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Keywords:
Schofield sequence; exceptional sheaf; projective cover; injective hull; weighted projective line
Schofield sequence; exceptional sheaf; projective cover; injective hull; weighted projective line
Summary:
We focus on the Schofield sequences over weighted projective lines in the sense of Geigle and Lenzing. We give a classification of all the Schofield sequences whose middle terms are line bundles or extension bundles for any weight type $(2,2,n)$. In particular, for the latter case such Schofield sequences are related to the distinguished exact structure on the subcategory of vector bundles. More precisely, the projective covers and injective hulls of extension bundles will provide Schofield sequences, and all the Schofield sequences in the subcategory of vector bundles can be obtained in this way.
References:
[1] Chen, B.: The Gabriel-Roiter Measure for Representation-Finite Hereditary Algebras: Dissertation. Bielefeld University, Bielefeld (2006). Zbl 1248.16014
[2] Crawley-Boevey, W.: Exceptional sequences of representations of quivers. Representations of Algebras CMS Conference Proceedings 14. AMS, Providence (1993), 117-124. MR 1265279 | Zbl 0828.16012
[3] Geigle, W., Lenzing, H.: A class of weighted projective curves arising in representation theory of finite dimensional algebras. Singularities, Representations of Algebras and Vector Bundles Lecture Notes in Mathematics 1273. Springer, Berlin (1987), 265-297. DOI 10.1007/BFb0078849 | MR 0915180 | Zbl 0651.14006
[4] Happel, D.: Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. London Mathematical Society Lecture Note Series 119. Cambridge University Press, Cambridge (1988). DOI 10.1017/CBO9780511629228 | MR 0935124 | Zbl 0635.16017
[5] Kedzierski, D., Meltzer, H.: Schofield induction for sheaves on weighted projective lines. Commun. Algebra 41 (2013), 2033-2039. DOI 10.1080/00927872.2011.642042 | MR 3225254 | Zbl 1314.14035
[6] Kerner, O.: Elementary stones. Commun. Algebra 22 (1994), 1797-1806. DOI 10.1080/00927879408824936 | MR 1264742 | Zbl 0819.16008
[7] Kerner, O.: Representations of wild quivers. Representation Theory of Algebras and Related Topics CMS Conference Proceedings 19. AMS, Providence (1996), 65-107. MR 1388560 | Zbl 0863.16010
[8] Kussin, D., Lenzing, H., Meltzer, H.: Triangle singularities, ADE-chains, and weighted projective lines. Adv. Math. 237 (2013), 194-251. DOI 10.1016/j.aim.2013.01.006 | MR 3028577 | Zbl 1273.14075
[9] Kussin, D., Meltzer, H.: The braid group action for exceptional curves. Arch. Math. 79 (2002), 335-344. DOI 10.1007/PL00012455 | MR 1951302 | Zbl 1062.16016
[10] Lenzing, H.: Weighted projective lines and applications. Representations of Algebras and Related Topics EMS Series of Congress Reports. EMS, Zürich (2011), 153-187. DOI 10.4171/101-1/5 | MR 2931898 | Zbl 1316.14037
[11] Lenzing, H., Meltzer, H.: Exceptional pairs in hereditary categories. Commun. Algebra 37 (2009), 2547-2556. DOI 10.1080/00927870802174140 | MR 2543502 | Zbl 1179.14016
[12] Meltzer, H.: Exceptional sequences for canonical algebras. Arch. Math. 64 (1995), 304-312. DOI 10.1007/BF01198084 | MR 1318999 | Zbl 0818.16016
[13] Meltzer, H.: Exceptional vector bundles, tilting sheaves and tilting complexes for weighted projective lines. Mem. Am. Math. Soc. 808 (2004), 138 pages. DOI 10.1090/memo/0808 | MR 2074151 | Zbl 1055.14016
[14] Ringel, C. M.: The braid group action on the set of exceptional sequences of a hereditary Artin algebra. Abelian Group Theory and Related Topics Contemporary Mathematics 171. AMS, Providence (1994), 339-352. DOI 10.1090/conm/171 | MR 1293154 | Zbl 0851.16010
[15] Ringel, C. M.: Exceptional objects in hereditary categories. An. Ştiinţ. Univ. ``Ovidius'' Constanţa, Ser. Mat. 4 (1996), 150-158. MR 1428463 | Zbl 0888.16005
[16] Ringel, C. M.: Exceptional modules are tree modules. Linear Algebra Appl. 275-276 (1998), 471-493. DOI 10.1016/S0024-3795(97)10046-5 | MR 1628405 | Zbl 0964.16014
[17] (ed.), A. N. Rudakov: Helices and Vector Bundles. London Mathematical Society Lecture Note Series 148. Cambridge University Press, Cambridge (1990). DOI 10.1017/CBO9780511721526 | MR 1074776 | Zbl 0727.00022
[18] Schofield, A.: Semi-invariants of quivers. J. Lond. Math. Soc., II. Ser. 43 (1991), 383-395. DOI 10.1112/jlms/s2-43.3.385 | MR 1113382 | Zbl 0779.16005
[19] Szántó, C., Szöllősi, I.: Schofield sequences in the Euclidean case. J. Pure Appl. Algebra 225 (2021), Article ID 106586, 124 pages. DOI 10.1016/j.jpaa.2020.106586 | MR 4163146 | Zbl 1483.16021
[20] Unger, L.: Schur modules over wild, finite dimensional path algebras with three simple modules. J. Pure Appl. Algebra 64 (1990), 205-222. DOI 10.1016/0022-4049(90)90157-D | MR 1055030 | Zbl 0701.16012
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