  Let $F$ be an infinite field of characteristic different from 2, and let $E$
be the Grassmann algebra of an infinite dimensional $F$-vector space $L$. In
this paper we study the $\mathbb{Z}$-graded polynomial identities of $E$ with
respect to certain $\mathbb{Z}$-grading such that the vector space $L$ is
homogeneous in the grading. More precisely, we construct three types of
$\mathbb{Z}$-gradings on $E$, denoted by $E^{\infty}$, $E^{k^\ast}$ and
$E^{k}$, and we give the explicit form of the corresponding $\mathbb{Z}$-graded
polynomial identities. We show that the homogeneous superalgebras $E_{\infty}$,
$E_{k^\ast}$ and $E_{k}$ studied in \cite{disil} can be obtained from
$E^{\infty}$, $E^{k^\ast}$ and $E^{k}$ as quotient gradings. Moreover we
exhibit several other types of homogeneous $\mathbb{Z}$-gradings on $E$, and
describe their graded identities.
