Russian Mathematical Surveys, 2025, Volume 80, Issue 4, Pages 723–725 DOI: https://doi.org/10.4213/rm10252e (Mi rm10252)

DOI: https://doi.org/10.4213/rm10252e

Funding agency	Grant number
Ministry of Science and Higher Education of the Russian Federation	075-15-2025-303
This work was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2025-303).

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Let ${\mathcal T}$ be a triangulated category, and let ${\mathcal I}_1$ and ${\mathcal I}_2$ be two full subcategories of ${\mathcal T}$. Denote by ${\mathcal I}_1*{\mathcal I}_2$ the full subcategory of ${\mathcal T}$ consisting of all objects such that there is a distinguished triangle $M_1\to M\to M_2$ with $M_i\in {\mathcal I}_i$. For any subcategory ${\mathcal I}\subset{\mathcal T}$, denote by $\langle {\mathcal I}\rangle$ (respectively, $\overline{\langle {\mathcal I}\rangle}$) the smallest full subcategory of ${\mathcal T}$ containing ${\mathcal I}$ and closed under finite direct sums, direct summands, and shifts (respectively, all direct sums and shifts). We put ${\mathcal I}_1 \diamond{\mathcal I}_2= \langle {\mathcal I}_1*{\mathcal I}_2\rangle$, and we define inductively $\langle {\mathcal I}\rangle_k=\langle{\mathcal I}\rangle_{k-1}\diamond \langle {\mathcal I}\rangle$ and $\overline{\langle {\mathcal I}\rangle}_k= \overline{\langle{\mathcal I}\rangle}_{k-1}\diamond \overline{\langle {\mathcal I}\rangle}$. If ${\mathcal I}$ consists of a single object $E$, we denote $\langle {\mathcal I}\rangle$ by $\langle E\rangle$ and put inductively $\langle E\rangle _k=\langle E\rangle_{k-1}\diamond\langle E\rangle$.

Let ${\mathscr A}$ be a differential graded algebra ( = DGA) over a base field $\Bbbk$. Denote by ${\mathcal D}({\mathscr A})$ the derived category of all right DG ${\mathscr A}$-modules, and by $\mathrm{perf}\text{-}{\mathscr A}\subset {\mathcal D}({\mathscr A})$ the triangulated subcategory of perfect modules. The latter is generated by ${\mathscr A}$, that is, $\mathrm{perf}\text{-}{\mathscr A}= \bigcup_{k}\langle{\mathscr A}\rangle_k$, and coincides with the subcategory of compact objects ${\mathcal D}({\mathscr A})^c\subset {\mathcal D}({\mathscr A})$ (see [3]).

A DGA ${\mathscr A}$ is called regular if $\mathrm{perf}\text{-}{\mathscr A}=\langle {\mathscr A}\rangle _k$ for some $k$. It is equivalent to the property ${\mathcal D}({\mathscr A})= \overline{\langle {\mathscr A}\rangle}_k$ (see [1], Proposition 2.2.4). A DGA ${\mathscr A}$ is called $\Bbbk$- smooth if ${\mathscr A}\in \mathrm{perf}\text{-}({\mathscr A}^{\circ}\otimes_{\Bbbk}{\mathscr A})$. A DGA ${\mathscr A}$ is called proper if ${\mathscr A}\in\mathrm{perf}\text{-}\Bbbk$. Denote by ${\mathcal D}_{\mathsf{cf}}({\mathscr A})\subset {\mathcal D}({\mathscr A})$ the full triangulated subcategory of all DG modules which are perfect as DG $\Bbbk$-modules. By [1], Theorem 1.3, for any regular and proper DGA ${\mathscr A}$ we have an equivalence ${\mathcal D}_{\mathsf{cf}}({\mathscr A})\cong \mathrm{perf}\text{-}{\mathscr A}$.

Let $h\colon{\mathscr A}\to{\mathscr B}$ be a homomorphism ( = map) of DGAs. Denote by $h^*\colon{\mathcal D}({\mathscr A})\to{\mathcal D}({\mathscr B})$ and $h_*\colon{\mathcal D}({\mathscr B})\to {\mathcal D}({\mathscr A})$ the induction and restriction functors, respectively. Every DG ${\mathscr A}\text{-}{\mathscr B}$-bimodule ${\mathsf T}$ defines a derived tensor functor $F_{\mathsf T}:=(-)\overset{\mathbf L} {\otimes}_{\mathscr A} {\mathsf T}\colon {\mathcal D}({\mathscr A}) \to {\mathcal D}({\mathscr B})$, which commutes with arbitrary direct sums. Denote by $\operatorname{Im} F_{\mathsf T}$ the full subcategory of ${\mathcal D}({\mathscr B})$ that consists of all objects isomorphic to $F_{\mathsf T}(X)$ for some $X\in{\mathcal D}({\mathscr A})$.

Note that $\langle \operatorname{Im} F_{\mathsf T}\rangle_{n}= \overline{\langle \operatorname{Im} F_{\mathsf T}\rangle}_{n}$ since $\operatorname{Im} F_{\mathsf T}\subset{\mathcal D}({\mathscr B})$ is closed under direct sums and shifts.

Let ${\mathsf I}\subset{\mathscr A}$ be a two-sided DG ideal. Denote by ${\mathscr R}$ the quotient DGA ${\mathscr A}/{\mathsf I}$ and by $q\colon{\mathscr A}\to{\mathscr R}$ the natural map of DGAs.

Let $h\colon{\mathscr A}\to{\mathscr B}$ be a map of DGAs, and let ${\mathsf T}_{h}$ be the DG ${\mathscr A}\text{-}{\mathscr A}$-bimodule which is the cone of the map $h$ in the DG category of all DG ${\mathscr A}\text{-}{\mathscr A}$-bimodules. Denote by ${\mathsf T}_h^{\otimes k}\in{\mathcal D}({\mathscr A}^{\circ} \otimes_{\Bbbk}{\mathscr A})$ the derived tensor product ${\mathsf T}_h\overset{\mathbf L}{\otimes}_{\mathscr A}\cdots \overset{\mathbf L}{\otimes}_{\mathscr A}{\mathsf T}_h$ of $k$ copies of ${\mathsf T}_h$.

Bibliography
1.	A. Bondal and M. Van den Bergh, Mosc. Math. J., 3:1 (2003), 1–36
2.	V. Hinich, Comm. Algebra, 25:10 (1997), 3291–3323
3.	B. Keller, Ann. Sci. École Norm. Sup. (4), 27:1 (1994), 63–102
4.	B. Toën and M. Vaquié, Ann. Sci. École Norm. Sup. (4), 40:3 (2007), 387–444

Citation in format AMSBIB

\Bibitem{EfiOrl25}
\by A.~I.~Efimov, D.~O.~Orlov
\paper Remarks on regular and smooth DG algebras
\jour Russian Math. Surveys
\yr 2025
\vol 80
\issue 4
\pages 723--725
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\crossref{https://doi.org/10.4213/rm10252e}
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