-modules on rigid analytic spaces III: weak holonomicity and operations

Abstract

AbstractWe 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$X$.