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When I am reading Lunts' Categorical Resolution of Singularities, section 3.2, I found the following:

Let $A$ and $B$ be DG algebras and $N$ is a $A$-$B$-bimodule. Then we obtain a new DG algebra $$C=\begin{pmatrix}B&0\\N&A\end{pmatrix}$$

However, I cannot find the definition for this notation. So what is the new DG algebra, as well as its differential? Moreover, what is the motivation of this construction?

Harold Finch
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    Well, I would guess it's $A \oplus N \oplus B$ with multiplication defined the only way you can: with the multiplications of $A$ and $B$ and the actions of $A$ and $B$ on $N$. (Any products that don't have an obvious definition are zero.) – Zhen Lin Feb 11 '22 at 10:18
  • @ZhenLin Thanks for your comment! I find the answer and it is quite close to your guess! – Harold Finch Feb 22 '22 at 11:05

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I find the DG-category version of the definition of gluing in the section 4 of Categorical resolutions of irrational singularities. If we treat the DG algebra as a DG-category with one object, the construction can be easily applied to DG algebras.

Harold Finch
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