Artículo
Abstract:
In [8] we introduced the notion of multi-Koszul algebra: it is an extension of the definition of generalized Koszul algebra given by R. Berger in [1] for homogeneous algebras (see also [7]) that can be applied to any nonnegatively graded connected algebra over a field k. The goal of this article is on the one hand to properly extend the notion of multi-Koszul algebra to the case where the base ring K is a product of copies of a field k, which a priori allows us to treat quiver algebras, and on the other hand to introduce the notion of multi-Koszul module such that it extends the usual definition of generalized Koszul module over a generalized Koszul algebra. We show eventually that multi-Koszul algebras and multi-Koszul modules are strongly linked via the notion of one-point extensions, as in the case of generalized Koszul algebras. Moreover, we describe the complete structure of right A∞-module on ExtA •(M,K) over ExtA •(K,K), where M is a multi-Koszul module over a multi-Koszul algebra A, extending a result in [9]. As a corollary, we obtain that the underlying right module structure of ExtA •(M,K) over ExtA •(K,K) is generated by the component of cohomological degree zero, as in the case of generalized Koszul modules over generalized Koszul algebras. © 2017 Elsevier Inc.
Registro:
| Documento: | Artículo |
| Título: | On the definition of multi-Koszul modules |
| Autor: | Herscovich, E. |
| Filiación: | Institut Joseph Fourier, Université Grenoble Alpes, Grenoble, France Departamento de Matemática, FCEyN, UBA, Buenos Aires, Argentina |
| Palabras clave: | A∞-algebras; Homological algebra; Koszul algebra; Yoneda algebra |
| Año: | 2018 |
| Volumen: | 499 |
| Página de inicio: | 478 |
| Página de fin: | 505 |
| DOI: | http://dx.doi.org/10.1016/j.jalgebra.2017.12.015 |
| Título revista: | Journal of Algebra |
| Título revista abreviado: | J. Algebra |
| ISSN: | 00218693 |
| CODEN: | JALGA |
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00218693_v499_n_p478_Herscovich |
Referencias:
- Berger, R., Koszulity for nonquadratic algebras (2001) J. Algebra, 239 (2), pp. 705-734
- Berger, R., La catégorie des modules gradués sur une algèbre graduée (nouvelle version du chapitre 5 d'un cours de Master 2 à Lyon 1) (2008), (French); Berger, R., Dubois-Violette, M., Wambst, M., Homogeneous algebras (2003) J. Algebra, 261 (1), pp. 172-185
- Berger, R., Ginzburg, V., Higher symplectic reflection algebras and non-homogeneous N-Koszul property (2006) J. Algebra, 304 (1), pp. 577-601
- Cassidy, T., Shelton, B., Generalizing the notion of Koszul algebra (2008) Math. Z., 260 (1), pp. 93-114
- Cheng, Z., Ye, Y., One-point extensions of t-Koszul algebras (2007) Acta Math. Sin. (Engl. Ser.), 23 (6), pp. 965-972
- Green, E.L., Marcos, E.N., Martínez-Villa, R., Zhang, P., D-Koszul algebras (2004) J. Pure Appl. Algebra, 193 (1-3), pp. 141-162
- Herscovich, E., On the multi-Koszul property for connected algebras (2013) Doc. Math., 18, pp. 1301-1347
- Herscovich, E., Applications of one-point extensions to compute the A∞-(co)module structure of several Ext (resp., Tor) groups https://www-fourier.ujf-grenoble.fr/~eherscov/Articles/Applications-of-one-point-extensions.pdf, Preprint, available at; Herscovich, E., Solotar, A., Suárez-Álvarez, M., PBW-deformations and deformations à la Gerstenhaber of N-Koszul algebras (2014) J. Noncommut. Geom., 8 (2), pp. 505-539
- Keller, B., Introduction to A-infinity algebras and modules (2001) Homology, Homotopy Appl., 3 (1), pp. 1-35
- Lemaire, J.-M., Algèbres connexes et homologie des espaces de lacets (1974) Lecture Notes in Mathematics, 422. , Springer-Verlag Berlin (French)
- Martínez Villa, R., Saorín, M., Koszul equivalences and dualities (2004) Pacific J. Math., 214 (2), pp. 359-378
Citas:
---------- APA ----------
(2018) . On the definition of multi-Koszul modules. Journal of Algebra, 499, 478-505.http://dx.doi.org/10.1016/j.jalgebra.2017.12.015
---------- CHICAGO ----------
Herscovich, E. "On the definition of multi-Koszul modules" . Journal of Algebra 499 (2018) : 478-505.http://dx.doi.org/10.1016/j.jalgebra.2017.12.015
---------- MLA ----------
Herscovich, E. "On the definition of multi-Koszul modules" . Journal of Algebra, vol. 499, 2018, pp. 478-505.http://dx.doi.org/10.1016/j.jalgebra.2017.12.015
---------- VANCOUVER ----------
Herscovich, E. On the definition of multi-Koszul modules. J. Algebra. 2018;499:478-505.http://dx.doi.org/10.1016/j.jalgebra.2017.12.015
