$\widehat {\mathcal {D}}$-modules on rigid analytic spaces III: weak holonomicity and operations. (28th December 2021)

• Record Type: Journal Article
• Authors: Ardakov, Konstantin; Bode, Andreas; Wadsley, Simon
• Is Part Of: Compositio mathematica. Volume 157:Number 12(2021)
• Page Start: 2553
• Page End: 2584
• Publication Date: 2021-12-28
• Subjects: ${\mathcal {D}}$-modules -- rigid analytic spaces -- holonomicity -- local cohomology; 14G22 -- 32C38; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X21007521
• Languages: English
• ISSNs: 0010-437X

Abstract

Abstract: We develop a dimension theory for coadmissible $\widehat {\mathcal {D}}$ -modules on rigid analytic spaces and study those which are of minimal dimension, in analogy to the theory of holonomic $\mathcal {D}$ -modules in the algebraic setting. We discuss a number of pathologies contained in this subcategory (modules of infinite length, infinite-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 $\underline {H}^{i}_Z(\mathcal {M})$ with support in a closed analytic subset $Z$ of $X$ are also coadmissible of minimal dimension for any integrable connection $\mathcal {M}$ on $X$ .