Epimorphisms of local cohomology modules, a general Peskine-Szpiro theorem, and an application to sheaf cohomology vanishing for thickenings

Abstract: We study the surjectivity of certain maps involving local cohomology modules, which we can realize as a dual version of part of the investigation developed by Bhatt, Blickle, Lyubeznik, Singh and Zhang on the sheaf cohomology of thickenings (i.e., subschemes defined by powers of ideals), where injectivity played a central role. To this end, we introduce and investigate properties of cohomologically Mittag-Leffler (cML) rings, associated to a given flat local endomorphism (for instance the Frobenius map of a regular ring of prime characteristic), a class which we show to contain, in our setting, the so-called cohomologically full rings of Dao, De Stefani and Ma (in particular, Cohen-Macaulay, Stanley-Reisner, and Du Bois singularities) as well as rings with an ideal inducing a pure endomorphism of the quotient. Our two major specific goals rely upon the prime characteristic setting. First, we extend for the class of cML rings a classical result of Peskine and Szpiro that relates the cohomological dimension and the height of a given Cohen-Macaulay ideal. Second, we prove and illustrate a Kodaira type vanishing result on the sheaf cohomology of thickenings.

• The paper establishes a framework for epimorphisms of local cohomology maps using cML rings, generalizing the classical Peskine-Szpiro theorem.
• It employs flat endomorphisms and duality theories, notably Matlis and local duality, to manage associated primes and cohomological dimensions.
• An application yields Kodaira-type vanishing results for sheaf cohomology on thickenings, enhancing our understanding of deformation phenomena in prime characteristic.

Epimorphisms of Local Cohomology Modules, a General Peskine-Szpiro Theorem, and Sheaf Cohomology Vanishing for Thickenings

Overview

The paper "Epimorphisms of local cohomology modules, a general Peskine-Szpiro theorem, and an application to sheaf cohomology vanishing for thickenings" (2604.20150) establishes a comprehensive framework around the surjectivity properties of local cohomology maps within prime characteristic commutative algebra and algebraic geometry. It introduces the concept of cohomologically Mittag-Leffler (cML) rings, which generalize several significant classes—Cohen-Macaulay, Stanley-Reisner, Du Bois, and cohomologically full rings. The authors develop a robust theory around this notion and derive a generalization of the classical Peskine-Szpiro theorem on cohomological dimension versus depth for this extended class. A pivotal application is provided in the context of vanishing results for sheaf cohomology on thickenings in positive characteristic, paralleling ongoing research on the geometry of thickenings and their cohomology.

Cohomologically Mittag-Leffler Rings: Definition and Examples

Central to the paper is the introduction of cML rings, which are defined with respect to a flat local endomorphism $\varphi$ (such as the Frobenius in prime characteristic $p$), and an ideal $I$ in a regular local ring or standard graded polynomial ring $R$. For a "cofinal flat triple" $(R, I, \varphi)$—where $\{\varphi^t(I)R\}_{t \geq 0}$ is cofinal with $\{I^t\}_{t \geq 0}$—$R/I$ is called $\varphi$-cML if, for all $i$ and all $p$0, the natural map

$p$1

is surjective. The authors prove this property is stable under completion and demonstrates that the cML property encapsulates or extends (via functorial properties and epimorphism criteria) several classical and modern ring classes including:

• Cohen-Macaulay rings: All CM rings are cohomologically full by Dao-De Stefani-Ma and thus cML.
• Stanley-Reisner rings: These arise as quotients by square-free monomial ideals and inherit cML via purity of the Frobenius action.
• Rings with Du Bois singularities over $p$2: These are included as cML due to their cohomological fullness.
• Rings with pure endomorphisms: cML status is transferred provided the induced endomorphism of the quotient is pure.

The class is strictly larger than any of the aforementioned, which is evidenced by the discussion and open questions at the end regarding the proper inclusions.

Local Cohomology, Epimorphisms, and Associated Primes

The technical foundation is the functorial behavior (relative to a flat endomorphism $p$3) of functors such as $p$4, and their effects on Ext and local cohomology systems. Surjectivity between local cohomology modules is transferred via the exactness of these functors and the dualities involved (notably Matlis duality and local duality), leading to refined control over the system of associated or attached primes: $p$5 with equality for $p$6-pure quotients in positive characteristic.

These results connect with classical finiteness theorems for associated primes of local cohomology (Huneke-Sharp) and are instrumental for tracking invariants under thickenings and deformations.

Generalization of the Peskine-Szpiro Theorem

A highlight is the extension of the classical positive characteristic theorem of Peskine-Szpiro, which originally stated that for a regular local ring $p$7 of characteristic $p$8, if $p$9 is Cohen-Macaulay, then the cohomological dimension $I$0 equals the height $I$1. The authors prove:

If $I$2 is $I$3-cML (with respect to a cofinal flat triple), then

$I$4

This result is shown to be optimal within the cML class. Additional corollaries provide formulas for the projective dimension and establish that cML rings of dimension one must be CM, but for higher dimension the property may not enforce Cohen-Macaulayness.

The interplay between projective dimension, depth, and generator number is elucidated via these relationships, and the necessity and sufficiency for a cML ring to be CM is fully addressed, including reductions to radicals and transfer properties under surjective maps of local cohomology.

Application: Kodaira-type Vanishing for Sheaf Cohomology on Thickenings

Leveraging the new cML framework, the authors obtain Kodaira-type vanishing theorems for sheaf cohomology of thickenings in characteristic $I$5. Letting $I$6 be the $I$7-th thickening, and assuming $I$8 is CM with depth of Frobenius thickenings sufficiently large, they prove: $I$9 whenever $R$0 for infinitely many $R$1. This result sits within, and extends, the context of Bhatt et al.'s work on the surjectivity (here, injectivity) of various maps in thickenings and provides concrete vanishing results that are verified by computational examples on determinantal and generic ideals.

Regularity Criteria in Prime Characteristic

The cofinal flat triple framework also enables new homological and Ext-theoretic criteria for regularity of rings in positive characteristic, extending Kunz's flatness characterization via Frobenius to conditions involving vanishing of Tor or Ext modules over thickenings, provided finite (co-)homological dimension.

Conclusion

This work establishes a robust, functorial framework—via the notion of cohomologically Mittag-Leffler rings—for the study of the surjectivity of local cohomology maps and the propagation of cohomological and homological properties under thickenings and Frobenius actions. The generalization of the Peskine-Szpiro Theorem, the connection to cohomologically full rings, and the transfer/vanishing results for sheaf cohomology on thickenings considerably expand the applicability of local cohomology and homological algebra in both commutative algebra and explicit computations in algebraic geometry. The introduced cML property opens avenues for further investigation of singularities and deformation-theoretic phenomena in both positive and mixed characteristic, with immediate consequences for vanishing theorems and homological conjectures.