On generalized deformation problems
Let $(R,m)$ be a Noetherian local ring and $I$ an ideal with finite projective dimension. If $R/I$ satisfies some property $\mathcal{P}$, it is natural to ask whether $R$ would also satisfy this property $\mathcal{P}$. This is called the generalized deformation problem. In this paper we discuss some properties that would satisfy this problem. There are two main parts for this paper. In the first part we focus on F-singularities of characteristic $p$. We show that F-injective satisfies this problem for the Cohen-Macaulay ring case and F-rational satisfies this problem for the excellent ring case. In the second part there is no restriction on the characteristic of $R$, we show that when $R$ is catenary and equidimensional with $I$ perfect, then the Serre's Condition $R_k$ would satisfy the problem. And the Serre's Condition $S_k$, $R_k+S_{k+1}$, normal rings, reduced rings and domains would always satisfy this problem.