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Let      A = k [ [ X , Y , Z ] ] A = k[[X,Y,Z]]       and      k [ [ T ] ] k[[T]]       be formal power series rings over a field      k k      , and let      n ⩾  4 n \geqslant 4       be an integer such that      n ≢ 0 mod 3 n\not \equiv 0\;\bmod \;3      . Let      φ  : A →  k [ [ T ] ] \varphi :A \to k[[T]]       denote the homomorphism of      k k      -algebras defined by      φ  ( X ) = T 7 n −  3 , φ  ( Y ) = T ( 5 n −  2 ) n \varphi (X) = {T^{7n - 3}},\;\varphi (Y) = {T^{(5n - 2)n}}      , and      φ  ( Z ) = T 8 n −  3 \varphi (Z) = {T^{8n - 3}}      . We put      p = Ker φ  {\mathbf {p}} = \operatorname {Ker} \,\varphi      . Then      R s ( p ) = ⊕  i ⩾  0 p ( i ) {R_s}({\mathbf {p}}) = { \oplus _{i \geqslant 0}}{{\mathbf {p}}^{(i)}}       is a Noetherian ring if and only if      ch k &gt; 0 \operatorname {ch} \,k &gt; 0      . Hence on Cowsik’s question there are counterexamples among the prime ideals defining space monomial curves, too.

Non-Cohen-Macaulay symbolic blow-ups for space monomial curves and counterexamples to Cowsik’s question