Pascal Translation and Hasse Divided-Power Modules in Mixed Characteristic

Abstract

This paper develops the characteristic-free algebraic theory of the generalized Pascal translation matrices Lₙ(a) and identifies their Hasse action with the regular module of a finite divided-power algebra. The principal result converts invariant-submodule questions for Pascal translation into ideal theory and gives an explicit mixed-characteristic classification unavailable from the ordinary exponential description. For every commutative unital ring R, the Hasse operators Dₙ⁽ʳ⁾ span a finite commutative divided-power algebra 𝒟ₙ(R). The map A ↦ Ae₁ identifies the regular 𝒟ₙ(R)-module with Rⁿ. Consequently, Hasse-stable R-submodules of Rⁿ are exactly the ideals of 𝒟ₙ(R). For coordinate submodules ⊕ⱼ Iⱼeⱼ₊₁, stability is characterized by the complete system of inclusions C(k,j) Iⱼ ⊆ Iₖ for 0 ≤ j ≤ k < n, where C(k,j) denotes the binomial coefficient. In characteristic p, the paper determines the divided-power algebra explicitly for every truncation n over every commutative 𝔽ₚ-algebra. If pˢ ≥ n and xᵢ = Dₙ⁽ᵖⁱ⁾, then 𝒟ₙ is a quotient of the polynomial algebra in x₀,…,xₛ₋₁ by the relations xᵢᵖ = 0 together with exactly those digit monomials whose base-p weighted value is at least n. The surviving digit monomials, indexed by 0,…,n−1, form a basis. This gives a complete p-typical monomial presentation of the characteristic-p Hasse algebra at arbitrary truncation. Over a field K of characteristic p, the socle dimension of 𝒟ₙ(K) is exactly the number of nonzero base-p digits of n. It follows that 𝒟ₙ(K) is Gorenstein ⇔ n = cpᵃ with a ≥ 0 and 1 ≤ c < p. For n = pˢ, the algebra takes the particularly symmetric complete-intersection form K[X₀,…,Xₛ₋₁]/(X₀ᵖ,…,Xₛ₋₁ᵖ), with one-dimensional socle and Hilbert series (1 + t + ⋯ + tᵖ⁻¹)ˢ. The first factorial boundary is classified completely. If n ≤ p, then 𝒟ₙ(K) ≅ K[t]/(tⁿ), and the Hasse-stable subspaces form a single tail chain. At n = p + 1 the algebra becomes K[y,z]/(yᵖ, yz, z²), and the invariant-subspace lattice acquires explicit parameter families. Thus the passage through the factorial boundary is accompanied by a precise algebraic change from a chain of invariant subspaces to a richer ideal geometry. The mixed-characteristic coordinate theory is also exact over ℤ/pᵉℤ. Writing Iⱼ = pᵅʲR, Hasse stability is equivalent to αₖ ≤ min(e, αⱼ + νₚ(C(k,j))) for every 0 ≤ j ≤ k < n. By Kummer's theorem, the additional valuation is exactly the number of base-p carries in j + (k − j) = k. Over fields, Lucas' theorem similarly converts Hasse stability of coordinate subspaces into the condition that their index sets are upper sets in the base-p digitwise partial order. The surrounding theory establishes Lₙ(a) as a universal finite polynomial-translation representation. The additive law Lₙ(a)Lₙ(b) = Lₙ(a + b) holds as an integral polynomial identity and therefore over every commutative unital base ring. The paper derives characteristic-free Hasse expansions, exact nilpotency and kernel structure, polynomial translation formulas, scaling identities, minor formulas, total-nonnegativity criteria, scalar interpolation, logarithmic powers, and exact order and root conditions. When the relevant factorials are invertible, the Hasse system collapses to the ordinary nilpotent-generator description Lₙ(a) = exp(aNₙ). The theory further extends to weighted function spaces, polynomial one-parameter groups, locally nilpotent actions, and multivariate tensor-product translations. In several variables, tensor products of the one-dimensional Pascal representations give the exact simultaneous translation action on rectangular polynomial coefficient arrays, together with multivariate Hasse expansions and, over ℚ-algebras, the corresponding commuting-derivative exponential formula. The resulting framework places generalized Pascal matrices, Hasse derivatives, finite divided-power algebras, base-p digit structure, Artinian Gorenstein theory, and mixed-characteristic invariant modules within a single algebraic theory of finite translation.