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$\widehat {\mathcal {D}}$-modules on rigid analytic spaces III: weak holonomicity and operations
- Part of
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- Journal:
- Compositio Mathematica / Volume 157 / Issue 12 / December 2021
- Published online by Cambridge University Press:
- 28 October 2021, pp. 2553-2584
- Print publication:
- December 2021
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- Article
- Export citation
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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 
$X$.
