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As the definition of free class of differential modules over a commutative ring in [1], we define DG free class for semifree DG modules over an Adams connected DG algebra A. For any DG A-modules M, we define its cone length as the least DG free classes of all semifree resolutions of M. The cone length of a DG A-module plays a similar role as projective dimension of a module over a ring does in homological ring theory. The left (resp., right) global dimension of an Adams connected DG algebra A is defined as the supremum of the set of cone lengths of all DG A-modules (resp., Aop-modules). It is proved that the definition is a generalization of that of graded algebras. Some relations between the global dimension of H(A) and the left (resp. right) global dimension of A are discovered. When A is homologically smooth, we prove that the left (right) global dimension of A is finite and the dimension of D(A) and Dc(A) are not bigger than the DG free class of a minimal semifree resolution X of the DG Ae-module A. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]