Text complete and PDF
Services on Demand
Journal
Article
Indicators
- Cited by SciELO
- Access statistics
Related links
- Cited by Google
- Similars in SciELO
- Similars in Google
Share
Revista Colombiana de Matemáticas
Print version ISSN 0034-7426
Rev.colomb.mat. vol.53 supl.1 Bogotá Dec. 2019 Epub Mar 24, 2020
https://doi.org/10.15446/recolma.v53nsupl.84097
Artículos originales
A note on deformations of Gorenstein-projective modules over finite dimensional algebras
Una nota acerca de deformaciones de módulos Gorenstein-proyectivos sobre álgebras de dimensión finita
1 Valdosta State University, United States of America
In this note, we present a survey of results concerning universal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional algebras.
Keywords: Deformations of modules; universal deformation rings; Gorenstein algebras; Gorenstein-projective modules
En esta nota, nosotros damos una revisión de resultados concernientes a anillos universales de deformación de módulos Gorenstein-proyectivos finitamente generados sobre álgebras de dimensión finita.
Palabras clave: Deformaciones de módulos; anillos universales de deformación; álgebras de Gorenstein; módulos Gorenstein-proyectivos
References
[1] M. Auslander and M. Brigder, Stable Module Theory, Memoirs of the American Mathematical Society, no. 94, American Mathematical Society, 1969. [ Links ]
[2] M. Auslander and I. Reiten, Applications of contravariantly finite subcategories, Adv. in Math. 86 (1991), 111-152. [ Links ]
[3] ______, Cohen-Macaulay and Gorenstein Artin algebras, Representation Theory of Finite Groups and Finite-Dimensional Algebras (G. O. Michler and C. M. Ringel, eds.), Progress in Mathematics, no. 95, Birkhäuser Verlag Basel, 1991, pp. 221-245. [ Links ]
[4] M. Auslander, I. Reiten, and S. Smal(, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, no. 36, Cambridge University Press, 1995. [ Links ]
[5] V. Bekkert, H. Giraldo, and J. A. Vélez-Marulanda, Universal deformation rings of finitely generated Gorenstein-projective modules over finite dimensional algebras, Submitted. Available in https://arxiv.org/abs/1705.05230. [ Links ]
[6] D. J. Benson, Representations and Cohomology I: Basic representation theory of finite groups and associative algebras, Cambridge Studies in Advanced Mathematics 30, Cambridge University Press, 1991. [ Links ]
[7] F. M. Bleher and S. N. Talbott, Universal deformation rings of modules for algebras of dihedral type of polynomial growth, Algebr. Represent. Theory 17 (2014), 289-303. [ Links ]
[8] F. M. Bleher and J. A. Vélez-Marulanda, Universal deformation rings of modules over Frobenius algebras, J. Algebra 367 (2012), 176-202. [ Links ]
[9]______, Deformations of complexes for finite dimensional algebras, J. Algebra 491 (2017), 90-140 [ Links ]
[10] F. M. Bleher and D. J. Wackwitz, Universal deformation rings and self-injective Nakayama algebras, J. Pure Appl. Algebra 223 (2019), no. 1, 218-244. [ Links ]
[11] M. Broué, Equivalences of blocks of group algebras, Finite Dimensional Algebras and Related Topics (V. Dlab and L. L. Scott, eds.), NATO ASI Series, no. 424, Springer Netherlands, 1994, pp. 1-26. [ Links ]
[12] X. Chen and M. Lu, Cohen-Macaulay Auslander algebras of skewed-gentle algebras, Comm. Algebra. 45 (2017), no. 2, 849-865. [ Links ]
[13] X. W. Chen, Algebras with radical square zero are either self-injective or CM-free, Proc. Amer. Math. Soc. 140 (2012), no. 1, 93-98. [ Links ]
[14] X. W. Chen, D. Shen, and G. Zhou, The Gorenstein-projective modules over a monomial algebra, Proc. Roy. Soc. Edinburgh Sect. A. 148 (2018), no. 6, 1115-1134. [ Links ]
[15] X. W. Chen and L. G. Sun, Singular equivalences of Morita type, Preprint, 2012. [ Links ]
[16] X. W. Chen and Y. Ye, Retractions and Gorenstein homological properties, Algebr. Represent. Theor. 17 (2014), 713-733 [ Links ]
[17] E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995), 611-633. [ Links ]
[18] K. Erdmann, Blocks of Tame Representation Type and Related Algebras, Lectures Notes in Mathematics, no. 1428, Springer-Verlag, 1990. [ Links ]
[19] Ch. Gei( and J. A. de la Peña, Auslander-Reiten components for clans, Bol. Soc. Mat. Mexicana 5 (1999), no. 3, 307-326. [ Links ]
[20] T. Holm, Derived equivalence classification of algebras of dihedral, semidihedral, and quaternion type, J. Algebra 211 (1999), 159-205. [ Links ]
[21] M. Kalck, Singularity categories of gentle algebras, Bull. London Math. Soc. 47 (2015), 65-74. [ Links ]
[22] B. Mazur, An introduction to the deformation theory of Galois representations, Modular Forms and Fermat's Last Theorem (G. Cornell, J. H. Silverman, and G. Stevens, eds.), Springer-Verlag, Boston, MA, 1997, pp. 243-311. [ Links ]
[23] J. Rickard, Derived categories as derived functors, J. London Math. Soc. 43 (1991), no. 2, 37-48. [ Links ]
[24] C. M. Ringel, The Gorenstein projective modules for the Nakayama algebras. I, J. Algebra 385 (2013), 241-261. [ Links ]
[25] M. Schlessinger, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208-222. [ Links ]
[26] Ø. Skartsaeterhagen, Singular equivalence and the (Fg) condition, J. Algebra 452 (2016), 66-93. [ Links ]
[27] A. Skowronski and K. Yamagata, Selfinjective algebras of quasitilted type, Trends in Representation Theory of Algebras and Related Topics (A. Skowronski, ed.), EMS Series of Congress Reports, European Mathematical Society, 2008, pp. 639-708. [ Links ]
[28]______, Frobenius Algebras I: Basic Representation Theory, EMS Textbooks in Mathematics, vol. 12, European Mathematical Society, 2011. [ Links ]
[29] J. A. Vélez-Marulanda, Universal deformation rings of strings modules over a certain symmetric special biserial algebra, Beitr. Algebra Geom. 56 (2015), no. 1, 129-146. [ Links ]
[30] Z. Wang, Singular equivalence of Morita type with level, J. Algebra 439 (2015), 245-269. [ Links ]
[31] C. A. Weibel, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics, no. 38, Cambridge University Press, 1994. [ Links ]
[32] A. Zaks, Injective dimensions of semi-primary rings, J. Algebra 13 (1969), 73-86. [ Links ]
[33] G. Zhou and A. Zimmermann, On singular equivalences of Morita type, J. Algebra 385 (2013), 64-79. [ Links ]
Received: July 07, 2018; Accepted: November 16, 2018