Weakly Gorensteinness of tensor algebras and Morita algebras
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915684268638208 |
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| author | Gao, Nan Zhang, Pu Zhu, Shijie |
| author_facet | Gao, Nan Zhang, Pu Zhu, Shijie |
| contents | An algebra $A$ is left weakly Gorenstein if any semi-Gorenstein-projective left $A$-modules is Gorenstein-projective. The weakly Gorensteinness of two kinds of algebras are answered. Using the method of the monomorphism category, it is proved that the tensor algebra $A\otimes B$ with ${\rm gl.dim} B< \infty$ is left weakly Gorenstein if and only if so is $A$. For a class of Morita algebras $Λ=\begin{pmatrix}\begin{smallmatrix} A & N \\ M & B \\ \end{smallmatrix}\end{pmatrix}$, the (semi-)Gorenstein-projective left $Λ$-modules are computed and described; and then it is proved that $Λ$ is left weakly Gorenstein if and only if so are $A$ and $B$. As an application, the upper triangular matrix algebra $T_n(A)$ is left weakly Gorenstein if and only if so is $A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_16072 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weakly Gorensteinness of tensor algebras and Morita algebras Gao, Nan Zhang, Pu Zhu, Shijie Representation Theory 16G10, 16E65, 18G15, 16E30 An algebra $A$ is left weakly Gorenstein if any semi-Gorenstein-projective left $A$-modules is Gorenstein-projective. The weakly Gorensteinness of two kinds of algebras are answered. Using the method of the monomorphism category, it is proved that the tensor algebra $A\otimes B$ with ${\rm gl.dim} B< \infty$ is left weakly Gorenstein if and only if so is $A$. For a class of Morita algebras $Λ=\begin{pmatrix}\begin{smallmatrix} A & N \\ M & B \\ \end{smallmatrix}\end{pmatrix}$, the (semi-)Gorenstein-projective left $Λ$-modules are computed and described; and then it is proved that $Λ$ is left weakly Gorenstein if and only if so are $A$ and $B$. As an application, the upper triangular matrix algebra $T_n(A)$ is left weakly Gorenstein if and only if so is $A$. |
| title | Weakly Gorensteinness of tensor algebras and Morita algebras |
| topic | Representation Theory 16G10, 16E65, 18G15, 16E30 |
| url | https://arxiv.org/abs/2512.16072 |