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

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