Algebras and Representation Theory

, Volume 14, Issue 1, pp 57–74

# Derived Equivalences of Triangular Matrix Rings Arising from Extensions of Tilting Modules

Open Access
Article

## Abstract

A triangular matrix ring Λ is defined by a triplet (R,S,M) where R and S are rings and RMS is an S-R-bimodule. In the main theorem of this paper we show that if TS is a tilting S-module, then under certain homological conditions on the S-module MS, one can extend TS to a tilting complex over Λ inducing a derived equivalence between Λ and another triangular matrix ring specified by (S′, R, M′), where the ring S′ and the R-S′-bimodule M′ depend only on M and TS, and S′ is derived equivalent to S. Note that no conditions on the ring R are needed. These conditions are satisfied when S is an Artin algebra of finite global dimension and MS is finitely generated. In this case, (S′,R,M′) = (S, R, DM) where D is the duality on the category of finitely generated S-modules. They are also satisfied when S is arbitrary, MS has a finite projective resolution and ExtSn(MS, S) = 0 for all n > 0. In this case, (S′,R,M′) = (S, R, HomS(M, S)).

### Keywords

Triangular matrix ring Derived equivalence Tilting complex

### Mathematics Subject Classifications (2000)

18E30 16S50 18A25

### References

1. 1.
Auslander, M., Reiten, I., Smalø, S.O.: Representation theory of Artin algebras. Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1995)
2. 2.
Barot, M., Lenzing, H.: One-point extensions and derived equivalence. J. Algebra 264(1), 1–5 (2003)
3. 3.
Beĭlinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analysis and Topology on Singular Spaces, I (Luminy, 1981), Astérisque, vol. 100, pp. 5–171. Soc. Math. France, Paris (1982)Google Scholar
4. 4.
Chase, S.U.: A generalization of the ring of triangular matrices. Nagoya Math. J. 18, 13–25 (1961)
5. 5.
Cline, E., Parshall, B., Scott, L.: Algebraic stratification in representation categories. J. Algebra 117(2), 504–521 (1988)
6. 6.
Cline, E., Parshall, B., Scott, L.: Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988)
7. 7.
Fields, K.L.: On the global dimension of residue rings. Pac. J. Math. 32, 345–349 (1970)
8. 8.
Fossum, R.M., Griffith, P.A., Reiten, I.: Trivial extensions of abelian categories. Lecture Notes in Mathematics, vol. 456. Springer, Berlin (1975)
9. 9.
Happel, D.: Triangulated categories in the representation theory of finite-dimensional algebras. London Mathematical Society Lecture Note Series, vol. 119. Cambridge University Press, Cambridge (1988)
10. 10.
Happel, D., Reiten, I., Smalø, S.O.: Tilting in abelian categories and quasitilted algebras. Mem. Am. Math. Soc. 120(575) (1996)Google Scholar
11. 11.
Hughes, D., Waschbüsch, J.: Trivial extensions of tilted algebras. Proc. Lond. Math. Soc. (3) 46(2), 347–364 (1983)
12. 12.
Keller, B.: On the construction of triangle equivalences. In: Derived Equivalences for Group Rings, Lecture Notes in Math., vol. 1685, pp. 155–176. Springer, Berlin (1998)
13. 13.
König, S.: Tilting complexes, perpendicular categories and recollements of derived module categories of rings. J. Pure Appl. Algebra 73(3), 211–232 (1991)
14. 14.
Ladkani, S.: On derived equivalences of categories of sheaves over finite posets. J. Pure Appl. Algebra 212(2), 435–451 (2008)
15. 15.
Mac Lane, S.: Categories for the working mathematician. Graduate Texts in Mathematics, vol. 5. Springer, New York (1998)
16. 16.
Michelena, S., Platzeck, M.I.: Hochschild cohomology of triangular matrix algebras. J. Algebra 233(2), 502–525 (2000)
17. 17.
Miyachi, J.I.: Recollement and tilting complexes. J. Pure Appl. Algebra 183(1–3), 245–273 (2003)
18. 18.
Palmér, I., Roos, J.E.: Explicit formulae for the global homological dimensions of trivial extensions of rings. J. Algebra 27, 380–413 (1973)
19. 19.
Rickard, J.: Morita theory for derived categories. J. Lond. Math. Soc. (2) 39(3), 436–456 (1989)
20. 20.
Tachikawa, H., Wakamatsu, T.: Applications of reflection functors for self-injective algebras. In: Representation Theory, I (Ottawa, Ont., 1984), Lecture Notes in Math., vol. 1177, pp. 308–327. Springer, Berlin (1986)Google Scholar