is a similar reconstruction property for kernels of surjective morphisms in $\cR$. Let $f:\cA\ra \cB$ be a morphism in $\cR$ with kernel $\cJ$, such that $f$ maps $\cA_{(k)}$ onto $\cB_{(k)}$ for all $k\geq 0$. The kernel $J$ of the induced map $gr(f): gr(\cA)\ra gr(\cB)$ is a homogeneous $\partial$-ideal (i.e., $\partial J \subset J$). A set $\{a_i|~i\in I\}$ such that $a_i$ is homogeneous of degree $d_i$ is said to generate $J$ as a $\partial$-ideal if $\{\partial^k a_i|~i\in I,~k\geq 0\}$ generates $J$ as an ideal.

\[idealrecon\] Let $\{a_i| i\in I\}$ be a generating set for $J$ as a $\partial$-ideal, where $a_i$ is homogeneous of degree $d_i$. Then there exist vertex operators $a_i(z)\in \cA_{(d_i)}$ with $\phi_{d_i}(a_i(z)) = a_i$, such that $\{a_i(z)|~i\in I\}$ generates $\cJ$ as a vertex algebra ideal.

The structure of $\cH(n)^{O(n)}$ {#secortho}
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The ring of Laurent polynomials $\mathbb{C}[t,t^{-1}]$ may be regarded as an abelian Lie algebra. It has a central extension $\gh = \mathbb{C}[t,t^{-1}]\oplus \mathbb{C}\kappa$ with bracket $[t^n,t^m] = n \delta_{n+m,0} \kappa$, and $\mathbb{Z}$-gradation $deg(t^n) = n$, $deg(\kappa) = 0$. Let $\gh_{\geq 0} = \oplus_{n\geq 0} \gh_n$, and let $C$ be the one-dimensional $\gh_{\geq 0}$-module on which $t^n$ acts trivially for $n\geq 0$, and $\kappa$ acts by the identity. Define 