F-Stable Secondary Representations and Deformation of F-Injectivity

Abstract

We prove that deformation of F-injectivity holds for local rings \((R,\mathfrak {m})\) that admit secondary representations of \({H}_{\mathfrak {m}}^{i}(R)\) which are stable under the natural Frobenius action. As a consequence, F-injectivity deforms when \((R,\mathfrak {m})\) is sequentially Cohen–Macaulay (or more generally when all the local cohomology modules \({H}_{\mathfrak {m}}^{i}(R)\) have no embedded attached primes). We obtain some additional cases if \(R/\mathfrak {m}\) is perfect or if R is \(\mathbb {N}\)-graded.