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

Abstract

Let $F$ be a finite field with 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 of the $\mathbb{Z}_{2}$-graded polynomial identities for any non-trivial $\mathbb{Z}_{2}$-grading such that a basis of $V$ is homogeneous in this grading.