This paper establishes a comprehensive exterior differential-homological framework for explicit computations in arithmetic algebraic geometry, extending previous differential-homological methods to incorporate exterior algebra structures. We introduce arithmetic exterior differential graded algebras that combine p-adic and archimedean derivations with exterior differential operators through explicit L∞-morphisms, resolving the homotopy-coherent compatibility between arithmetic derivations and the exterior differential. We construct arithmetic exterior differential polynomial rings with universal properties, develop filtered colimit constructions for arithmetic exterior closures with explicit height control using Arakelov theory, and provide detailed convergence analysis with explicit error bounds derived from spectral properties of exterior differential operators. The framework is validated through nontrivial numerical examples including elliptic curve differential form families and Grassmannian local parameterizations, and its compatibility with classical theories including Weil conjectures and p-adic Hodge theory is rigorously established with explicit comparison maps. We present algorithms for computing arithmetic exterior local parameterizations with detailed data structures and complexity analysis distinguishing form degree from series order. All constructions are mathematically verified with complete proofs addressing the graded Leibniz rules, L∞-coherence conditions, spectral estimates for exterior differential operators, and the precise relationship between arithmetic derivations and the exterior differential. We provide complete proofs for all key theorems, including the explicit construction of L∞-morphisms capturing the non-commutativity of δp and d, the universal property of exterior differential polynomial rings, the directedness of the index set for arithmetic closures with precise height estimates, the convergence analysis via Picard iteration and majorant series, and the compatibility with Weil conjectures and p-adic Hodge theory for exterior cohomology. The framework provides a solid foundation for explicit computations in arithmetic exterior algebraic geometry.
shifa liu (Wed,) studied this question.