Permanence properties of $F$-injectivity

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.