Gorenstein properties of split-by-nilpotent extension algebras


  • Pamela Suarez Centro Marplatense de Investigaciones Matemáticas (CEMIM), Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, 7600 Mar del Plata, Argentina




Let $A$ be a finite-dimensional $k$-algebra over an algebraically closed field $k$. In this note, we study the Gorenstein homological properties of a split-by-nilpotent extension algebra. Let $R$ be a split-by-nilpotent extension of $A$. We provide sufficient conditions to ensure when a Gorenstein-projective module over $A$ induces a similar structure over $R$. We also study when a Gorenstein-projective $R$-module induces a Gorenstein-projective $A$-module. Moreover, we study the relationship between the Gorensteinness of $A$ and $R$.


Download data is not yet available.


I. Assem and N. Marmaridis, Tilting modules over split-by-nilpotent extensions, Comm. Algebra 26 no. 5 (1998), 1547–1555.  DOI  MR  Zbl

I. Assem, D. Simson, and A. Skowroński, Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts 65, Cambridge University Press, Cambridge, 2006.  DOI  MR  Zbl

M. Auslander and M. Bridger, Stable module theory, Mem. Amer. Math. Soc. no. 94 (1969).  DOI  MR  Zbl

L. L. Avramov and A. Martsinkovsky, Absolute, relative, and Tate cohomology of modules of finite Gorenstein dimension, Proc. London Math. Soc. (3) 85 no. 2 (2002), 393–440.  DOI  MR  Zbl

A. Beligiannis and I. Reiten, Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc. 188 no. 883 (2007).  DOI  MR  Zbl

X.-W. Chen, Singularity categories, Schur functors and triangular matrix rings, Algebr. Represent. Theory 12 no. 2-5 (2009), 181–191.  DOI  MR  Zbl

X.-W. Chen, D. Shen, and G. Zhou, The Gorenstein-projective modules over a monomial algebra, Proc. Roy. Soc. Edinburgh Sect. A 148 no. 6 (2018), 1115–1134.  DOI  MR  Zbl

E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 no. 4 (1995), 611–633.  DOI  MR  Zbl

E. E. Enochs and O. M. G. Jenda, Relative homological algebra, De Gruyter Expositions in Mathematics 30, Walter de Gruyter, Berlin, 2000.  DOI  MR  Zbl

C. Geiss and I. Reiten, Gentle algebras are Gorenstein, in Representations of algebras and related topics, Fields Inst. Commun. 45, Amer. Math. Soc., Providence, RI, 2005, pp. 129–133.  MR  Zbl

D. Happel, On Gorenstein algebras, in Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), Progr. Math. 95, Birkhäuser, Basel, 1991, pp. 389–404.  DOI  MR  Zbl

B. Keller and I. Reiten, Cluster-tilted algebras are Gorenstein and stably Calabi-Yau, Adv. Math. 211 no. 1 (2007), 123–151.  DOI  MR  Zbl

M. Lu, Gorenstein properties of simple gluing algebras, Algebr. Represent. Theory 22 no. 3 (2019), 517–543.  DOI  MR  Zbl

R. Marczinzik, On stable modules that are not Gorenstein projective, 2017. arXiv:1709.01132v3 [math.RT].

C. M. Ringel, The Gorenstein projective modules for the Nakayama algebras. I, J. Algebra 385 (2013), 241–261.  DOI  MR  Zbl

C. M. Ringel and P. Zhang, Gorenstein-projective and semi-Gorenstein-projective modules, Algebra Number Theory 14 no. 1 (2020), 1–36.  DOI  MR  Zbl

P. Suarez, Split-by-nilpotent extensions and support $tau$-tilting modules, Algebr. Represent. Theory 23 no. 6 (2020), 2295–2313.  DOI  MR  Zbl

B.-L. Xiong and P. Zhang, Gorenstein-projective modules over triangular matrix Artin algebras, J. Algebra Appl. 11 no. 4 (2012), 1250066, 14 pp.  DOI  MR  Zbl