6 papers · 1 filter
Regular rings over valuation rings
Shiji Lyu
Bertin (1972) defined regularity for coherent local rings, and Knaf (2004) studied the property for a local ring essentially finitely presented over a valuation ring . We di…
-ifications, semi-Nagata rings, and the lifting problem
Shiji Lyu
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 such that every finite -algebra that is an…
Uniform bounds in excellent -algebras and applications to semi-continuity
Shiji Lyu
We study two important numerical invariants, Hilbert--Kunz multiplicity and -signature, on the spectrum of a Noetherian -algebra that is not necessarily -fi…
The Japanese and universally Japanese properties for valuation rings and Prüfer domains
Shiji Lyu
We discuss the Japanese and universally Japanese properties for valuation rings and Prüfer domains. These properties, regarding finiteness of integral closure, have been studied ex…
Formal lifting of dualizing complexes and consequences
Shiji Lyu
We show that for a Noetherian ring that is -adically complete for an ideal , if admits a dualizing complex, so does . This gives an alternative proof of the fact…
Permanence properties of splinters via ultrapower
Shiji Lyu
We show that the splinter property ascends along regular residue field extensions, and along arbitrary regular maps in equal characteristic. We also study the splinter property of…