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. … For this reason, and because the word 'tilt' inflects easily, we call our functors tilting functors or simply tilts.
Brenner & Butler (1980, p. 103)
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 JosephBernšteĭn , Israel Gelfand ,andV. A. Ponomarev ( 1973 ); these functors were used to relate representations of two quivers. These functors were reformulated by MauriceAuslander , María Inés Platzeck ,and Idun Reiten ( 1979 ), and generalized by SheilaBrennerandMichael C. R. Butler ( 1980 ) who introduced tilting functors. DieterHappelandClaus Michael Ringel ( 1982 ) 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 = EndA(T ). This is another finite-dimensional algebra, and T is a finitely-generated left B-module. The tilting functors HomA(T,−), Ext1
A(T,−), −⊗BT and Tor B
1(−,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 = EndA(T ). Write F = HomA(T,−), F′ = Ext1
A(T,−), G = −⊗BT, and G′ = TorB
1(−,T). F is right adjoint to G and F′ is right adjoint to G′.
Brenner & Butler (1980) 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 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 splits, i.e. every indecomposable object of B-mod is either in or in .
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. 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. 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 allR-modules.
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