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## Fundamenta Mathematicae

1993 | 142 | 1 | 59-84
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### On tame repetitive algebras

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Let A be a finite dimensional algebra over an algebraically closed field, and denote by T(A) (respectively, Â) the trivial extension of A by its minimal injective cogenerator bimodule (respectively, the repetitive algebra of A). We characterise the algebras A such that Â is tame and exhaustive, that is, the push-down functor mod Â → mod T(A) associated with the covering functor Â → T(A)\nsimto Â/(ν_A)$is dense. We show that, if Â is tame and exhaustive, then A is simply connected if and only if A is not an iterated tilted algebra of type$Â_m$. Then we prove that Â is tame and exhaustive if and only if A is tilting-cotilting equivalent to an algebra which is either hereditary of Dynkin or Euclidean type or is tubular canonical. Słowa kluczowe Kategorie tematyczne Czasopismo Rocznik Tom Numer Strony 59-84 Opis fizyczny Daty wydano 1993 otrzymano 1991-09-30 poprawiono 1992-06-03 Twórcy autor • Département de Mathématiques et d'Informatique, Université de Sherbrooke, Sherbrooke, Québec, Canada, J1K 2R1 autor • Institute of Mathematics, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland Bibliografia • [1] I. Assem, Tilted algebras of type$\sym A_n$, Comm. Algebra 10 (1982), 2121-2139. • [2] I. Assem and D. Happel, Generalized tilted algebras of type$\sym A_n$, ibid. 9 (1981), 2101-2125. • [3] I. Assem, D. Happel and O. Roldán, Representation-finite trivial extension algebras, J. Pure Appl. Algebra 33 (1984), 235-242. • [4] I. Assem, J. Nehring and A. Skowroński, Domestic trivial extensions of simply connected algebras, Tsukuba J. Math. 13 (1989), 31-72. • [5] I. Assem and A. Skowroński, Iterated tilted algebras of type$Â_n$, Math. Z. 195 (1987), 269-290. • [6] I. Assem and A. Skowroński, On some classes of simply connected algebras, Proc. London Math. Soc. (3) 56 (1988), 417-450. • [7] I. Assem and A. Skowroński, Algebras with cycle-finite derived categories, Math. Ann. 280 (1988), 441-463. • [8] I. Assem and A. Skowroński, Quadratic forms and iterated tilted algebras, J. Algebra 128 (1990), 55-85. • [9] M. Auslander and I. Reiten, Representation theory of artin algebras III, IV, Comm. Algebra 3 (1975), 239-294 and 5 (1977), 443-518. • [10] K. Bongartz and P. Gabriel, Covering spaces in representation theory, Invent. Math. 65 (1981/82), 331-378. • [11] S. Brenner and M. Butler, Generalizations of the Bernstein-Gelfand-Ponomarev reflection functors, in: Proc. ICRA II (Ottawa 1979), Lecture Notes in Math. 832, Springer, Berlin 1980, 103-169. • [12] O. Bretscher, C. Läser and C. Riedtmann, Self-injective and simply connected algebras, Manuscripta Math. 36 (1981/82), 253-307. • [13] B. Conti, Simply connected algebras of tree-class$\sym A_n$and$ⅅ_n$, in: Proc. ICRA IV (Ottawa 1984), Lecture Notes in Math. 1177, Springer, Berlin 1986, 60-90. • [14] V. Dlab and C. M. Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 173 (1976). • [15] P. Dowbor and A. Skowroński, On Galois coverings of tame algebras, Arch. Math. (Basel) 44 (1985), 522-529. • [16] P. Dowbor and A. Skowroński, Galois coverings of representation-infinite algebras, Comment. Math. Helv. 62 (1987), 311-337. • [17] Yu. A. Drozd, Tame and wild matrix problems, in: Proc. ICRA II (Ottawa 1979), Lecture Notes in Math. 832, Springer, Berlin 1980, 242-258. • [18] P. Gabriel, Auslander-Reiten sequences and representation-finite algebras, in: Proc. ICRA II (Ottawa 1979), Lecture Notes in Math. 831, Springer, Berlin 1980, 1-71. • [19] P. Gabriel, The universal cover of a representation-finite algebra, in: Proc. ICRA III (Puebla 1980), Lecture Notes in Math. 903, Springer, Berlin 1981, 68-105. • [20] D. Happel, Tilting sets on cylinders, Proc. London Math. Soc. (3) 51 (1985), 21-55. • [21] D. Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), 339-389. • [22] D. Happel, J. Rickard and A. Schofield, Piecewise hereditary algebras, Bull. London Math. Soc. 20 (1988), 23-28. • [23] D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. ath. Soc. 274 (1982), 399-443. • [24] D. Happel and C. M. Ringel, Construction of tilted algebras, in: Proc. ICRA III (Puebla 1980), Lecture Notes in Math. 903, Springer, Berlin 1981, 125-144. • [25] D. Happel and C. M. Ringel, The derived category of a tubular algebra, in: Proc. ICRA IV (Ottawa 1984), Lecture Notes in ath. 1177, Springer, Berlin 1986, 156-180. • [26] D. Happel and D. Vossieck, inimal algebras of infinite representation type with preprojective component, Manuscripta ath. 42 (1983), 221-243. • [27] D. Hughes and J. Waschbüsch, Trivial extensions of tilted algebras, Proc. London Math. Soc. (3) 46 (1983), 347-364. • [28] R. Martínez-Villa and J. A. de la Pe na, The universal cover of a quiver with relations, J. Pure Appl. Algebra 30 (1983), 277-292. • [29] J. Nehring, Polynomial growth trivial extensions of non-simply connected algebras, Bull. Polish Acad. Sci. 36 (1988), 441-445. • [30] J. Nehring and A. Skowroński, Polynomial growth trivial extensions of simply connected algebras, Fund. Math. 132 (1989), 117-134. • [31] C. M. Ringel, Tame algebras, in: Proc. ICRA II (Ottawa 1979), Lecture Notes in Math. 831, Springer, Berlin 1980, 137-287. • [32] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, Berlin 1984. • [33] C. M. Ringel, Representation theory of finite-dimensional algebras, in: Representations of Algebras, Proc. Durham 1985, Cambridge University Press, 1986, 7-80. • [34] O. Roldán, Tilted algebras of type$\widetilde\sym A_n$,$\widetilde\sym B_n$,$\widetilde\sym C_n$and$\widetilde\sym B\sym C_n\$, Ph.D. thesis, Carleton University, 1983.
• [35] A. Skowroński, Generalization of Yamagata's theorem on trivial extensions, Arch. Math. (Basel) 48 (1987), 68-76.
• [36] A. Skowroński, On algebras of finite strong global dimension, Bull. Polish Acad. Sci. 35 (1987), 539-547.
• [37] A. Skowroński, Group algebras of polynomial growth, anuscripta Math. 59 (1987), 499-516.
• [38] A. Skowroński, Selfinjective algebras of polynomial growth, ath. Ann. 285 (1989), 177-199.
• [39] A. Skowroński, Algebras of polynomial growth, in: Topics in Algebra, Banach Center Publ. 26, Part 1, PWN, Warszawa 1990, 535-568.
• [40] A. Skowroński and J. Waschbüsch, Representation-finite biserial algebras, J. Reine Angew. Math. 345 (1983), 172-181.
• [41] H. Tachikawa and T. Wakamatsu, Tilting functors and stable equivalences for selfinjective algebras, J. Algebra 109 (1987), 138-165.
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• [44] T. Wakamatsu, Stable equivalence between universal covers of trivial extension self-injective algebras, Tsukuba J. Math. 9 (1985), 299-316.
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