M. Lizbeth Shaid Sandoval Miranda, Valente Santiago Vargas, Edgar O. Velasco Páez
{"title":"梯度三角矩阵类别","authors":"M. Lizbeth Shaid Sandoval Miranda, Valente Santiago Vargas, Edgar O. Velasco Páez","doi":"arxiv-2409.03910","DOIUrl":null,"url":null,"abstract":"This paper focuses on defining an analog of differential-graded triangular\nmatrix algebra in the context of differential-graded categories. Given two\ndg-categories $\\mathcal{U}$ and $\\mathcal{T}$ and $M \\in\n\\text{DgMod}(\\mathcal{U} \\otimes \\mathcal{T}^{\\text{op}})$, we construct the\ndifferential graded triangular matrix category $\\Lambda := \\left(\n\\begin{smallmatrix} \\mathcal{T} & 0 \\\\ M & \\mathcal{U} \\end{smallmatrix}\n\\right)$. Our main result is that there is an equivalence of dg-categories\nbetween the dg-comma category\n$(\\text{DgMod}(\\mathcal{T}),\\text{GDgMod}(\\mathcal{U}))$ and the category\n$\\text{DgMod}\\left( \\left( \\begin{smallmatrix} \\mathcal{T} & 0 \\\\ M &\n\\mathcal{U} \\end{smallmatrix} \\right)\\right)$. This result is an extension of a\nwell-known result for Artin algebras (see, for example, [2,III.2].","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"1 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Diferential graded triangular matrix categories\",\"authors\":\"M. Lizbeth Shaid Sandoval Miranda, Valente Santiago Vargas, Edgar O. Velasco Páez\",\"doi\":\"arxiv-2409.03910\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This paper focuses on defining an analog of differential-graded triangular\\nmatrix algebra in the context of differential-graded categories. Given two\\ndg-categories $\\\\mathcal{U}$ and $\\\\mathcal{T}$ and $M \\\\in\\n\\\\text{DgMod}(\\\\mathcal{U} \\\\otimes \\\\mathcal{T}^{\\\\text{op}})$, we construct the\\ndifferential graded triangular matrix category $\\\\Lambda := \\\\left(\\n\\\\begin{smallmatrix} \\\\mathcal{T} & 0 \\\\\\\\ M & \\\\mathcal{U} \\\\end{smallmatrix}\\n\\\\right)$. Our main result is that there is an equivalence of dg-categories\\nbetween the dg-comma category\\n$(\\\\text{DgMod}(\\\\mathcal{T}),\\\\text{GDgMod}(\\\\mathcal{U}))$ and the category\\n$\\\\text{DgMod}\\\\left( \\\\left( \\\\begin{smallmatrix} \\\\mathcal{T} & 0 \\\\\\\\ M &\\n\\\\mathcal{U} \\\\end{smallmatrix} \\\\right)\\\\right)$. This result is an extension of a\\nwell-known result for Artin algebras (see, for example, [2,III.2].\",\"PeriodicalId\":501136,\"journal\":{\"name\":\"arXiv - MATH - Rings and Algebras\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Rings and Algebras\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.03910\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.03910","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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 $\Lambda := \left(
\begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}
\right)$. 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].