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Let $R$ be a domain and \begin{align*} T \,\colon= R[[X_1,\ldots,X_d]]. \end{align*} Suppose that we have $d$ elements $f_1,\ldots,f_d \in T$ and let us consider an ideal $J$ of $T$ such that $(f_1,\ldots,f_d) \subset J$ and the following three conditions$\colon$ \begin{align*} & 1. \quad \overline{f_i} \,\colon \overset{{\mathrm{def}}}{=} f_i~{\mathrm{mod}}(X_1,\ldots,X_d)~{\mathrm{is~ an~ irreducible~ element~ of~}} R ~{\mathrm{for~each}}~ 1 \leq i \leq d. \\ & 2. \quad T/(f_1,\ldots,f_d) \phantom{a} {\mathrm{is}}~not~{\mathrm{finite~over}}~ R. \\ & 3. \quad T/J \phantom{a} {\mathrm{is~finite~over}}~R. \end{align*}

Q. Does the following equality hold$\colon$$\phantom{A}$${\mathrm{ht}}(J) > d$$\,$?

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