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

Abstract

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 > 0 \operatorname {ch} \,k > 0 . Hence on Cowsik’s question there are counterexamples among the prime ideals defining space monomial curves, too.