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.
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 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. Dieter Happel and Claus Michael Ringel (1982) defined tilted algebras and tilting modules as further generalizations of this.
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 algebraB = 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.
Facts
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 .)
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 .
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 ringsR 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 fieldk. 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.
From the theory of cluster algebras came the definition of cluster category (from Buan et al. (2006)) and cluster tilted algebra (Buan, Marsh & Reiten (2007)) 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.
References
Angeleri Hügel, Lidia; Happel, Dieter; Krause, Henning, eds. (2007), Handbook of tilting theory(PDF), London Mathematical Society Lecture Note Series, vol. 332, Cambridge University Press, doi:10.1017/CBO9780511735134, ISBN978-0-521-68045-5, MR 2385175
Assem, Ibrahim (1990). "Tilting theory–an introduction" (PDF). In Balcerzyk, Stanisław; Józefiak, Tadeusz; Krempa, Jan; Simson, Daniel; Vogel, Wolfgang (eds.). Topics in algebra, Part 1 (Warsaw, 1988). Banach Center Publications. Vol. 26. Warsaw: PWN. pp. 127–180. doi:10.4064/-26-1-127-180. MR 1171230.
Bernšteĭn, Iosif N.; Gelfand, Izrail M.; Ponomarev, V. A. (1973), "Coxeter functors, and Gabriel's theorem", Russian Mathematical Surveys, 28 (2): 17–32, Bibcode:1973RuMaS..28...17B, CiteSeerX10.1.1.642.2527, doi:10.1070/RM1973v028n02ABEH001526, ISSN 0042-1316, MR 0393065
Brenner, Sheila; Butler, Michael 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., vol. 832, Berlin, New York: Springer-Verlag, pp. 103–169, doi:10.1007/BFb0088461, ISBN978-3-540-10264-9, MR 0607151
Cline, Edward; Parshall, Brian; Scott, Leonard (1986), "Derived categories and Morita theory", Algebra, 104 (2): 397–409, doi:10.1016/0021-8693(86)90224-3, MR 0866784
Colpi, Riccardo; Fuller, Kent R. (February 2007), "Tilting Objects in Abelian Categories and Quasitilted Rings" (PDF), Transactions of the American Mathematical Society, 359 (2): 741–765, doi:10.1090/s0002-9947-06-03909-2
Happel, Dieter (1988), Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Notes Series, vol. 119, Cambridge University Press, doi:10.1017/CBO9780511629228, ISBN9780521339223
Happel, Dieter (2001), "A characterization of hereditary categories with tilting object", Invent. Math., 144 (2): 381–398, Bibcode:2001InMat.144..381H, doi:10.1007/s002220100135, S2CID 120437744
Rickard, Jeremy (1989), "Morita theory for derived categories", Journal of the London Mathematical Society, 39 (2): 436–456, doi:10.1112/jlms/s2-39.3.436
tilting, theory, turns, that, there, applications, functors, which, make, analogous, transformations, which, like, think, change, basis, fixed, root, system, tilting, axes, relative, roots, which, results, different, subset, roots, lying, positive, cone, this,. 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 amp 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 Joseph Bernsteĭn Israel Gelfand and V A Ponomarev 1973 these functors were used to relate representations of two quivers These functors were reformulated by Maurice Auslander Maria Ines Platzeck and Idun Reiten 1979 and generalized by Sheila Brenner and Michael C R Butler 1980 who introduced tilting functors Dieter Happel and Claus Michael Ringel 1982 defined tilted algebras and tilting modules as further generalizations of this Contents 1 Definitions 2 Facts 3 Generalizations and extensions 4 ReferencesDefinitions EditSuppose 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 T has projective dimension at most 1 in other words it is a quotient of a projective module by a projective submodule Ext1A 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 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 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 EditSuppose A is a finite dimensional algebra T is a tilting module over A and B EndA T Write F HomA T F Ext1A T G BT and G TorB1 T F is right adjoint to G and F is right adjoint to G Brenner amp Butler 1980 showed that tilting functors give equivalences between certain subcategories of mod A and mod B Specifically if we define the two subcategories F ker F displaystyle mathcal F ker F and T ker F displaystyle mathcal T ker F of A mod and the two subcategories X ker G displaystyle mathcal X ker G and Y ker G displaystyle mathcal Y ker G of B mod then T F displaystyle mathcal T mathcal F is a torsion pair in A mod i e T displaystyle mathcal T and F displaystyle mathcal F are maximal subcategories with the property Hom T F 0 displaystyle operatorname Hom mathcal T mathcal F 0 this implies that every M in A mod admits a natural short exact sequence 0 U M V 0 displaystyle 0 to U to M to V to 0 with U in T displaystyle mathcal T and V in F displaystyle mathcal F and X Y displaystyle mathcal X mathcal Y is a torsion pair in B mod Further the restrictions of the functors F and G yield inverse equivalences between T displaystyle mathcal T and Y displaystyle mathcal Y while the restrictions of F and G yield inverse equivalences between F displaystyle mathcal F and X displaystyle mathcal X Note that these equivalences switch the order of the torsion pairs T F displaystyle mathcal T mathcal F and X Y displaystyle mathcal X mathcal Y Tilting theory may be seen as a generalization of Morita equivalence which is recovered if T is a projective generator in that case T mod A displaystyle mathcal T operatorname mod A and Y mod B displaystyle mathcal Y operatorname mod B 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 X Y displaystyle mathcal X mathcal Y splits i e every indecomposable object of B mod is either in X displaystyle mathcal X or in Y displaystyle mathcal Y Happel 1988 and Cline Parshall amp 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 Generalizations and extensions EditA 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 ExtiA T T 0 for all i gt 0 There is an exact sequence 0 A T 1 T n 0 displaystyle 0 to A to T 1 to dots to T n to 0 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 amp Smalo 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 amp 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 from Buan et al 2006 and cluster tilted algebra Buan Marsh amp Reiten 2007 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 References EditAngeleri Hugel Lidia Happel Dieter Krause Henning eds 2007 Handbook of tilting theory PDF London Mathematical Society Lecture Note Series vol 332 Cambridge University Press doi 10 1017 CBO9780511735134 ISBN 978 0 521 68045 5 MR 2385175 Assem Ibrahim 1990 Tilting theory an introduction PDF In Balcerzyk Stanislaw Jozefiak Tadeusz Krempa Jan Simson Daniel Vogel Wolfgang eds Topics in algebra Part 1 Warsaw 1988 Banach Center Publications Vol 26 Warsaw PWN pp 127 180 doi 10 4064 26 1 127 180 MR 1171230 Auslander Maurice Platzeck Maria Ines Reiten Idun 1979 Coxeter functors without diagrams Transactions of the American Mathematical Society 250 1 46 doi 10 2307 1998978 ISSN 0002 9947 JSTOR 1998978 MR 0530043 Bernsteĭn Iosif N Gelfand Izrail M Ponomarev V A 1973 Coxeter functors and Gabriel s theorem Russian Mathematical Surveys 28 2 17 32 Bibcode 1973RuMaS 28 17B CiteSeerX 10 1 1 642 2527 doi 10 1070 RM1973v028n02ABEH001526 ISSN 0042 1316 MR 0393065 Brenner Sheila Butler Michael 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 vol 832 Berlin New York Springer Verlag pp 103 169 doi 10 1007 BFb0088461 ISBN 978 3 540 10264 9 MR 0607151 Buan Aslak Marsh Robert Reineke Markus Reiten Idun Todorov Gordana 2006 Tilting theory and cluster combinatorics Advances in Mathematics 204 2 572 618 arXiv math 0402054 doi 10 1016 j aim 2005 06 003 MR 2249625 S2CID 15318919 Buan Aslak Marsh Robert Reiten Idun 2007 Cluster tilted algebras Transactions of the American Mathematical Society 359 1 323 332 doi 10 1090 s0002 9947 06 03879 7 MR 2247893 Cline Edward Parshall Brian Scott Leonard 1986 Derived categories and Morita theory Algebra 104 2 397 409 doi 10 1016 0021 8693 86 90224 3 MR 0866784 Colpi Riccardo Fuller Kent R February 2007 Tilting Objects in Abelian Categories and Quasitilted Rings PDF Transactions of the American Mathematical Society 359 2 741 765 doi 10 1090 s0002 9947 06 03909 2 Happel Dieter Reiten Idun Smalo Sverre O 1996 Tilting in abelian categories and quasitilted algebras Memoirs of the 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 JSTOR 1999116 MR 0675063 Happel Dieter 1988 Triangulated categories in the representation theory of finite dimensional algebras London Mathematical Society Lecture Notes Series vol 119 Cambridge University Press doi 10 1017 CBO9780511629228 ISBN 9780521339223 Happel Dieter 2001 A characterization of hereditary categories with tilting object Invent Math 144 2 381 398 Bibcode 2001InMat 144 381H doi 10 1007 s002220100135 S2CID 120437744 Rickard Jeremy 1989 Morita theory for derived categories Journal of the London Mathematical Society 39 2 436 456 doi 10 1112 jlms s2 39 3 436 Unger L 2001 1994 Tilting theory Encyclopedia of Mathematics EMS Press Retrieved from https en wikipedia org w index php title Tilting theory amp oldid 1095683644, wikipedia, wiki, book, books, library,
