$(S_2)$-ifications, semi-Nagata rings, and the lifting problem

This is a two-part article. In the first part, we study an alternative notion to Nagata rings. A Nagata ring is a Noetherian ring $R$ such that every finite $R$-algebra that is an integral domain has finite normalization. We replace the normalization by an $(S_2)$-ification, study new phenomena, and prove parallel results. In particular, we show a Nagata domain has a finite $(S_2)$-ification. In the second part, we study the local lifting problem. We show that for a semilocal Noetherian ring $R$ that is $I$-adically complete for an ideal $I$, if $R/I$ has $(S_k)$ (resp. Cohen--Macaulay, Gorenstein, lci) formal fibers, so does $R$. As a consequence, we show if $R/I$ is a quotient of a Cohen--Macaulay ring, so is $R$. We also discuss difficulties in lifting geometrically $(R_k)$ formal fibers.