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 DC A-module with a free underlying graded A^#-module may not be semi-projective.
In this paper, we introduce a new numerical invariant complete level for a DG module over a local chain DG algebra and give a characterization of it in terms of ghost length. We also study some of its upper bounds. The cone length of a DG module is an invariaut closely related with the invariant level. We discover some important results on it.
In this paper, we introduce and study differential graded(DG for short) polynomial algebras. In brief, a DG polynomial algebra A is a connected cochain DG algebra such that its underlying graded algebra A~# is a polynomial algebra K[x_1, x_2,..., x_n] with |xi| = 1 for any i ∈ {1, 2,..., n}. We describe all possible differential structures on DG polynomial algebras, compute their DG automorphism groups, study their isomorphism problems, and show that they are all homologically smooth and Gorenstein DG algebras. Furthermore, it is proved that the DG polynomial algebra A is a Calabi-Yau DG algebra when its differential ?_A≠ 0 and the trivial DG polynomial algebra(A, 0) is Calabi-Yau if and only if n is an odd integer.
