereditary algebra gives rise to the notion of a tilted algebra, which nowadays (as of 2000) plays a very important role in the representation theory of algebras. One can also consider the dual notion of cotilting modules.

Tilting theory goes back to the work of I.N. Bernshtein, I.M. Gel'fand and V.A. Ponomarev on the characterization of representation-finite hereditary algebras through their ordinary quivers (cf. also Quiver). Their reflection functors on quivers has led to a module-theoretical interpretation by M. Auslander, M.I. Platzeck and I. Reiten. Next steps in this theory are the work by Brenner and Butler and Happel and Ringel, which gave the basis for all its further development. Worthwhile mentioning is the connection of tilting theory with derived categories established by Happel (cf. also Derived category).

The success of this strategy to study a bigger class of algebras through tilting theory has led to several generalizations. On one hand, one can relax the condition on the projective dimension and consider tilting modules of finite projective dimension. In this way it was possible to show the connection between tilting theory and some other homological problems in the representation theory of algebras. On the other hand, this concept can be generalized to a so-called tilting object in more general Abelian categories. For instance, this has led to the notion of a quasi-tilted algebra. Recently (as of 2000), there has been much work also on exploring such notions in categories of (not necessarily finitely-generated) modules over arbitrary rings.

For references, see also Tilting theory; Tilted algebra.