It turns out that there are applications of our functors which make use of the analogous transformations which we like to think of as a change of basis for a fixed root-system — a tilting of the axes relative to the roots which results in a different subset of roots lying in the positive cone.
Tilting theory was motivated by the introduction of reflection functors by Joseph Bernšteĭn, Israel Gelfand, and V. A. Ponomarev (1973); these functors were used to relate representations of two quivers.
These functors were reformulated by Maurice Auslander, María Inés Platzeck, and Idun Reiten (1979), and generalized by Sheila Brenner and Michael C. R. Butler (1980) who introduced tilting functors.
Suppose that A is a finite-dimensional unital associative algebra over some field.
A finitely-generated right A-module T is called a tilting module if it has the following three properties: Given such a tilting module, we define the endomorphism algebra B = EndA(T ).
This is another finite-dimensional algebra, and T is a finitely-generated left B-module.
The tilting functors HomA(T,−), Ext1A(T,−), −⊗BT and TorB1(−,T) relate the category mod-A of finitely-generated right A-modules to the category mod-B of finitely-generated right B-modules.
Suppose A is a finite-dimensional algebra, T is a tilting module over A, and B = EndA(T ).
Brenner & Butler (1980) showed that tilting functors give equivalences between certain subcategories of mod-A and mod-B.
; this implies that every M in A-mod admits a natural short exact sequence
Further, the restrictions of the functors F and G yield inverse equivalences between
(Note that these equivalences switch the order of the torsion pairs
Tilting theory may be seen as a generalization of Morita equivalence which is recovered if T is a projective generator; in that case
In case A is hereditary (i.e. B is a tilted algebra), the global dimension of B is at most 2, and the torsion pair
Happel (1988) and Cline, Parshall & Scott (1986) showed that in general A and B are derived equivalent (i.e. the derived categories Db(A-mod) and Db(B-mod) are equivalent as triangulated categories).
A generalized tilting module over the finite-dimensional algebra A is a right A-module T with the following three properties: These generalized tilting modules also yield derived equivalences between A and B, where B = EndA(T ).
Rickard (1989) extended the results on derived equivalence by proving that two finite-dimensional algebras R and S are derived equivalent if and only if S is the endomorphism algebra of a "tilting complex" over R. Tilting complexes are generalizations of generalized tilting modules.
A version of this theorem is valid for arbitrary rings R and S. Happel, Reiten & Smalø (1996) defined tilting objects in hereditary abelian categories in which all Hom- and Ext-spaces are finite-dimensional over some algebraically closed field k. The endomorphism algebras of these tilting objects are the quasi-tilted algebras, a generalization of tilted algebras.
Happel (2001) classified the hereditary abelian categories that can appear in the above construction.
Colpi & Fuller (2007) defined tilting objects T in an arbitrary abelian category C; their definition requires that C contain the direct sums of arbitrary (possibly infinite) numbers of copies of T, so this is not a direct generalization of the finite-dimensional situation considered above.
Given such a tilting object with endomorphism ring R, they establish tilting functors that provide equivalences between a torsion pair in C and a torsion pair in R-Mod, the category of all R-modules.
A cluster tilted algebra arises from a tilted algebra as a certain semidirect product, and the cluster category of A summarizes all the module categories of cluster tilted algebras arising from A.