2016/08/15 by René Marczinzik, Marczinzik, Rene · 5 citations
Mathematics · Materials Science · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Porphyrin and Phthalocyanine Chemistry
paper · pdf · doi:10.48550/arxiv.1608.04212
We prove that a finite dimensional algebra A with representation-finite\nsubcategory consisting of modules that are semi-Gorenstein-projective and\nn-th syzygy modules is left weakly Gorenstein. This generalises a theorem of\nRingel and Zhang who proved the result in the case n=1. As an application we\nshow that monomial algebras and endomorphism rings of modules over\nrepresentation-finite algebras are weakly Gorenstein. We then give a new\nconnection between the theory of dominant dimension and Gorenstein homological\nalgebra for gendo-symmetric algebras. As an application, we will see that the\nexistence of a non-projective Gorenstein-projective-injective module in a\ngendo-symmetric algebra already implies that this algebra is not CM-finite. We\napply out methods to give a first systematic construction of non-weakly\nGorenstein algebras using the theory of gendo-symmetric algebras. In\nparticular, we can construct non-weakly Gorenstein algebras with an arbitrary\nnumber of simple modules from certain quantum exterior algebras such as the\nLiu-Schulz algebra.\n