M. Lizbeth Shaid Sandoval Miranda, Valente Santiago Vargas, Edgar O. Velasco Páez
{"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}
引用次数: 0
Abstract
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].