Applications of one-point extensions to compute the A∞-(co)module structure of several Ext (resp., Tor) groups
Let A be a nonnegatively graded connected algebra over a noncommutative separable k-algebra K, and let M be a bounded below graded right A-module. If we denote by T the A∞-coalgebra Tor• A(K,K), we know that there exists an A∞-comodule structure on T′=Tor• A(M,K) over T. The structure of the A∞-alge...
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todo:paper_00224049_v223_n3_p1054_Herscovich2023-10-03T14:32:45Z Applications of one-point extensions to compute the A∞-(co)module structure of several Ext (resp., Tor) groups Herscovich, E. Let A be a nonnegatively graded connected algebra over a noncommutative separable k-algebra K, and let M be a bounded below graded right A-module. If we denote by T the A∞-coalgebra Tor• A(K,K), we know that there exists an A∞-comodule structure on T′=Tor• A(M,K) over T. The structure of the A∞-algebra E=ExtA •(K,K) and the corresponding A∞-module on E′=ExtA •(M,K) are just obtained by taking the bigraded dual. In this article we prove that there is partial description of the A∞-comodule T′ over T and of the structure of A∞-module E′ over E, similar to and also generalizing the partial description of the A∞-algebra structure on E given by Keller's higher-multiplication theorem in [19]. We also provide a criterion to check if a given A∞-comodule structure on T′ is a model by regarding if the associated twisted tensor product is a minimal projective resolution of M, analogous to a theorem of B. Keller explained by the author of this article in [9]. Finally, we give an application of this result by computing the A∞-module structure on E′ for any generalized Koszul algebra A and any generalized Koszul module M. © 2018 Elsevier B.V. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00224049_v223_n3_p1054_Herscovich |
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Universidad de Buenos Aires |
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I-28 |
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R-134 |
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| description |
Let A be a nonnegatively graded connected algebra over a noncommutative separable k-algebra K, and let M be a bounded below graded right A-module. If we denote by T the A∞-coalgebra Tor• A(K,K), we know that there exists an A∞-comodule structure on T′=Tor• A(M,K) over T. The structure of the A∞-algebra E=ExtA •(K,K) and the corresponding A∞-module on E′=ExtA •(M,K) are just obtained by taking the bigraded dual. In this article we prove that there is partial description of the A∞-comodule T′ over T and of the structure of A∞-module E′ over E, similar to and also generalizing the partial description of the A∞-algebra structure on E given by Keller's higher-multiplication theorem in [19]. We also provide a criterion to check if a given A∞-comodule structure on T′ is a model by regarding if the associated twisted tensor product is a minimal projective resolution of M, analogous to a theorem of B. Keller explained by the author of this article in [9]. Finally, we give an application of this result by computing the A∞-module structure on E′ for any generalized Koszul algebra A and any generalized Koszul module M. © 2018 Elsevier B.V. |
| format |
JOUR |
| author |
Herscovich, E. |
| spellingShingle |
Herscovich, E. Applications of one-point extensions to compute the A∞-(co)module structure of several Ext (resp., Tor) groups |
| author_facet |
Herscovich, E. |
| author_sort |
Herscovich, E. |
| title |
Applications of one-point extensions to compute the A∞-(co)module structure of several Ext (resp., Tor) groups |
| title_short |
Applications of one-point extensions to compute the A∞-(co)module structure of several Ext (resp., Tor) groups |
| title_full |
Applications of one-point extensions to compute the A∞-(co)module structure of several Ext (resp., Tor) groups |
| title_fullStr |
Applications of one-point extensions to compute the A∞-(co)module structure of several Ext (resp., Tor) groups |
| title_full_unstemmed |
Applications of one-point extensions to compute the A∞-(co)module structure of several Ext (resp., Tor) groups |
| title_sort |
applications of one-point extensions to compute the a∞-(co)module structure of several ext (resp., tor) groups |
| url |
http://hdl.handle.net/20.500.12110/paper_00224049_v223_n3_p1054_Herscovich |
| work_keys_str_mv |
AT herscoviche applicationsofonepointextensionstocomputetheacomodulestructureofseveralextresptorgroups |
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1807319709190717440 |