$\mathbb{Z}_{2}$-graded identities of the Grassmann algebra over a finite field (update version)

Abstract: 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.