Differential graded triangular matrix categories
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910596279042048 |
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| author | Miranda, M. Lizbeth Shaid Sandoval Vargas, Valente Santiago Páez, Edgar O. Velasco |
| author_facet | Miranda, M. Lizbeth Shaid Sandoval Vargas, Valente Santiago Páez, Edgar O. Velasco |
| contents | This paper focuses on defining an analog of differential-graded triangular matrix algebra in the context of differential-graded categories. Given two dg-categories $\mathcal{U}$ and $\mathcal{T}$ and $M \in \text{DgMod}(\mathcal{U} \otimes \mathcal{T}^{\text{op}})$, we construct the differential graded triangular matrix category $Λ:= \left( \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix} \right)$. Our main result is that there is an equivalence of dg-categories between the dg-comma category $(\text{DgMod}(\mathcal{T}),\text{GDgMod}(\mathcal{U}))$ and the category $\text{DgMod}\left( \left( \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix} \right)\right)$. This result is an extension of a well-known result for Artin algebras (see, for example, [2,III.2]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03910 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Differential graded triangular matrix categories Miranda, M. Lizbeth Shaid Sandoval Vargas, Valente Santiago Páez, Edgar O. Velasco Representation Theory Category Theory Rings and Algebras This paper focuses on defining an analog of differential-graded triangular matrix algebra in the context of differential-graded categories. Given two dg-categories $\mathcal{U}$ and $\mathcal{T}$ and $M \in \text{DgMod}(\mathcal{U} \otimes \mathcal{T}^{\text{op}})$, we construct the differential graded triangular matrix category $Λ:= \left( \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix} \right)$. Our main result is that there is an equivalence of dg-categories between the dg-comma category $(\text{DgMod}(\mathcal{T}),\text{GDgMod}(\mathcal{U}))$ and the category $\text{DgMod}\left( \left( \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix} \right)\right)$. This result is an extension of a well-known result for Artin algebras (see, for example, [2,III.2]. |
| title | Differential graded triangular matrix categories |
| topic | Representation Theory Category Theory Rings and Algebras |
| url | https://arxiv.org/abs/2409.03910 |