Tilting theory

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.
Definitions.
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"A"("T"). This is another finite-dimensional algebra, and "T" is a finitely-generated left "B"-module. 
The tilting functors Hom"A"("T",−), Ext("T",−), −⊗"B""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.
Facts.
Suppose "A" is a finite-dimensional algebra, "T" is a tilting module over "A", and "B" = End"A"("T"). Write "F"=Hom"A"("T",−), "F′"=Ext("T",−), "G"=−⊗"B""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 formula_1 and formula_2 of "A"-mod, and the two subcategories formula_3 and formula_4 of "B"-mod, then formula_5 is a torsion pair in "A"-mod (i.e. formula_6 and formula_7 are maximal subcategories with the property formula_8; this implies that every "M" in "A"-mod admits a natural short exact sequence formula_9 with "U" in formula_6 and "V" in formula_7) and formula_12 is a torsion pair in "B"-mod. Further, the restrictions of the functors "F" and "G" yield inverse equivalences between formula_6 and formula_14, while the restrictions of "F′" and "G′" yield inverse equivalences between formula_7 and formula_16. (Note that these equivalences switch the order of the torsion pairs formula_5 and formula_12.)
Tilting theory may be seen as a generalization of Morita equivalence which is recovered if "T" is a projective generator; in that case formula_19 and formula_20.
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 K0("A") and K0("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 formula_12 splits, i.e. every indecomposable object of "B"-mod is either in formula_16 or in formula_14.
 and 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).
Generalizations and extensions.
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"A"("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 ≤ 2 such that every indecomposable module either has projective dimension ≤ 1 or injective dimension ≤ 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".