Remarks on 1-motivic sheaves

We generalize the construction of the category of 1-motives with torsion M due to Barbieri-Viale, Rosenschon and Saito, as well as the construction of the category of 1-motivic sheaves Shv due to Barbieri-Viale and Kahn to perfect fields k (without inverting the exponential characteristic). For k transcendental over the prime field we extend showing that M and Shv have equivalent bounded derived categories.