M-regularity of $\mathbb{Q}$-twisted sheaves and its application to linear systems on abelian varieties

G. Pareschi and M. Popa give criterions for global generations and surjectivity of multiplication maps of global sections of coherent sheaves on abelian varieties in the theory of M-regularity. In this paper, we generalize some of their criterions via the M-regularity of $\mathbb{Q}$-twisted sheaves. As an application, we show that the M-regularity of a suitable $\mathbb{Q}$-twisted sheaf implies property $(N_p)$ and jet-ampleness for ample line bundles on abelian varieties.