  We prove that $F$-injectivity localizes, descends under faithfully flat
homomorphisms, and ascends under flat homomorphisms with Cohen-Macaulay and
geometrically $F$-injective fibers, all for arbitrary Noetherian rings of prime
characteristic. As a consequence, we show that the $F$-injective locus is open
on most rings arising in arithmetic and geometry. As a geometric application,
we prove that over an algebraically closed field of characteristic $p > 3$,
generic projection hypersurfaces associated to suitably embedded smooth
projective varieties of dimension $\le 5$ are $F$-pure, and hence
$F$-injective. This geometric result is the positive characteristic analogue of
a theorem of Doherty.
