Injektive Moduln über Ringen linearer Differentialoperatoren

Abstract

LetR be a division ring of characteristic 0 withn commuting (partial) differentiationsd _i . DefineR[d]=R[d ₁, ...,d _n ] to be all polynomials ind _i with coefficients inR. A typical element ofR[d] has the form Σ(r _α d ^α(1)₁ ...d ^α(n) _n ∣α∈ℕⁿ) withr _αεR. Equality and addition are defined as in commuting polynomial rings, with multiplication induced by the relationsd _i r=rd _i +d _i (r) forrεR and 1≤i≤n. The power series ring

$$[[X_1 ,...,X_n ]]R = [[X]]R = \{ \Sigma (X^\alpha r_\alpha \left| {\alpha \in \mathbb{N}^n )} \right|r_\alpha \in R\} $$

is anR[d]-module,d _i acting as partial differentiation ∂/∂X _i on[[X]]R andR acting via the ring homomorphism

$$R \mathrel\backepsilon r \mapsto \Sigma (1/\alpha !) X^\alpha d_1^{\alpha (1)} ...d_n^{\alpha (n)} (r) \in [[X]]R$$

. Then the module[[X]]R is a big injective cogenerator in the sense of Roos [11]. This result is in a certain sense a dual of the Hilbert Basis Theorem: For each left idealL ofR[d] there exists afinite number of power seriesf ₁, ...,f _m such thatL is the annihilator of thef _i inR[d]. For commutative rings of differential operators the minimal injective cogeneratorM is explicitly described as a submodule of[[X]]R, especially forR=ℂ we haveM=Σ(R[X] exp (ΣX _i a _i )|(a ₁, ...,a _n )εR ⁿ) and all these power series are convergent.