  In this paper we undertake a basic study on connective commutative
differential graded algebras (CDGA), more precisely, piecewise Noetherian CDGA,
which is a DG-counter part of commutative Noetherian algebra. We establish
basic results for example, Auslaner-Buchsbaum formula and Bass formula without
any unnecessary assumptions.
  The key notion is the sup-projective (sppj) and inf-injective (ifij)
resolutions introduced by the author, which are DG-versions of the projective
and injective resolution for ordinary modules. These are different from
DG-projective and DG-injective resolutions which is known DG-version of the
projective and injective resolution. In the paper, we show that sppj and ifij
resolutions are powerful tools to study DG-modules. Many classical result about
the projective and injective resolutions can be generalized to DG-setting by
using sppj and ifij resolutions. . Among other things we prove a DG-version of
Bass's structure theorem of a minimal injective resolution holds for a minimal
ifij resolution and a DG-version of the Bass numbers introduced by the same
formula with the classical case. We also prove a structure theorem of a minimal
ifij resolution of a dualizing complex $D$, which is completely analogues to
the structure theorem of a minimal injective resolution of a dualizing complex
over an ordinary commutative algebra.
  Specializing to results about a dualizing complex, we study a Gorenstein
CDGA. We generalize a result by Felix-Halperin-Felix-Thomas and Avramov-Foxby
which gives conditions that a CDGA $R$ is Gorenstein in terms of its cohomology
algebra $\text{H}(R)$.
