2025/05/01 by Lyu, Shiji
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2505.00521
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 (S2)-ification, study new phenomena, and prove parallel results. In particular, we show a Nagata domain has a finite (S2)-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 (Sk) (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.