Tauvel's height formula in iterated differential operator rings

Thomas Guédénon

Abstract: Let $k$ be a field of positive characteristic, $R$ an associative algebra over $k$ and let $\Delta_{1, n}=\{\delta_1, \ldots, \delta_n\}$ be a finite set of $k$-linear derivations from $R$ to $R$. Let $A=R_n=R[\theta _1, \delta_1]\cdots[\theta _n,\delta_n]$ be an iterated differential operator $k$-algebra over $R$ such that $\delta_j(\theta_i)\in R_{i-1}\theta_i+R_{i-1}; 1\leq i<j\leq n$. As central result we show that if $R$ is noetherian affine $\Delta_{1, n}$-hypernormal and if Tauvel's height formula holds for the $\Delta_{1, n}$-prime ideals of $R$, then Tauvel's height formula holds in $A$. In particular, let $g$ be a completely solvable finite-dimensional $k$-Lie algebra acting by derivations on $R$ and let $U(g)$ be the enveloping algebra of $g$. If $R$ is noetherian affine $g$-hypernormal and if Tauvel's height formula holds for the $g$-prime ideals of $R$, then Tauvel's height formula holds in the crossed product of $R$ by $U(g)$.

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