Hauptteilmodul/Verknüpfung von Derivationen/Beziehung zu symmetrischem Produkt/Fakt/en

Let ${}R$ be a commutative ${}K$-Algebra over a field ${}K$ and let ${}\delta _{1},\ldots ,\delta _{n}$ be derivations.

Then the composition ${}\delta _{n}\circ \cdots \circ \delta _{1}$ is sent under the natural map

$\operatorname {Diff} _{K}^{n}(R,R)\longrightarrow \operatorname {Hom} _{R}(\operatorname {Sym} _{R}^{n}(\Omega _{R{|K}}),R)$

to the image of the symmetric product ${}\delta _{n}\cdots \delta _{1}$ under the natural map

$\operatorname {Sym} _{R}^{n}(\operatorname {Der} _{K}(R,R))\cong \operatorname {Sym} _{R}^{n}(\operatorname {Hom} _{R}(\Omega _{R{|K}},R))\longrightarrow \operatorname {Hom} _{R}(\operatorname {Sym} _{R}^{n}(\Omega _{R{|K}}),R).$