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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


Authors: Mayumi Morimoto and Shiro Goto
Journal: Proc. Amer. Math. Soc. 116 (1992), 305-311
MSC: Primary 13A30; Secondary 13H10, 14H50, 14M05
MathSciNet review: 1095226
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Abstract: Let $ \mathfrak{p} = \mathfrak{p}({n_1},{n_2},{n_3})$ denote the prime ideal in the formal power series ring $ A = k[[X,Y,Z]]$ over a field $ k$ defining the space monomial curve $ X = {T^{{n_1}}},Y = {T^{{n_2}}}$, and $ Z = {T^{{n_3}}}$ with $ \operatorname{GCD} ({n_1},{n_2},{n_3}) = 1$. Then the symbolic Rees algebra $ {R_s}(\mathfrak{p}) = { \oplus _{n \geq 0}}{\mathfrak{p}^{(n)}}$ for $ \mathfrak{p} = \mathfrak{p}({n^2} + 2n + 2,{n^2} + 2n + 1,{n^2} + n + 1)$ is Noetherian but not Cohen-Macaulay if $ {\text{ch}}k = p > 0$ and $ n = {p^e}$ with $ e \geq 1$. The same is true for $ \mathfrak{p} = \mathfrak{p}({n^2},{n^2} + 1,{n^2} + n + 1)$ if $ {\text{ch}}k = p > 0$ and $ n = {p^e} \geq 3$ .


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1095226-6
PII: S 0002-9939(1992)1095226-6
Article copyright: © Copyright 1992 American Mathematical Society