[Paper Review] Frobenius actions on local cohomology modules and deformation

Linquan Ma, Phạm Hùng Quý|arXiv (Cornell University)|Jun 7, 2016

Commutative Algebra and Its Applications19 references3 citations

TL;DR

This paper introduces and studies two new classes of singularities in positive characteristic: $F$-full and $F$-anti-nilpotent, defined via Frobenius actions on local cohomology modules. It proves that both properties deform under reduction by a regular element, providing new evidence for the deformation of $F$-injectivity and generalizing earlier results in the field.

ABSTRACT

Let $(R,m)$ be a Noetherian local ring of characteristic $p>0$. We introduce and study $F$-full and $F$-anti-nilpotent singularities, both are defined in terms of the Frobenius actions on the local cohomology modules of $R$ supported at the maximal ideal. We prove that if $R/(x)$ is $F$-full or $F$-anti-nilpotent for a nonzerodivisor $x\in R$, then so is $R$. We use these results to obtain new cases on the deformation of $F$-injectivity.

Motivation & Objective

• To define and study new classes of singularities—$F$-full and $F$-anti-nilpotent—using Frobenius actions on local cohomology modules in Noetherian local rings of characteristic $p > 0$.
• To investigate whether these new singularities deform when passing from $R$ to $R/(x)$ for a regular element $x$.
• To provide new evidence toward the long-standing open problem of whether $F$-injectivity deforms in characteristic $p > 0$.
• To generalize previous results on deformation of $F$-injectivity, particularly those of [12], by introducing stronger conditions that imply $F$-injectivity.
• To establish connections between $F$-fullness, $F$-anti-nilpotency, and other $F$-singularities, especially in relation to Du Bois singularities and stable Frobenius finiteness.

Proposed method

• Define $F$-full and $F$-anti-nilpotent singularities via the behavior of the Frobenius map on local cohomology modules $H^i_{rak m}(R)$.
• Use the long exact sequence induced by the short exact sequence $0 \to R \xrightarrow{x} R \to R/(x) \to 0$ to relate cohomology modules of $R$ and $R/(x)$.
• Analyze the cokernel of multiplication by $x$ on $H^i_{rak m}(R)$, showing that finite length implies injectivity of $x^{p-1}F$ under certain conditions.
• Apply the commutative diagram involving Frobenius and multiplication maps to transfer injectivity properties from $R/(x)$ to $R$.
• Use duality and the fact that the residue field is perfect to show that kernels and cokernels of $x$-multiplication have finite length.
• Leverage the equivalence between $F$-anti-nilpotency and stable Frobenius finiteness to prove deformation results.

Experimental results

Research questions

• RQ1 Does $F$-anti-nilpotency deform? That is, if $R/(x)$ is $F$-anti-nilpotent, is $R$ also $F$-anti-nilpotent?
• RQ2 Does $F$-fullness deform? That is, if $R/(x)$ is $F$-full, is $R$ also $F$-full?
• RQ3 Under what conditions does the injectivity of the Frobenius map on $H^i_{rak m}(R)$ follow from the injectivity on $H^i_{rak m}(R/(x))$?
• RQ4 Can the deformation of $F$-injectivity be established under weaker assumptions than previously known?
• RQ5 Is there a meaningful connection between $F$-fullness and the failure of $F$-injectivity to deform?

Key findings

• If $R/(x)$ is $F$-anti-nilpotent, then $R$ is also $F$-anti-nilpotent, proving that this property deforms.
• If $R/(x)$ is $F$-full, then $R$ is also $F$-full, establishing the deformation of $F$-fullness.
• If $R/(x)$ is both $F$-full and $F$-injective, then $R$ is $F$-injective, providing a new sufficient condition for $F$-injectivity.
• When the residue field is perfect and $\operatorname{Coker}(H^i_{\frak m}(R) \xrightarrow{x} H^i_{\frak m}(R))$ has finite length for all $i$, then $x^{p-1}F: H^i_{\frak m}(R) \to H^i_{\frak m}(R)$ is injective for all $i$, implying $R$ is $F$-injective.
• The map $x^{p-1}F$ is injective on $H^i_{\frak m}(R)$ for all $i \leq f_{\frak m}(R/(x)) + 1$ when $R/(x)$ is $F$-injective and the residue field is perfect, generalizing results from [12].
• Strong $F$-injectivity (i.e., $F$-injective and $F$-full) deforms: if $R/(x)$ is strongly $F$-injective, then $R$ is also strongly $F$-injective.