Generic vanishing theory in positive characteristic

We simplify and improve the main fundamental theorems of positive characteristic generic vanishing theory. As a quick corollary of the theory, we prove that a normal variety $X$ of maximal Albanese dimension satisfies $H^0(X,{\omega }_X)\mathrm{\ne }0$ and that if $\mathrm{A}\mathrm{l}\mathrm{b}(X)$ is ordinary, then $S^0(X,{\omega }_X)\mathrm{\ne }0$.