${\mathbb{Z}}_2$-graded identities of the Grassmann algebra over a finite field (update version)

Let $F$ be a finite field with the characteristic $p>2$ and let $G$ be the unitary Grassmann algebra generated by an infinite dimensional vector space $V$ over $F$. In this paper, we determine a basis for ${\mathbb{Z}}_2$-graded polynomial identities for any non-trivial ${\mathbb{Z}}_2$-grading such that its underlying vector space is homogeneous.