AIMS Mathematics 2025, 10(8): 19958-19973
Published: 15 August 2025
This work establishes the categorical completeness of the category of left -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 -submodules of categorical products with compatible diagonal -actions governed by transition morphisms.
Key innovations include the following:
● Canonical extensions to the bounded derived category , proving homotopy limits preserve cohomology: .
● Geometric compatibility: Limits commute with the forgetful functor to -modules and preserve holonomicity, with characteristic varieties satisfying in .
These results provide a unified framework for limit constructions across abelian and derived categories of -modules, with immediate applications to microlocal analysis, arithmetic -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.