Differential Algebraic Framework for Coherent Sheaves: Explicit Constructions and Computational Methods

This paper develops a comprehensive differential algebraic framework for coherent sheaves on algebraic varieties and complex manifolds. We construct the coherent sheaf differential closure Kcoh(F), a differentially closed sheaf extension that contains explicit representations of cohomology classes, sheaf morphisms, and solutions to sheaf-theoretic differential equations. Our approach provides constructive proofs for fundamental results in sheaf theory, including explicit representations of sheaf cohomology groups, computational methods for extension groups, and certified algorithms for sheaf operations. We establish rigorous convergence criteria, complexity bounds, and validation protocols using interval arithmetic and computational algebraic geometry. The framework bridges differential algebra, algebraic geometry, and computational mathematics, offering new tools for explicit calculations in sheaf theory while maintaining full mathematical rigor. Applications include explicit Riemann-Roch theorems, computational deformation theory, and certified sheaf cohomology calculations.