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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: Shiro Goto, Koji Nishida and Keiichi Watanabe
Journal: Proc. Amer. Math. Soc. 120 (1994), 383-392
MSC: Primary 13A30; Secondary 13E15, 13H10
MathSciNet review: 1163334
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1163334-9
PII: S 0002-9939(1994)1163334-9
Article copyright: © Copyright 1994 American Mathematical Society