Tilting theory
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 algebra, tilting theory uses a tilting module T over an algebra A to construct tilting functors relating modules over A to modules over the tilted algebra EndA(T) of endomorphisms of T.
Tilting theory was motivated by the introduction of Coxeter functors by Bernšteĭn, Gelfand & Ponomarev (1973), which 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.
Definitions
Happel & Ringel (1982) defined tilting modules and tilted algebras as follows. Suppose that A is a finite-dimensional algebra over a field. Then a right A-module T is called a tilting module if it has the following 3 properties:
- T has projective dimension at most 1, in other words it is a quotient of a projective module by a projective submodule.
- Ext1(T,T) = 0
- The right A-module A is the kernel of a surjective morphism between finite sums of summands of T.
A tilted algebra B is an algebra of endomorphisms of a tilting module T over a hereditary finite dimensional algebra A.
The tilting functors are the 4 functors HomA(T,*), Ext1
A(T,*), *⊗BT, and TorB
1(*,T), where T is considered as a right A module and a left B module. Brenner & Butler (1980) showed that tilting functors give equivalences between certain subcategories of right A-modules and right B-modules.
See also
References
- 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, MR2385175, http://books.google.com/books?isbn=052168045X
- Assem, Ibrahim (1990), Tilting theory---an introduction, in Balcerzyk, Stanisław; Józefiak, Tadeusz; Krempa, Jan et al., "Topics in algebra, Part 1 (Warsaw, 1988)", Banach Center Publications, Banach Center Publ. (Warszawa: PWN) 26: 127–180, MR1171230, http://journals.impan.gov.pl/BC/oldindex.html
- 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, MR530043
- 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, MR0393065
- 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, MR607151
- Coelho, Flávio Ulhoa (2001), "Tilting functor", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=t/t130110
- Coelho, Flávio Ulhoa (2001), "Tilting module", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=t/t130120
- 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, MR675063
- Kerner, O. (2001), "Tilted algebra", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=t/t130100
- Unger, L. (2001), "Tilting theory", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=T/t130130