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Gendo-symmetric algebras, dominant dimen...
Marczinzik, Rene...

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Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra by Marczinzik, Rene ( Author )
Lemonjar Open Access
Australian National University
09-08-2023
We prove that a finite dimensional algebra $A$ with representation-finite subcategory consisting of modules that are semi-Gorenstein-projective and $n$-th syzygy modules is left weakly Gorenstein. This generalises a theorem of Ringel and Zhang who proved the result in the case $n=1$. As an application we show that monomial algebras and endomorphism rings of modules over representation-finite algebras are weakly Gorenstein. We then give a new connection between the theory of dominant dimension and Gorenstein homological algebra for gendo-symmetric algebras. As an application, we will see that the existence of a non-projective Gorenstein-projective-injective module in a gendo-symmetric algebra already implies that this algebra is not CM-finite. We apply out methods to give a first systematic construction of non-weakly Gorenstein algebras using the theory of gendo-symmetric algebras. In particular, we can construct non-weakly Gorenstein algebras with an arbitrary number of simple modules from certain quantum exterior algebras such as the Liu-Schulz algebra.
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Article
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29.34 KB
English
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MYR 0.01
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http://arxiv.org/abs/1608.04212
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