[Paper Review] Annihilators of $D$-modules in mixed characteristic

Rankeya Datta, Nicholas Switala+1 more|arXiv (Cornell University)|Jul 23, 2019

Commutative Algebra and Its Applications16 references4 citations

TL;DR

This paper establishes that in mixed characteristic, the annihilator of any nonzero $$\mathscr{D}(R,V)$-module over a power series or polynomial ring $R$ over a DVR $V$ is either zero or a power of the uniformizer $\pi$. It provides a counterexample to Hochster's question on faithfulness of top local cohomology modules in mixed characteristic, showing that $H^4_I(\mathbb{Z}_2[[x_0,\dots,x_5]])$ is annihilated by 2, and further refutes a conjecture of Lyubeznik and Yildirim on associated primes of duals of local cohomology modules.

ABSTRACT

Let $R$ be a polynomial or formal power series ring with coefficients in a DVR $V$ of mixed characteristic with a uniformizer $π$. We prove that the $R$-module annihilator of any nonzero $\D(R,V)$-module is either zero or is generated by a power of $π$. In contrast to the equicharacteristic case, nonzero annihilators can occur; we give an example of a top local cohomology module of the ring $\mathbb{Z}_2[[x_0, \ldots, x_5]]$ that is annihilated by $2$, thereby answering a question of Hochster in the negative.

Motivation & Objective

• To classify annihilators of $\mathscr{D}(R,V)$-modules over polynomial or power series rings in mixed characteristic.
• To investigate the faithfulness of top local cohomology modules in mixed characteristic, particularly in response to Hochster's question.
• To test the validity of the Lyubeznik-Yildirim conjecture on associated primes of duals of local cohomology modules.
• To construct explicit examples where local cohomology modules are annihilated by a non-zero element of the base ring.

Proposed method

• Use of $\mathscr{D}(R,V)$-modules, defined as rings of $V$-linear differential operators on $R$, to analyze module structure.
• Application of the theory of local cohomology and its compatibility with completion to relate cohomology over $\mathbb{Z}[x_0,\dots,x_5]$ to its $\mathfrak{m}$-adic completion.
• Construction of a specific ideal $I$ in $\mathbb{Z}_2[[x_0,\dots,x_5]]$ generated by 10 monomials with arithmetic rank 4 and cohomological dimension 4.
• Proof that the transition maps in the direct limit defining $H^4_I(R)$ are injective, implying $H^4_I(R) \cong H^6_\mathfrak{m}(R/(2))$.
• Identification of $H^4_I(R)$ as isomorphic to the injective hull $E_{\bar{R}}(\bar{R}/\mathfrak{m})$ over $\bar{R} = R/(2)$, showing it is annihilated by 2.
• Use of the dual module $D(H^4_I(R)) \cong R/(2)$ to show that 0 is not an associated prime, contradicting the Lyubeznik-Yildirim conjecture.

Experimental results

Research questions

• RQ1 Is the top local cohomology module of a regular local ring in mixed characteristic always faithful?
• RQ2 Can a nonzero annihilator occur for a $\mathscr{D}(R,V)$-module in mixed characteristic?
• RQ3 Does the Lyubeznik-Yildirim conjecture on associated primes of duals of local cohomology modules hold in mixed characteristic?
• RQ4 What is the structure of the top local cohomology module $H^4_I(\mathbb{Z}_2[[x_0,\dots,x_5]])$ for a specific ideal $I$?
• RQ5 Can the annihilator of a $\mathscr{D}(R,V)$-module be a nontrivial power of the uniformizer $\pi$?

Key findings

• The annihilator of any nonzero $\mathscr{D}(R,V)$-module over $R = V[[x_1,\dots,x_n]]$ or $R = V[x_1,\dots,x_n]$ is either zero or a power of the uniformizer $\pi$.
• The top local cohomology module $H^4_I(\mathbb{Z}_2[[x_0,\dots,x_5]])$ is annihilated by 2, so $\operatorname{Ann}_R(H^4_I(R)) = (2)$, providing a negative answer to Hochster's question.
• The ideal $I$ in $\mathbb{Z}_2[[x_0,\dots,x_5]]$ has both arithmetic rank and cohomological dimension equal to 4.
• The module $H^4_I(R)$ is isomorphic to the injective hull $E_{\bar{R}}(\bar{R}/\mathfrak{m})$, confirming its structure and support at the maximal ideal.
• The dual module $D(H^4_I(R)) \cong R/(2)$, so 0 is not an associated prime of $D(H^4_I(R))$, contradicting the Lyubeznik-Yildirim conjecture.
• The example shows that the mixed-characteristic analogue of the Hochster-Jeffries result on faithfulness does not hold.