@article{Lei2025, 
author = {Huang-Rui Lei and Jian-Gang Tang},
title = {Limits in        D  -module categories: Completeness and derived geometric extensions},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {8},
pages = {19958-19973},
keywords = {D-modules, complete categories, homotopy limits, derived categories, holonomic modules, microlocal analysis},
url = {https://www.sciopen.com/article/10.3934/math.2025891},
doi = {10.3934/math.2025891},
abstract = {This work establishes the categorical completeness of the category        M    o    d    (            D              X        ) of left        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              O              X      -submodules of categorical products with compatible diagonal              D              X      -actions governed by transition morphisms.Key innovations include the following:● Canonical extensions to the bounded derived category        D          b        (      M    o    d    (            D              X        )  ), proving homotopy limits preserve cohomology:         H          n            (                      lim        ←                            h        o              ⁡            M              i              ∙            )    ≅            lim      ←        ⁡      H          n        (            M              i              ∙        ).● Geometric compatibility: Limits commute with the forgetful functor to              O              X      -modules and preserve holonomicity, with characteristic varieties satisfying        C    h        (              lim      ←        ⁡            M              i            )    ⊆            lim      ←        ⁡      C    h    (            M              i        ) in        T          ∗        X.These results provide a unified framework for limit constructions across abelian and derived categories of        D  -modules, with immediate applications to microlocal analysis, arithmetic        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.}
}