In mathematics, specifically representation theory, tilting theory describes a way to relate the module categories of two algebras using so-called tilting modules and associated tilting functors. Here, the second algebra is the endomorphism algebra of a tilting module over the first algebra.
Tilting theory was motivated by the introduction of reflection functors by ; these functors were used to relate representations of two quivers. These functors were reformulated by , and generalized by who introduced tilting functors. defined tilted algebras and tilting modules as further generalizations of this.
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 = End<sub>A</sub>(T ). This is another finite-dimensional algebra, and T is a finitely-generated left B-module. The tilting functors Hom<sub>A</sub>(T,−), Ext(T,−), −âÂÂ<sub>B</sub>T and Tor(−,T) relate the category mod-A of finitely-generated right A-modules to the category mod-B of finitely-generated right B-modules.
In practice one often considers hereditary finite-dimensional algebras A because the module categories over such algebras are fairly well understood. The endomorphism algebra of a tilting module over a hereditary finite-dimensional algebra is called a tilted algebra.
Suppose A is a finite-dimensional algebra, T is a tilting module over A, and B = End<sub>A</sub>(T ). Write F = Hom<sub>A</sub>(T,−), F′ = Ext(T,−), G = −âÂÂ<sub>B</sub>T, and G′ = Tor(−,T). F is right adjoint to G and F′ is right adjoint to G′.
showed that tilting functors give equivalences between certain subcategories of mod-A and mod-B. Specifically, if we define the two subcategories and of A-mod, and the two subcategories and of B-mod, then is a torsion pair in A-mod (i.e. and are maximal subcategories with the property ; this implies that every M in A-mod admits a natural short exact sequence with U in and V in ) and is a torsion pair in B-mod. Further, the restrictions of the functors F and G yield inverse equivalences between and , while the restrictions of F′ and G′ yield inverse equivalences between and . (Note that these equivalences switch the order of the torsion pairs and .)
Tilting theory may be seen as a generalization of Morita equivalence which is recovered if T is a projective generator; in that case and .
If A has finite global dimension, then B also has finite global dimension, and the difference of F and F induces an isometry between the Grothendieck groups K<sub>0</sub>(A) and K<sub>0</sub>(B).
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 splits, i.e. every indecomposable object of B-mod is either in or in .
and showed that in general A and B are derived equivalent (i.e. the derived categories D<sup>b</sup>(A-mod) and D<sup>b</sup>(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 = End<sub>A</sub>(T ).
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.
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. The quasi-tilted algebras over k are precisely the finite-dimensional algebras over k of global dimension â¤ 2 such that every indecomposable module either has projective dimension â¤ 1 or injective dimension â¤ 1. classified the hereditary abelian categories that can appear in the above construction.
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.
From the theory of cluster algebras came the definition of cluster category (from ) and cluster tilted algebra () associated to a hereditary algebra A. 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.