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Regular rings over valuation rings

Shiji Lyu

Abstract

Bertin (1972) defined regularity for coherent local rings, and Knaf (2004) studied the property for a local ring $A$ essentially finitely presented over a valuation ring $V$. We discuss several properties of this notion of regularity for such $A$, obtaining results parallel to results for regularity of Noetherian local rings. We include classical and modern topics: openness of loci, perfectoid big Cohen--Macaulay algebras, and cotangent complexes. We also give an application to Noetherian rings, showing a version of Kodaira's vanishing theorem in large enough residue characteristics.

Regular rings over valuation rings

Abstract

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 discuss several properties of this notion of regularity for such , obtaining results parallel to results for regularity of Noetherian local rings. We include classical and modern topics: openness of loci, perfectoid big Cohen--Macaulay algebras, and cotangent complexes. We also give an application to Noetherian rings, showing a version of Kodaira's vanishing theorem in large enough residue characteristics.

Paper Structure

This paper contains 40 sections, 60 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Openness of loci
  4. The direct summand theorem
  5. Variants of Kunz's Theorem
  6. Cotangent complexes
  7. Vanishing theorems
  8. Acknowlegement
  9. Preliminary facts
  10. Ideals of valuation rings
  11. Essentially finitely generated algebras over Prüfer domains
  12. Derived category of coherent rings
  13. Finite rank valuation rings
  14. An elementary reduction
  15. Openness of loci
  16. ...and 25 more sections

Key Result

Theorem 1.2.1

Let $R$ be a Noetherian ring. The following are equivalent. If $R$ satisfies the equivalent conditions, then every finitely generated $R$-algebra has open normal locus.

Theorems & Definitions (120)

  • Theorem 1.2.1: Mat-CA and EGA4_2
  • Theorem 1.2.2: Theorem \ref{['thm:RegQCopen']} and Corollary \ref{['cor:NorQCopen']}
  • Theorem 1.2.3: =Theorem \ref{['thm:ReducedLocusQCOpen']}
  • Corollary 1.2.4: =Corollary \ref{['cor:RegNorQCopenForFlat']}
  • Corollary 1.2.5: =Corollary \ref{['cor:RegNorQCopenForFiniteRank']}
  • Theorem 1.3.1: =Theorem \ref{['thm:RLRisSplinter']}
  • Theorem 1.4.1: =Theorem \ref{['thm:PerfdKunz']}
  • Theorem 1.4.2: =Theorem \ref{['thm:KunzALL']} and Corollary \ref{['cor:OneFeFiniteTorDim=Regular']}
  • Theorem 1.5.2
  • Theorem 1.5.3
  • ...and 110 more