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

Abstract:

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