  The paper explores the indecomposable submodule structures of quantum divided
power algebra $\mathcal{A}_q(n)$ defined in \cite{HU} and its truncated objects
$\mathcal{A}_q(n, \bold m)$. An "intertwinedly-lifting" method is established
to prove the indecomposability of a module when its socle is non-simple. The
Loewy filtrations are described for all homogeneous subspaces
$\mathcal{A}^{(s)}_q(n)$ or $\mathcal{A}_q^{(s)}(n, \bold m)$, the Loewy layers
and dimensions are determined. The rigidity of these indecomposable modules is
proved. An interesting combinatorial identity is derived from our realization
model for a class of indecomposable $\mathfrak{u}_q(\mathfrak{sl}_n)$-modules.
Meanwhile, the quantum Grassmann algebra $\Omega_q(n)$ over $\mathcal{A}_q(n)$
is constructed, together with the quantum de Rham complex $(\Omega_q(n),
d^\bullet)$ via defining the appropriate $q$-differentials, and its subcomplex
$(\Omega_q(n,\bold m), d^\bullet)$. For the latter, the corresponding quantum
de Rham cohomology modules are decomposed into the direct sum of some
sign-trivial $\mathfrak{u}_q(\mathfrak{sl}_n)$-modules.
