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Limits in D -module categories: Completeness and derived geometric extensions
AIMS Mathematics 2025, 10(8): 19958-19973
Published: 15 August 2025
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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.

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