三角范畴偏silting对象的补
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摘要
研究了带silting对象T的Krull-Schmidt三角范畴中预silting对象P的补的存在性。证明了如果P的直和项中的每个不可分解对象关于T都是广义两项的,则P存在补,从而P是偏silting对象。作为应用,说明了与遗传代数导出等价的代数上的预silting复形都是偏silting复形。
The existence of the complements of the presilting object P in a Krull-Schmidt triangulated category with a silting object T is investigated. It is shown that if all the indecomposable direct summands of P are generalized two-term related to T, then there exist complements of P. Consequently, P is partial silting. As an application, the presilting complexes over algebras which are derived equivalent to hereditary algebras are partial silting complexes.
The existence of the complements of the presilting object P in a Krull-Schmidt triangulated category with a silting object T is investigated. It is shown that if all the indecomposable direct summands of P are generalized two-term related to T, then there exist complements of P. Consequently, P is partial silting. As an application, the presilting complexes over algebras which are derived equivalent to hereditary algebras are partial silting complexes.
引文
[1] HAPPEL D, RINGEL C M. Tilted algebras[J]. Trans Amer Math Soc, 1982, 274(2):339-443.
[2] KELLER B, VOSSIECK D. Aisles in derived categories[J]. Bull Soc Math Belg: Ser A, 1988, 40(2):239-253.
[3] AIHARA T, IYAMA O. Silting mutation in triangulated categories[J]. J Lond Math Soc, 2012, 85(3):633-668.
[4] AIHARA T, IYAMA O, REITEN I. τ-tilting theory[J]. Compos Math, 2014, 150(3):415-452.
[5] IYAMA O, JORGENSEN P, YANG D. Intermediate co-t-structures, two-term silting objects, τ-tilting modules, and torsion classes[J]. Algebra Number Theory, 2014, 8(10):2413-2431.
[6] AIHARA T, MIZUNO Y. Classifying tilting complexes over preprojective algebras of Dynkin type[J]. Algebra Number Theory, 2017, 11(6):1287-1315.
[7] BONGARTZ K. Tilted algebras[M]// Proc ICRA Ⅲ(Puebla, 1980), Lecture Notes in Math, vol. 903. New York: Springer, 1981: 26-38.
[8] LI Shen, ZHANG Shunhua. Some applications of τ-tilting theory[J/OL].[2015-12-11]. arXiv: 1512.03613v1. https://arxiv.org/pdf/1512.03613.pdf.
[9] AIHARA T. Tilting-connected symmetric algebras[J]. Algebra Representation Theory, 2013, 16(3):873-894.
[10] BUAN A B. Subcategories of the derived category and cotilting complexes[J]. Colloq Math, 2001, 88(1):1-11.
[11] RICKARD J. Morita theory for derived categories[J]. J Lond Math Soc, 1989, 39(2):436-456.
[12] HAPPEL D. Triangulated categories in the representation theory of finite-dimensional algebras[M]// London Mathematical Society Lecture Note Series, 119. Cambridge: Cambridge University Press, 1988.
[2] KELLER B, VOSSIECK D. Aisles in derived categories[J]. Bull Soc Math Belg: Ser A, 1988, 40(2):239-253.
[3] AIHARA T, IYAMA O. Silting mutation in triangulated categories[J]. J Lond Math Soc, 2012, 85(3):633-668.
[4] AIHARA T, IYAMA O, REITEN I. τ-tilting theory[J]. Compos Math, 2014, 150(3):415-452.
[5] IYAMA O, JORGENSEN P, YANG D. Intermediate co-t-structures, two-term silting objects, τ-tilting modules, and torsion classes[J]. Algebra Number Theory, 2014, 8(10):2413-2431.
[6] AIHARA T, MIZUNO Y. Classifying tilting complexes over preprojective algebras of Dynkin type[J]. Algebra Number Theory, 2017, 11(6):1287-1315.
[7] BONGARTZ K. Tilted algebras[M]// Proc ICRA Ⅲ(Puebla, 1980), Lecture Notes in Math, vol. 903. New York: Springer, 1981: 26-38.
[8] LI Shen, ZHANG Shunhua. Some applications of τ-tilting theory[J/OL].[2015-12-11]. arXiv: 1512.03613v1. https://arxiv.org/pdf/1512.03613.pdf.
[9] AIHARA T. Tilting-connected symmetric algebras[J]. Algebra Representation Theory, 2013, 16(3):873-894.
[10] BUAN A B. Subcategories of the derived category and cotilting complexes[J]. Colloq Math, 2001, 88(1):1-11.
[11] RICKARD J. Morita theory for derived categories[J]. J Lond Math Soc, 1989, 39(2):436-456.
[12] HAPPEL D. Triangulated categories in the representation theory of finite-dimensional algebras[M]// London Mathematical Society Lecture Note Series, 119. Cambridge: Cambridge University Press, 1988.