[Paper Review] Injective modules over some rings of Differential operators

Tony J. Puthenpurakal|arXiv (Cornell University)|Jan 7, 2013

Commutative Algebra and Its Applications7 references3 citations

TL;DR

This paper establishes that injective left modules over certain rings of differential operators—specifically the Weyl algebra, power series ring, and convergent power series ring over a field of characteristic zero—are also injective as modules over the base polynomial or power series ring. The key result relies on showing that the ring of differential operators is projective as a module over the base ring, enabling transfer of injectivity. This result is applied to prove that the connecting homomorphism in the Mayer-Vietoris sequence for local cohomology is $A_n(K)$-linear.

ABSTRACT

Let $R$ be a regular domain containing a field $K$ of characteristic zero and let $D$ be the ring of $K$-linear differential operators on $R$. Let $E$ be an injective left $D$-module. We ask the question, when is $E$ injective as a $R$-module? We show that this is indeed the case when $R = K[X_1,...,X_n]$ or $R = K[[X_1,...,X_n]]$ or $R = \mathbb{C}\{z_1,...,z_n\}$. We also give an application of our result to local cohomology.

Motivation & Objective

• To determine when an injective left module over a ring of differential operators remains injective as a module over the base ring.
• To address a key open question in local cohomology: whether the connecting homomorphism in the Mayer-Vietoris sequence for local cohomology modules is $A_n(K)$-linear.
• To generalize conditions under which injectivity over a larger ring implies injectivity over a subring, particularly in the context of differential operators.
• To analyze the structure of indecomposable injective modules over rings of differential operators when viewed as modules over the base regular ring.

Proposed method

• Proves that if $S$ is a left Noetherian ring containing a regular commutative ring $R$ as a subring, and $S$ is projective as a right $R$-module, then any injective left $S$-module is also injective as an $R$-module.
• Uses a technical criterion: an $R$-module $E$ is injective if $\operatorname{Ext}^1_R(R/J, E) = 0$ for all ideals $J$ generated by regular sequences.
• Establishes that the ring of differential operators on $R = K[X_1,\dots,X_n]$, $R = K[[X_1,\dots,X_n]]$, or $R = \mathbb{C}\{z_1,\dots,z_n\}$ satisfies the required hypotheses, including the key condition $(*)$ that $I^r s \subseteq S I$ for some $r$.
• Applies the main theorem to show that the Mayer-Vietoris sequence for local cohomology modules over $R = K[X_1,\dots,X_n]$ is a complex of $A_n(K)$-modules.
• Analyzes the associated primes of injective $S$-modules when viewed as $R$-modules, showing that $\operatorname{Ass}_R E_S(M)$ has a unique maximal element.
• Uses the structure of Koszul complexes and properties of derivations to verify the condition $(*)$ in examples such as differential polynomial rings.

Experimental results

Research questions

• RQ1 Under what conditions is an injective left module over a ring of differential operators also injective as a module over the base regular ring?
• RQ2 Is the connecting homomorphism $\delta^i$ in the Mayer-Vietoris sequence for local cohomology $A_n(K)$-linear?
• RQ3 What is the structure of the associated primes of an indecomposable injective $S$-module when viewed as an $R$-module?
• RQ4 How does the injectivity of $S$-modules as $R$-modules depend on the projectivity of $S$ as a right $R$-module?
• RQ5 Does the condition $(*)$, ensuring $I^r s \subseteq S I$, hold for rings of differential operators?

Key findings

• Injective left $A_n(K)$-modules are injective as $R$-modules when $R = K[X_1,\dots,X_n]$, $R = K[[X_1,\dots,X_n]]$, or $R = \mathbb{C}\{z_1,\dots,z_n\}$.
• The Mayer-Vietoris connecting homomorphism $\delta^i$ in the local cohomology sequence is $A_n(K)$-linear.
• For an indecomposable injective $S$-module $E_S(M)$, the set of associated primes $\operatorname{Ass}_R E_S(M)$ has a unique maximal element $P$, and $P \in \operatorname{Ass}_R M$.
• When $S$ is a finitely generated, Cohen-Macaulay $R$-algebra and $\mathfrak{m}$ is a maximal ideal of $S$, then $E_S(S/\mathfrak{m}) \cong E_R(R/\mathfrak{n})^{\ell_R(S/Q_1)}$ where $\mathfrak{n} = \mathfrak{m} \cap R$ and $Q_1$ is the $\mathfrak{m}$-primary component of $\mathfrak{n}S$.
• The condition $(*)$ holds for rings of differential operators, including Weyl algebras and differential polynomial rings, ensuring that $\Gamma_I(M)$ is a $S$-submodule for any $S$-module $M$.
• The Weyl algebra $A_n(K)$ is free (hence projective) as a right $R$-module for $R = K[X_1,\dots,X_n]$, which is essential for the main theorem.