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Let E be the infinite dimensional Grassmann algebra over a finite field F ofcharacteristic not 2. In this paper we deal with the homogeneous Z_2-gradingsof E. In particular, we compute an exact value for the Z_2-graded homogeneouscodimensions of E, and a lower and an upper bound for the Z_2-graded(non-homogeneous) codimensions of E for each of its Z_2-homogeneous grading.

On the Z_2-graded codimensions of the Grassmann algebra over a finite field