## Abstract

Let *A* be a connected cochain DG algebra, whose underlying graded algebra is an Artin-Schelter regular algebra of global dimension 2 generated in degree 1. We give a description of all possible differential of *A* and compute *H*(*A*). Such kind of DG algebras are proved to be strongly Gorenstein. Some of them serve as examples to indicate that a connected DG algebra with Koszul underlying graded algebra may not be a Koszul DG algebra defined in He and We (J Algebra, 2008, 320: 2934–2962). Unlike positively graded chain DG algebras, we give counterexamples to show that a bounded below DG *A*-module with a free underlying graded *A*#-module may not be semi-projective.

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Mao, X. DG algebra structures on AS-regular algebras of dimension 2.
*Sci. China Math.* **54**, 2235–2248 (2011). https://doi.org/10.1007/s11425-011-4256-z

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DOI: https://doi.org/10.1007/s11425-011-4256-z