## Abstract

When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular DG algebras.

When *A* is a regular DG algebra such that *H*(*A*) is a Koszul graded algebra, *H*(*A*) is proved to have the finite global dimension. And we give an example to illustrate that the global dimension of *H*(*A*) may be infinite, if the condition that *H*(*A*) is Koszul is weakened to the condition that *A* is a Koszul DG algebra. For a general regular DG algebra *A*, we give some equivalent conditions for the Gorensteiness.

For a finite connected DG algebra *A*, we prove that D^{c}(*A*) and D^{c}(*A*
^{op}) admit Auslander-Reiten triangles if and only if *A* and *A*
^{op} are Gorenstein DG algebras. When *A* is a non-trivial regular DG algebra such that *H*(*A*) is locally finite, D^{c}(*A*) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in D
^{b}_{lf}
(*A*) and D
^{b}_{lf}
(*A*
^{op}) instead, when *A* is a regular DG algebra.

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This work was supported by the National Natural Science Foundation of China (Grant No. 10731070) and the Doctorate Foundation of Ministry of Education of China (Grant No. 20060246003)

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Mao, X., Wu, Q. Compact DG modules and Gorenstein DG algebras.
*Sci. China Ser. A-Math.* **52**, 648–676 (2009). https://doi.org/10.1007/s11425-008-0175-z

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DOI: https://doi.org/10.1007/s11425-008-0175-z

### Keywords

- differential graded algebra
- Gorenstein DG algebra
- regular DG algebra
- Koszul DG algebra
- compact DG module
- Auslander-Reiten triangles
- amplitude
- projective dimension