Limits in $ \mathcal{D} $-module categories: Completeness and derived geometric extensions

This work establishes the categorical completeness of the category $ \mathsf{Mod}(\mathcal{D}_{X}) $ of left $ \mathcal{D} $-modules on smooth complex algebraic varieties, resolving a fundamental structural question in algebraic analysis. We explicitly construct all small limits, such as products, equalizers, pullbacks, and arbitrary limits, demonstrating they are realized as $ \mathcal{O}_{X} $-submodules of categorical products with compatible diagonal $ \mathcal{D}_{X} $-actions governed by transition morphisms.

Key innovations include the following:

● Canonical extensions to the bounded derived category $ D^{b}(\mathsf{Mod}(\mathcal{D}_{X})) $, proving homotopy limits preserve cohomology: $ H^{n}\big(\varprojlim^{\mathrm{ho}}\mathcal{M}^{\bullet}_{i}\big) \cong \varprojlim H^{n}(\mathcal{M}^{\bullet}_{i}) $.

● Geometric compatibility: Limits commute with the forgetful functor to $ \mathcal{O}_{X} $-modules and preserve holonomicity, with characteristic varieties satisfying $ \mathrm{Ch}\big(\varprojlim \mathcal{M}_{i}\big) \subseteq \varprojlim \mathrm{Ch}(\mathcal{M}_{i}) $ in $ T^{*}X $.

These results provide a unified framework for limit constructions across abelian and derived categories of $ \mathcal{D} $-modules, with immediate applications to microlocal analysis, arithmetic $ \mathcal{D} $-modules in positive characteristic, and the Riemann-Hilbert correspondence. The explicit formulations are adaptable to singular characteristic varieties and resolve foundational questions in geometric representation theory.