On the perturbations of noetherian local domains

Time: 9:30 -- 11: 00, March 12, 2025

Venue: Room 612, A6, Institute of Mathematics-VAST

Abstract: Let (R,m) be a noetherian local ring, with a presentation R=S/I, where (S,n) is, say, a regular local ring and I is an ideal of S generated by f_1,..,f_c. An m-adic perturbation of R is a quotient ring S/(f_1+e_1,...,f_c+e_c), where e_1,...,e_c are elements in a (high) power of n, namely a ring obtained by perturbing the defining equations of R. We study how the properties of being reduced, an integral domain, and normal, behave under small (m-adic) perturbations of the defining equations of a noetherian local ring. It is not hard to show that the property of being a local integral domain (reduced, normal ring  is not stable under small perturbations in general. We prove that perturbation stability holds in the following situations: (1) perturbation of being an integral domain for factorial excellent Henselian local rings; (2) perturbation of normality for excellent local complete intersections containing a field of characteristic zero; and (3) perturbation of reducedness for excellent local complete intersections containing a field of characteristic zero, and for factorial Nagata local rings. Joint work with N.H. Duc and P.H. Quy.