On generalized deformation problems

Abstract

Let (R,m) be a Noetherian local ring, and I an R-ideal such that pdRR/I<∞. If R/I satisfies a property P, it is natural to ask if R would also have the property P, we call this the generalized deformation problem. Our paper gives some properties that hold for the generalized deformation problems. There are two main parts in this paper. The first part is about F-singularities in the generalized deformation problems. Motivated by Aberbach's work, we show that if every maximal regular sequence on R/I is Frobenius closed, then every regular sequence on R is Frobenius closed. Moreover, under mild assumptions, Frobenius closed can be replaced by tightly closed. By these two results, we solve the generalized deformation problems for F-injectivity in the Cohen-Macaulay case and F-rationality under mild assumptions. Namely, if R is a Noetherian local ring of characteristic p, and I an R-ideal such that pdRR/I<∞, then if R is Cohen-Macaulay and R/I is F-injective, then R is F-injective. If R is excellent and R/I is F-rational, then R is F-rational. The second part is about some basic properties of rings in arbitrary characteristic. We prove that if R is a Noetherian local ring, I an R-ideal such that pdRR/I<∞, then if R/I is Sk, Rk+Sk+1, normal, reduced or a domain, then so is R.