Deformation of F-injectivity and local cohomology

Frobenius Actions on Local Cohomology Modules and Deformation

“…We prove that F -fullness and F -anti-nilpotency both deform, and we obtain more evidence on deformation of F -injectivity. Our results largely generalize earlier results of [12] in this direction. We list some of our main results here: (1) If R/(x) is F -anti-nilpotent, then so is R.…”

“…Now the map x p−1 F : H s+1 m (R) → H s+1 m (R) is injective by the same argument as in Theorem 5.11. The following immediate corollary of the above proposition recovers (and in fact generalizes) results in [12]. Because of the deep connections between F -injective and Du Bois singularities [21,2] and Remark 2.6, we believe that it is rarely the case that an F -injective ring fails to be F -full (again, the only example we know this happens is [18,Example 3.5], which is based on the construction of [4, Example 2.16]).…”

“…compare with[12], Main Theorem). Let (R, m) be a local ring of prime characteristic p and x a regular element of R. Suppose R/(x) is F -injective.…”

F-injectivity and Frobenius closure of ideals in Noetherian rings of characteristic p> 0

“…The first partial result concerning Problem 4 is obtained in [18]. It is shown in [27] that if (R, m) is a local ring essentially of finite type over C such that R/xR is of dense F -injective type for a regular element x ∈ m, then R is of dense F -injective type.…”

Nilpotence of Frobenius actions on local cohomology and Frobenius closure of ideals

“…Fedder proved that R is indeed F -injective under the assumptions R is Cohen-Macaulay and R/(x) is F -injective for some regular element x in [Fed83]. We refer the reader to [HMS14] and [MQ] for more recent developments on the deformation of F -injectivity problem.…”