The Second Vanishing Theorem in Ramified Mixed Characteristic

Abstract: We prove the Second Vanishing Theorem for local cohomology modules over regular local ramified mixed characteristic rings. A few applications are given.

• The paper establishes the precise equivalence between local cohomological dimension and connectedness conditions in regular local rings of ramified mixed characteristic.
• It innovates by employing geometric techniques such as formal schemes, resolution of singularities, and Gabber’s global generation result.
• The proof unifies previous cases, impacting the understanding of depth, Lyubeznik numbers, and connected components in punctured spectra.

The Second Vanishing Theorem in Ramified Mixed Characteristic

Introduction and Historical Context

The paper "The Second Vanishing Theorem in Ramified Mixed Characteristic" (2606.26867) establishes the validity of the Second Vanishing Theorem (SVT) for local cohomology over regular local ramified mixed characteristic rings. The SVT describes a precise relationship between local cohomological dimension, the dimension of supports, and the connectedness of certain loci in spectra of local rings. While the cases of equal characteristic (Ogus in char 0; Peskine-Szpiro in char $p$) and unramified mixed characteristic (Huneke-Lyubeznik; Zhang) are already settled, the ramified mixed characteristic case has remained notably resistant. This is in part due to the absence of structure theorems that facilitate power series presentations over nice coefficient rings in the ramified setting, as available in the unramified case.

Recent advances further complicated the landscape: Linquan Ma's construction of a regular ramified ring with a local cohomology module having infinitely many associated primes provided evidence that the ramified mixed characteristic case could exhibit behavior foreign to the unramified or equicharacteristic contexts. Despite these obstacles, the present work confirms the SVT for the ramified case, restoring a symmetry in the theory of local cohomological dimension.

Statement and Equivalences

Given a regular local ring $A$ and an ideal $I$, the local cohomological dimension $\mathrm{cd}(A, I)$ is the supremum of $i$ such that $H^i_I(M)\neq 0$ for some $A$-module $M$. The SVT for a pair $(A, I)$ asserts the equivalence:

• $\mathrm{cd}(A, I)\leq \dim A-2$
• $A$0 and the punctured spectrum $A$1 is connected,

where $A$2 denotes the completion of the strict henselization of $A$3.

In the context of ramified mixed characteristic, the absence of nice coefficient rings and subtleties in the behavior of normalization and associated primes made a conceptual proof out of reach using known algebraic tools. Prior partial results required restrictive hypotheses—either on the ideal $A$4 or on the regularity and structure of $A$5.

Methodology and Proof Structure

A key innovation of the paper is the pivot back to geometric methods, distinct from the largely algebraic techniques mainstreamed in recent decades. The proof is structured in a sequence of precise reductions:

1. Formal Schemes and Reduction to Surfaces: The SVT is reformulated in terms of extending formal functions from punctured formal neighborhoods to the whole formal completion, reducing the problem (via standard dimension shifting arguments) to the case where $A$6 is a two-dimensional integral domain.
2. Residue Field and Normalization Reductions: A reduction to complete local rings with separably closed (or algebraically closed) residue field is carried out, ensuring geometric connectedness can be argued strictly on the closed fiber. Further, normalization steps reduce the problem to the normal surface case.
3. Geometric Resolution and Gabber's Result: A resolution of singularities for the closed two-dimensional subscheme is constructed, embedded into projective space over $A$7. Crucially, Gabber's result that the normal bundle of a regular subscheme in mixed characteristic is globally generated is invoked—this is the only stage where ramification in the mixed characteristic is essential. This technical global generation result allows geometric lifting of sections in formal neighborhoods, circumventing the need for explicit presentation of $A$8 as a power series ring.
4. Cohomological Vanishing and Intersection Theory: The formal extension problem is analyzed using cohomological vanishing results. The negative definiteness of the intersection matrix (via Du Val and Lipman) for the components of the exceptional divisor in the resolution ensures the vanishing of $A$9 on the formal punctured neighborhood, thus completing the extension argument.

These reductions are made fully rigorous by a sequence of propositions establishing injectivity and isomorphism of formal global section maps, with careful analysis of how automorphisms and base changes preserve the relevant properties.

Main Results and Key Implications

The principal result is the proof that the SVT holds for all regular local rings in ramified mixed characteristic and all ideals $I$0.

• Numerical Strength: The conclusion is unconditional—there are no extra assumptions on the ideal $I$1 or on the structure of the ring beyond regularity. This matches the strongest form of the SVT as previously known in other characteristics.
• Methodological Distinction: The proof abandons the need for presenting the ring as a power series over a "nice" base (impossible in general in the ramified case), instead relying on geometric properties and intersection theory.

This completion impacts the structure theory of local cohomology, especially in connecting depth and cohomological dimension. For $I$2 (i.e., depth at least two), the SVT ensures that cohomological dimension drops accordingly, matching expectations from both geometric intuition and previous algebraic results.

Theoretical and Practical Applications

The SVT is a structural result with multiple downstream consequences:

• Depth vs. Cohomological Dimension: In regular local rings, sufficient depth on $I$3 implies an upper bound on $I$4. The SVT confirms this for depth at least two. The full relationship for greater depths remains known only in equal characteristic.
• Connected Components in Punctured Spectra: As an application (mirroring results in the unramified case by Hernández et al.), the module $I$5 reflects the number of connected components in the punctured spectrum, with explicit module-theoretic realizations. This generalizes classical connectedness theorems and provides explicit calculations for local cohomology.
• Implications for Lyubeznik Numbers: While not made explicit in the ramified case, the behavior confirmed here is anticipated to enable further progress on invariants such as mixed characteristic Lyubeznik numbers, and possibly connections to $I$6-singularities and $I$7-adic Hodge theory.
• Potential for Generalization: The techniques, especially the use of Gabber's global generation and negative definiteness, suggest that other open vanishing or connectedness problems in ramified mixed characteristic may be approachable via geometric methods, bypassing limitations of algebraic presentation theorems.

Conclusion

This paper definitively establishes the Second Vanishing Theorem for regular local rings in ramified mixed characteristic, using geometric techniques that leverage global generation properties and intersection theory. The results not only unify previous theorems across all characteristics but also clarify that, despite exceptional behavior such as infinite associated primes in ramified settings, the expected cohomological dimension connectedness theorems do hold. The geometric approach introduced may become a template for solving further problems in the ramified mixed characteristic context.

The work has significant consequences for the structure of local cohomology, depth-theoretic results, and their avatars in singularity theory, and will likely inform future attempts at understanding the intricate algebraic geometry of ramified mixed characteristic rings.