A note concerning the vanishing of local cohomology for roots in mixed characteristic

1. Problem Statement and Context

The paper addresses a specific problem in commutative algebra concerning the Cohen-Macaulayness of integral closures in mixed characteristic.

• Setting: Let $(S, \mathfrak{n})$ be an unramified regular local ring of mixed characteristic $p > 0$ with dimension $d$. Let $L$ be the quotient field of $S$. Consider a field extension $K = L(\omega)$ where $\omega^p \in L$ (a Kummer extension). Let $R$ be the integral closure of $S$ in $K$.
• The Core Question: In general, the integral closure of a regular local ring is not necessarily Cohen-Macaulay, especially in mixed characteristic. The paper investigates the specific case of modular Kummer extensions (where the degree of the extension is divisible by the characteristic $p$).
• Motivation: This setup serves as a counterexample to certain results by Roberts in the modular case and is relevant to the existence of small Cohen-Macaulay modules and singularity theory. The author aims to identify a precise, single obstruction to $R$ being Cohen-Macaulay.

2. Methodology

The proof relies on a combination of local cohomology, duality theory, and the theory of syzygies.

• Reduction to Hypersurfaces: The author reduces the problem to the ring $A = S[X]/(X^p - f)$, where $f = \omega^p \in S$. The integral closure $R$ is related to the conductor ideal $C$ of $A$ in $K$.
• Conductor and Duality: A central tool is the identification of the conductor ideal $C = \{x \in K \mid xR \subseteq A\}$ with the dual module $\text{Hom}_A(R, A)$. The paper utilizes the isomorphism $R \cong C^{-1}$ and $\text{Hom}_S(R, S) \cong C$ (as $S$-modules).
• Serre's Conditions: The analysis focuses on Serre's conditions $(S_k)$, which relate the depth of a module to its dimension. Specifically, the paper investigates when modules satisfy $(S_2)$ and $(S_3)$.
• Local Cohomology: The vanishing of specific local cohomology modules, particularly $H^{d-1}_{\mathfrak{n}}(R)$, is used as the primary criterion.
• Syzygy Theory: The proof connects the Cohen-Macaulay property to the module being a "syzygy" (a kernel in a free resolution), leveraging the Evans-Griffith Syzygy Theorem.

3. Key Contributions and Results

Theorem 1.1 (Main Equivalence)

The paper establishes the equivalence of four conditions for the integral closure $R$:

1. $R$ is Cohen-Macaulay.
2. The local cohomology module $H^{d-1}_{\mathfrak{n}}(R) = 0$.
3. The dual module $R^* := \text{Hom}_S(R, S)$ satisfies Serre's condition $(S_3)$ as an $S$-module.
4. $R^*$ satisfies $(S_3)$ for all prime ideals $Q$ containing $p$.

Significance: This result isolates the obstruction to Cohen-Macaulayness to a single local cohomology module ($H^{d-1}$). It implies that if the dual module satisfies a specific depth condition ($(S_3)$), the ring itself is Cohen-Macaulay.

Corollary 1.2 (Syzygy Connection)

Using the Evans-Griffith Syzygy Theorem (extended to mixed characteristic by André), the paper proves:

• $R$ is an $n$-th syzygy over $S$ for all $1 \leq n \leq d$ if and only if $R^*$ is a third syzygy (or an $n$-th syzygy for all $n$).
  This links the structural properties of $R$ directly to the homological properties of its dual.

Proof Strategy Highlights

• Case Analysis: The proof handles cases where $f$ is square-free or where the conductor $C$ lacks a primary component containing $p$ (where $R$ is known to be Cohen-Macaulay).
• Primary Decomposition: In the non-trivial case, the conductor $C$ is decomposed into primary components ($P$ containing $p$, and $J_i$ associated with divisors of $f$).
• Depth Calculations: The author reduces the problem to checking the depth of a quotient ring $S/I$. By showing that a specific ideal $I$ is free (using the fact that $S$ is factorial and $I$ is reflexive), the Cohen-Macaulayness of $R$ is established.
• Linkage Theory: The proof utilizes the concept of linkage between ideals (specifically between $C$ and $(xA :_A C)$) to show that $A/C$ satisfies $(S_2)$, which implies the vanishing of the relevant local cohomology.

4. Significance and Implications

• "Mirror Image" Result: The author notes that this result acts as a "mirror image" to the classical Evans-Griffith Syzygy Theorem. While the classical theorem states that a module with full support satisfying $(S_k)$ and rank $< k$ is free, this paper shows that for the integral closure $R$, the condition on the dual module ($R^*$ satisfying $(S_3)$) forces $R$ to be Cohen-Macaulay (and thus a syzygy).
• Mixed Characteristic Context: The results are significant because they hold in mixed characteristic, a setting where many classical results from equicharacteristic algebra fail or require complex modifications (e.g., the Direct Summand Theorem).
• Obstruction Theory: By pinpointing $H^{d-1}_{\mathfrak{n}}(R)$ as the sole obstruction, the paper provides a concrete computational target for determining the Cohen-Macaulayness of these specific rings.
• Applications: The findings are relevant to:
  • The failure of Roberts' results in the modular case.
  • The search for small Cohen-Macaulay modules.
  • Singularity theory, where the structure of integral closures determines the nature of singularities.

Conclusion

Prashanth Sridhar's note provides a sharp characterization of when the integral closure of an unramified regular local ring in a modular Kummer extension is Cohen-Macaulay. By establishing an equivalence between the Cohen-Macaulay property, the vanishing of a specific local cohomology group, and the $(S_3)$ condition on the dual module, the paper deepens the understanding of the interplay between duality, syzygies, and local cohomology in mixed characteristic.