$\widehat {\mathcal {D}}$-modules on rigid analytic spaces III: weak holonomicity and operations. (28th October 2021) - Ardakov, Konstantin; Bode, Andreas; Wadsley, Simon
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 $X$ .