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Differential graded triangular matrix categories (2409.03910v2)

Published 5 Sep 2024 in math.RT, math.CT, and math.RA

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].

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