Tilting theory
Encyclopedia
In algebra, tilting theory uses a tilting module T over an algebra A to construct tilting functors relating modules over A to modules over the tilted algebra EndA(T) of endomorphisms of T.
Tilting theory was motivated by the introduction of Coxeter functors by , which were reformulated by , and generalized by who introduced tilting functors. defined tilted algebras and tilting modules as further generalizations of this.
A tilted algebra B is an algebra of endomorphisms of a tilting module T over a hereditary finite dimensional algebra A.
The tilting functors are the 4 functors HomA(T,*), Ext(T,*), *⊗BT, and Tor(*,T), where T is considered as a right A module and a left B module. showed that tilting functors give equivalences between certain subcategories of right A-modules and right B-modules.
Tilting theory was motivated by the introduction of Coxeter functors by , which were reformulated by , and generalized by who introduced tilting functors. defined tilted algebras and tilting modules as further generalizations of this.
Definitions
defined tilting modules and tilted algebras as follows. Suppose that A is a finite-dimensional algebra over a field. Then a right A-module T is called a tilting module if it has the following 3 properties:- T has projective dimension at most 1, in other words it is a quotient of a projective module by a projective submodule.
- Ext1(T,T) = 0
- The right A-module A is the kernel of a surjective morphism between finite sums of summands of T.
A tilted algebra B is an algebra of endomorphisms of a tilting module T over a hereditary finite dimensional algebra A.
The tilting functors are the 4 functors HomA(T,*), Ext(T,*), *⊗BT, and Tor(*,T), where T is considered as a right A module and a left B module. showed that tilting functors give equivalences between certain subcategories of right A-modules and right B-modules.