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 Bernšteĭn, Gelfand & Ponomarev (1973); these functors were used to relate representations of two quivers. These functors were reformulated by Auslander, Platzeck & Reiten (1979), and generalized by Brenner & Butler (1980) who introduced tilting functors. Happel & Ringel (1982) defined tilted algebras and tilting modules as further generalizations of this.
- T has projective dimension at most 1, in other words it is a quotient of a projective module by a projective submodule.
A(T,T) = 0.
- The right A-module A is the kernel of a surjective morphism between finite direct sums of direct summands of T.
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 TorB
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 .)
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 .
Generalizations and extensions
A generalized tilting module over the finite-dimensional algebra A is a right A-module T with the following three properties:
- T has finite projective dimension.
A(T,T) = 0 for all i>0.
- There is an exact sequence where the Ti are finite direct sums of direct summands of T.
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 all R-modules.
From the theory of cluster algebras came the definition of cluster category 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.
- Angeleri Hügel, Lidia; Happel, Dieter; Krause, Henning, eds. (2007), Handbook of tilting theory, London Mathematical Society Lecture Note Series 332, Cambridge University Press, doi:10.1017/CBO9780511735134, ISBN 978-0-521-68045-5; 978-0-521-68045-5 Check
|isbn=value (help), MR 2385175
- Assem, Ibrahim (1990), Tilting theory---an introduction, in Balcerzyk, Stanisław; Józefiak, Tadeusz; Krempa, Jan; Simson, Daniel; Vogel, Wolfgang, "Topics in algebra, Part 1 (Warsaw, 1988)", Banach Center Publications, Banach Center Publ. (Warszawa: PWN) 26: 127–180, MR 1171230
- Auslander, Maurice; Platzeck, María Inés; Reiten, Idun (1979), "Coxeter functors without diagrams", Transactions of the American Mathematical Society 250: 1–46, doi:10.2307/1998978, ISSN 0002-9947, MR 530043
- Bernšteĭn, I. N.; Gelfand, I. M.; Ponomarev, V. A. (1973), "Coxeter functors, and Gabriel's theorem", Russian mathematical surveys 28 (2): 17–32, doi:10.1070/RM1973v028n02ABEH001526, ISSN 0042-1316, MR 0393065
- Brenner, Sheila; Butler, M. C. R. (1980), "Generalizations of the Bernstein-Gel'fand-Ponomarev reflection functors", Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math. 832, Berlin, New York: Springer-Verlag, pp. 103–169, doi:10.1007/BFb0088461, MR 607151
- Cline, E.; Parshall, B.; Scott, L. (1986), "Derived categories and Morita theory", Algebra 104: 397–409
- Colpi, Riccardo; Fuller, Kent R. (February 2007), "Tilting Objects in Abelian Categories and Quasitilted Rings", Transactions of the American Mathematical Society 359 (2): 741–765
- Happel, Dieter; Reiten, Idun; Smalø, S.O. (1996), "Tilting in abelian categories and quasitilted algebras", Memoirs American Mathematical Society 575
- Happel, Dieter; Ringel, Claus Michael (1982), "Tilted algebras", Transactions of the American Mathematical Society 274 (2): 399–443, doi:10.2307/1999116, ISSN 0002-9947, MR 675063
- Happel, Dieter (1988), Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Notes 119, Cambridge University Press
- Happel, Dieter (2001), "A characterization of hereditary categories with tilting object", Invent. Math. 144 (2): 381–398
- Rickard, Jeremy (1989), "Morita theory for derived categories", Journal London Mathematical Society 39 (2): 436–456
- Unger, L. (2001), "Tilting theory", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4