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Abstract:

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.

Registro:

Documento: Artículo
Título:Applications of one-point extensions to compute the A∞-(co)module structure of several Ext (resp., Tor) groups
Autor:Herscovich, E.
Filiación:Institut Joseph Fourier, Université Grenoble Alpes, Grenoble, France
Departamento de Matemática, FCEyN, UBA, Buenos Aires, Argentina
CONICET, Argentina
Año:2019
Volumen:223
Número:3
Página de inicio:1054
Página de fin:1072
DOI: http://dx.doi.org/10.1016/j.jpaa.2018.05.014
Título revista:Journal of Pure and Applied Algebra
Título revista abreviado:J. Pure Appl. Algebra
ISSN:00224049
CODEN:JPAAA
Registro:http://digital.bl.fcen.uba.ar/collection/paper/document/paper_00224049_v223_n3_p1054_Herscovich

Referencias:

  • Anderson, F.W., Fuller, K.R., Rings and Categories of Modules (1992) Grad. Texts Math., 13. , 2nd ed. Springer-Verlag New York
  • Berger, R., Koszulity for nonquadratic algebras (2001) J. Algebra, 239 (2), pp. 705-734
  • Berger, R., Ginzburg, V., Higher symplectic reflection algebras and non-homogeneous N-Koszul property (2006) J. Algebra, 304 (1), pp. 577-601
  • 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); Curtis, C.W., Reiner, I., Methods of Representation Theory, vol. I (1981) Pure Appl. Math., , John Wiley & Sons, Inc. New York with applications to finite groups and orders; Pure and Applied Mathematics; A Wiley-Interscience Publication
  • 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
  • Green, E.L., Martínez Villa, R., Koszul and Yoneda algebras (1996) Representation Theory of Algebras, Cocoyoc, 1994, CMS Conf. Proc., 18, pp. 247-297. , Amer. Math. Soc. Providence, RI
  • He, J.-W., Lu, D.-M., Higher Koszul algebras and A-infinity algebras (2005) J. Algebra, 293 (2), pp. 335-362
  • Herscovich, E., On the multi-Koszul property for connected algebras (2013) Doc. Math., 18, pp. 1301-1347
  • Herscovich, E., Notes on dg (co)algebras and their (co)modules (2016), https://www-fourier.ujf-grenoble.fr/~eherscov/Articles/Notes-on-dg(co)algebras.pdf, available at; Herscovich, E., On the Merkulov construction of A∞-(co)algebras (2018) Ukr. Math. J., , https://www-fourier.ujf-grenoble.fr/~eherscov/Articles/On-the-Merkulov-construction.pdf, in press; available at
  • Herscovich, E., Hochschild (co)homology of Koszul dual pairs https://www-fourier.ujf-grenoble.fr/~eherscov/Articles/Hochschild-(co)homology-of-Koszul-dual-pairs.pdf, available at; Herscovich, E., Using torsion theory to compute the algebraic structure of Hochschild (co)homology (2018) Homol. Homotopy Appl., 20 (1), pp. 117-139
  • 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
  • Huebschmann, J., On the construction of A∞-structures (2010) Georgian Math. J., 17 (1), pp. 161-202
  • Lefèvre-Hasegawa, K., Sur les A∞-catégories (2003), http://www.math.jussieu.fr/~keller/lefevre/TheseFinale/corrainf.pdf, Thesis (Ph.D.) Université Paris 7 Paris, France (in French); corrections at; Lu, D.M., Palmieri, J.H., Wu, Q.S., Zhang, J.J., A∞-algebras for ring theorists (2004) Proceedings of the International Conference on Algebra, pp. 91-128
  • Lu, D.M., Palmieri, J.H., Wu, Q.S., Zhang, J.J., Koszul equivalences in A∞-algebras (2008) N.Y. J. Math., 14, pp. 325-378
  • Lu, D.-M., Palmieri, J.H., Wu, Q.-S., Zhang, J.J., A-infinity structure on Ext-algebras (2009) J. Pure Appl. Algebra, 213 (11), pp. 2017-2037
  • Merkulov, S.A., Strong homotopy algebras of a Kähler manifold (1999) Int. Math. Res. Not., 3, pp. 153-164
  • Năstăsescu, C., Van Oystaeyen, F., Methods of Graded Rings (2004) Lect. Notes Math., 1836. , Springer-Verlag Berlin
  • Polishchuk, A., Positselski, L., Quadratic Algebras (2005) Univ. Lect. Ser., 37. , American Mathematical Society Providence, RI
  • Weibel, C.A., An Introduction to Homological Algebra (1994) Camb. Stud. Adv. Math., 38. , Cambridge University Press Cambridge

Citas:

---------- APA ----------
(2019) . Applications of one-point extensions to compute the A∞-(co)module structure of several Ext (resp., Tor) groups. Journal of Pure and Applied Algebra, 223(3), 1054-1072.
http://dx.doi.org/10.1016/j.jpaa.2018.05.014
---------- CHICAGO ----------
Herscovich, E. "Applications of one-point extensions to compute the A∞-(co)module structure of several Ext (resp., Tor) groups" . Journal of Pure and Applied Algebra 223, no. 3 (2019) : 1054-1072.
http://dx.doi.org/10.1016/j.jpaa.2018.05.014
---------- MLA ----------
Herscovich, E. "Applications of one-point extensions to compute the A∞-(co)module structure of several Ext (resp., Tor) groups" . Journal of Pure and Applied Algebra, vol. 223, no. 3, 2019, pp. 1054-1072.
http://dx.doi.org/10.1016/j.jpaa.2018.05.014
---------- VANCOUVER ----------
Herscovich, E. Applications of one-point extensions to compute the A∞-(co)module structure of several Ext (resp., Tor) groups. J. Pure Appl. Algebra. 2019;223(3):1054-1072.
http://dx.doi.org/10.1016/j.jpaa.2018.05.014