Regular rings over valuation rings

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.

• The paper presents a framework extending regularity to local rings over valuation bases by employing the concept of finite weak global dimension.
• It utilizes Noetherian approximation, perfectoid BCM algebras, and advanced flatness criteria to generalize the Direct Summand and Kunz theorems.
• Results include rigorous characterizations of openness properties and new vanishing theorems, with important implications for arithmetic geometry and commutative algebra.

Authoritative Summary of "Regular rings over valuation rings" (2603.29104)

Introduction and Scope

This work presents a comprehensive investigation into commutative algebraic structures of regular local rings in the setting where the base is a valuation ring, focusing primarily on local rings essentially of finite presentation over such bases. The main technical lens is the extension of Bertin's notion of regularity (finite weak global dimension for coherent local rings) to valuation ring contexts, building substantial parallels with the classical theory for Noetherian local rings. The themes include the behavior of loci (openness of normal and regular loci), generalizations of the Direct Summand Theorem, variants of Kunz's theorem, roles of big Cohen–Macaulay (BCM) algebras, and the behavior of cotangent complexes. The results have impact in both the theory of commutative algebra and the arithmetic and geometric applications involving non-Noetherian bases.

Openness of Loci and Structural Results

A substantial part of the paper analyzes the openness of the loci of various regularity properties. It establishes precise analogues of the well-known facts for Noetherian rings regarding the quasi-compactness and openness of the regular and normal loci, elucidating the complications that arise for general valuation rings. The equivalence of openness properties to certain algebraic conditions (liftability of regularity along finite extensions, isolation of primes) is developed in detail.

For valuation rings of finite rank, or spherically complete ones, the results show that open regular (and normal) loci of finitely generated algebras exist and are quasi-compact, establishing J-2-type properties in this broader class. Non-examples are discussed, demonstrating the necessity of the additional conditions by constructing rings where openness fails due to the topological complexity of the spectrum.

Technical reduction arguments based on approximation by finite rank valuation subrings play a central role, as does an adaptation of standard devissage to this coherent setting.

Direct Summand Theorem for Regular Rings over Valuation Rings

The paper proves that any essentially finitely presented local algebra over a valuation ring with finite weak global dimension is a splinter, extending Hochster and André’s direct summand theorem to this broad setting. This is achieved by constructing faithfully flat extensions using perfectoid BCM algebras, leveraging the functoriality under colimits of absolute integral closures, and deploying dimension reduction and Noetherian approximation arguments. The essential ingredient is a highly technical version of the BCM algebra construction adapted to the valuation ring case, producing modules and algebras with the correct regular sequence properties via ultraproducts or filtered colimits.

Kunz-Type Theorems and BCM Algebras

Several versions of Kunz's theorem are established. In positive or mixed characteristic, the equivalences between finite weak global dimension, flatness of certain perfectoid or absolute integral closures, low derived $\operatorname{Tor}$-dimension of $p$-adic completions, and existence of perfectoid BCM algebras or modules with specific properties are proven.

Furthermore, an extended commutative diagram of implications between various forms of module flatness and algebra regularity is rigorously justified in both mixed and equal characteristic, with explicit attention to derived completions. The arguments rely on new flatness criteria formulated for non-Noetherian settings using generalizations of regular sequences, BCM modules over valuation rings, and controlling $\operatorname{Tor}$-dimensions via Noetherian approximation.

Notably, for $F$-finite analogues, the paper shows that finite $\operatorname{Tor}$-dimension of iterated Frobenius modules in characteristic $p$ suffices to guarantee regularity in the sense of finite weak dimension, by employing refined descent results and the construction of suitable finite type approximations.

Cotangent Complex Characterizations

A full technical development is given for the behavior of cotangent complexes $L_{A/k}$ (in both absolute and relative forms) for local rings essentially finitely presented over valuation rings, with residue fields possibly not contained in the base. The key results show equivalence between the vanishing of $H^{-1}$ in the cotangent complex, generation of maximal ideals by regular sequences, and finite weak dimension, paralleling the classical results over fields and extending them to this coherent setting.

The proofs feature delicate arguments around the base change and descent of cotangent complexes, reduction to the case of absolute integral closures, and technical manipulations using the approximation by regular local rings, with detailed control of the ambient homotopical and cohomological dimension.

Vanishing Theorems and Applications

A novel application establishes versions of the Kodaira vanishing theorem for regular schemes over valuation rings of large residue characteristic, showing the vanishing of higher cohomology of twisted dualizing sheaves for ample line bundles on flat, regular, projectively embedded closed subschemes, provided the residue characteristic exceeds an explicit bound depending on the embedding data. This is significant because such vanishing theorems are generally known to fail in small or mixed characteristics for general bases.

The proofs involve constructing families over products of local rings, exploiting ultraproducts and the topological compactness of the spectrum to transfer vanishing from the residue characteristic zero case, combined with approximation theory and detailed analysis of the constructible topology and Fitting ideals.

Technical Developments and Further Directions

Throughout, an overarching technical apparatus is established around the approximation of non-Noetherian valuation rings by direct limits of Noetherian regular rings. Results on the functoriality and descent of Tor-independence, behavior of regular sequences, and preservation of homological properties are proven for these approximations.

The paper highlights that most results generalize to any Pr\"ufer or absolutely integrally closed domains satisfying certain isolating conditions for principal ideals.

Conclusion

This paper constructs a robust theory of regularity for coherent local rings essentially of finite presentation over valuation rings, extending structural, homological, and cohomological properties well beyond the classical Noetherian context. The framework established here enables new theorems (such as versions of the direct summand and vanishing theorems), and the technical machinery has implications for future research in both commutative algebra and the study of arithmetic schemes and birational geometry over non-Noetherian and valuation ring bases. The methods open avenues for further study of duality and Macaulayfication in the context of valuation rings, linking with recent advances in perfectoid and prismatic cohomology, and deepening the connection between commutative algebra, model theory, and algebraic geometry.