Permanence properties of $F$-injectivity

Abstract

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.

29 pages. v2: added Theorem 6.6, fixed Lemma A.2, more transparent proof of Lemma 4.5, other small additions and changes. v3: Replaced Theorem 6.6 with a reference, added references, shortened some proofs. v4: Minor changes

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