# zbMATH — the first resource for mathematics

Representable equivalences are represented by tilting modules. (English) Zbl 0706.16011
It was shown by C. Menini and A. Orsatti that, if R, S are two rings, Y a full subcategory of left R-modules closed under direct sums and factor modules, D a full subcategory of left S-modules containing $${}_ SS$$ and closed under submodules, such that there exists an inverse pair of additive equivalences F: $$Y\to D$$ and G: $$D\to Y$$ then there exists a module $${}_ RM$$ such that $$End_ RM=S$$, $$F\overset \sim \rightarrow Hom_ R(_ RM$$,-), Y is the full subcategory of all R- modules generated by $${}_ RM$$, while $$G\overset \sim \rightarrow M_ S\otimes$$-, and D is the full subcategory of all S-modules cogenerated by $$Hom_ R(_ RM_ S,_ RQ)$$ where $${}_ RQ$$ is an injective cogenerator of the category of R-modules [Rend. Semin. Mat. Univ. Padova 82, 203-231 (1989; Zbl 0701.16007)]. Thus, examples of modules with these properties are the tilting modules in the sense of D. Happel and C. M. Ringel [Trans. Am. Math. Soc. 274, 399- 443 (1982; Zbl 0503.16024)]. The aim of this paper is to show that, if R is a finite dimensional algebra over a field, and $${}_ RM$$ is a faithful finite dimensional module with the above properties, then $${}_ RM$$ is a tilting module.
Reviewer: I.Assem

##### MSC:
 16P10 Finite rings and finite-dimensional associative algebras 16D90 Module categories in associative algebras
Full Text:
##### References:
 [1] I. Assem , Torsion theories induced by tilting modules , Can. J. Math. , 36 ( 1984 ), pp. 899 - 913 . MR 762747 | Zbl 0539.16022 · Zbl 0539.16022 · doi:10.4153/CJM-1984-051-5 [2] R. Colpi , Some remarks on equivalences between categories of modules , to appear in Comm. Algebra . MR 1071082 | Zbl 0708.16002 · Zbl 0708.16002 · doi:10.1080/00927879008824002 [3] G. D’Este , Some remarks on representable equivalences , to appear in vol. 26 of the Banach Center Publications . Zbl 0722.16007 · Zbl 0722.16007 [4] M. Hoshino , Tilting modules and torsion theories , Bull. London Math. Soc. , 14 ( 1982 ), pp. 334 - 336 . MR 663483 | Zbl 0486.16019 · Zbl 0486.16019 · doi:10.1112/blms/14.4.334 [5] C. Menini - A. Orsatti , Representable equivalences between categories of modules and applications , Rend. Sem. Mat. Univ. Padova , 82 ( 1989 ), pp. 203 - 231 . Numdam | MR 1049594 | Zbl 0701.16007 · Zbl 0701.16007 · numdam:RSMUP_1989__82__203_0 · eudml:108160 [6] C.M. Ringel , Tame algebras and integral quadratic forms , Springer LMN 1099 ( 1984 ). MR 774589 | Zbl 0546.16013 · Zbl 0546.16013 · doi:10.1007/BFb0072870 [7] C.M. Ringel - H. TACHIKAWA, QF-3 rings , J. Reine Angew. Math. , 272 ( 1974 ), pp. 49 - 72 . MR 379578 | Zbl 0318.16006 · Zbl 0318.16006 · crelle:GDZPPN002189984 · eudml:151523 [8] S.O. Smalø , Torsion theories and tilting modules , Bull. London Math. Soc. , 16 ( 1984 ), pp. 518 - 522 . MR 751824 | Zbl 0519.16016 · Zbl 0519.16016 · doi:10.1112/blms/16.5.518
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.