[Paper Review] D-modules on rigid analytic spaces III: Weak holonomicity and operations

Konstantin Ardakov, Andreas Bode+1 more|arXiv (Cornell University)|Apr 30, 2019

Algebraic Geometry and Number Theory38 references4 citations

TL;DR

This paper develops a dimension theory for coadmissible $\wideparen{\mathcal{D}}$-modules on rigid analytic spaces, introducing the notion of weak holonomicity analogous to holonomicity in algebraic $\mathcal{D}$-modules. It proves Bernstein's inequality for these modules, establishes stability under closed immersions and duality, and shows that higher direct images and local cohomology of integrable connections are coadmissible and weakly holonomic, despite pathologies like infinite-dimensional fibers and modules of infinite length.

ABSTRACT

We develop a dimension theory for coadmissible D-cap-modules on rigid analytic spaces and study those which are of minimal dimension, in analogy to the theory of holonomic D-modules in the algebraic setting. We discuss a number of pathologies contained in this subcategory (modules of infinite length, infinte-dimensional fibres). We prove stability results for closed immersions and the duality functor, and show that all higher direct images of integrable connections restricted to a Zariski open subspace are coadmissible of minimal dimension. It follows that the local cohomology sheaves $H^i_Z(\mathcal{M})$ with support in a closed analytic subset $Z$ of $X$ are also coadmissible of minimal dimension for any integrable connection $\mathcal{M}$ on $X$.

Motivation & Objective

• To develop a dimension theory for coadmissible $\wideparen{\mathcal{D}}$-modules on rigid analytic spaces, generalizing the notion of holonomicity from algebraic $\mathcal{D}$-modules.
• To define and study weakly holonomic $\wideparen{\mathcal{D}}$-modules as those of minimal dimension, analogous to holonomic modules in the algebraic setting.
• To investigate pathologies in this category, such as modules of infinite length and infinite-dimensional fibers, which do not occur in the algebraic theory.
• To prove stability of weak holonomicity under key operations: closed immersions, duality, higher direct images, and local cohomology functors.
• To establish that higher direct images and local cohomology sheaves of integrable connections are coadmissible and weakly holonomic, extending classical results to the analytic setting.

Proposed method

• Adapts the homological dimension theory of Fréchet–Stein algebras by relaxing conditions from Auslander regular to Auslander–Gorenstein with universally bounded self-injective dimension.
• Defines the dimension $d(M)$ of a coadmissible $\wideparen{\mathcal{D}}(X)$-module $M$ via $d(M) = 2\dim X - j(M)$, where $j(M)$ is the homological grade of $M$, and proves Bernstein's inequality: $d(M) \geq \dim X$ for non-zero $M$.
• Uses the fact that $\wideparen{\mathcal{D}}(X)$ is a faithfully flat $\mathcal{D}(X)$-module to relate the analytic and algebraic theories.
• Applies the theory of derived pushforwards along Zariski open embeddings to show that $\mathrm{R}^i j_* (\mathcal{M}|_U)$ are coadmissible and weakly holonomic for integrable connections $\mathcal{M}$.
• Applies the local cohomology exact sequence and isomorphism $\underline{H}^i_Z(\mathcal{M}) \cong \mathrm{R}^{i-1}j_*(\mathcal{M}|_U)$ for $i \geq 2$ to deduce weak holonomicity of local cohomology sheaves.
• Employs Čech cohomology on admissible coverings to compute $\mathrm{R}^i j_*(\mathcal{M}|_U)(X)$ and applies stability results for minimal dimension modules under cohomology.

Experimental results

Research questions

• RQ1 Can a meaningful notion of dimension and minimal dimension (weak holonomicity) be defined for coadmissible $\wideparen{\mathcal{D}}$-modules on rigid analytic spaces, despite the absence of a characteristic variety theory?
• RQ2 Do the higher direct images $\mathrm{R}^i j_*(\mathcal{M}|_U)$ of integrable connections under Zariski open embeddings remain coadmissible and of minimal dimension?
• RQ3 Are local cohomology sheaves $\underline{H}^i_Z(\mathcal{M})$ with support in a closed analytic subset $Z$ coadmissible and weakly holonomic for integrable connections $\mathcal{M}$?
• RQ4 What pathologies arise in the category of weakly holonomic $\wideparen{\mathcal{D}}$-modules that are absent in the algebraic holonomic category?
• RQ5 Is weak holonomicity preserved under duality and closed immersions in the analytic setting?

Key findings

• The paper establishes Bernstein's inequality for coadmissible $\wideparen{\mathcal{D}}(X)$-modules: for any non-zero module $M$, the dimension satisfies $d(M) \geq \dim X$, with equality characterizing weakly holonomic modules.
• It proves that $\mathrm{R}^i j_*(\mathcal{M}|_U)$ is a coadmissible $\wideparen{\mathcal{D}}_X$-module of minimal dimension for any Zariski open embedding $j:U\to X$ and integrable connection $\mathcal{M}$ on $X$, extending classical results to the analytic setting.
• The local cohomology sheaves $\underline{H}^i_Z(\mathcal{M})$ are shown to be coadmissible and weakly holonomic for any closed analytic subset $Z$ and integrable connection $\mathcal{M}$ on a smooth rigid analytic space $X$, via the isomorphism $\underline{H}^i_Z(\mathcal{M}) \cong \mathrm{R}^{i-1}j_*(\mathcal{M}|_U)$.
• The category of weakly holonomic $\wideparen{\mathcal{D}}$-modules contains pathologies not present in the algebraic case, such as modules of infinite length and infinite-dimensional fibers, indicating that the analogues of properties $(*)$ and $(**)$ do not characterize weak holonomicity.
• The duality functor preserves weak holonomicity, and the category $\mathcal{C}_X^{\mathrm{wh}}$ of weakly holonomic modules is abelian, ensuring stability under kernels and cokernels.
• The natural map $\theta: \wideparen{\mathcal{D}}(X) \otimes_{\mathcal{D}(X)} M[f^{-1}] \to \mathcal{M}(U)$ is a surjection of coadmissible $\wideparen{\mathcal{D}}(X)$-modules, which is used to transfer dimension bounds from the algebraic to the analytic setting.