Derived Equivalences of Triangular Matrix Rings Arising from Extensions of Tilting Modules
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A triangular matrix ring Λ is defined by a triplet (R,S,M) where R and S are rings and R M S is an S-R-bimodule. In the main theorem of this paper we show that if T S is a tilting S-module, then under certain homological conditions on the S-module M S , one can extend T S 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 T S , 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 M S 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, M S has a finite projective resolution and Ext S n (M S , S) = 0 for all n > 0. In this case, (S′,R,M′) = (S, R, Hom S (M, S)).
KeywordsTriangular matrix ring Derived equivalence Tilting complex
Mathematics Subject Classifications (2000)18E30 16S50 18A25
- 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
- 10.Happel, D., Reiten, I., Smalø, S.O.: Tilting in abelian categories and quasitilted algebras. Mem. Am. Math. Soc. 120(575) (1996)Google Scholar
- 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