Homological Invariants for Connected DG Algebras

In this article, we study various homological invariants of differential graded (DG for short) modules over a connected DG algebra following Frankild–Jørgensen. Two different versions of homological dimensions (resolutional and functorial) are defined. In some cases, they are proved to be simply the bound of the cohomology of the DG module. Some homological identities, such as Auslander–Buchsbaum formula and Bass formula, are proved for compact DG modules over a connected DG algebra.