# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## The Gorensteinness of the symbolic blow-ups for certain space monomial curvesHTML articles powered by AMS MathViewer

by Shiro Goto, Koji Nishida and Yasuhiro Shimoda
Trans. Amer. Math. Soc. 340 (1993), 323-335 Request permission

## Abstract:

Let ${\mathbf {p}} = {\mathbf {p}}({n_1},{n_2},{n_3})$ 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 ${\text {GCD}}({n_1},{n_2},{n_3}) = 1$. Then the symbolic Rees algebras ${R_s}({\mathbf {p}}) = { \oplus _{n \geq 0}}{{\mathbf {p}}^{(n)}}$ are Gorenstein rings for the prime ideals ${\mathbf {p}} = {\mathbf {p}}({n_1},{n_2},{n_3})$ with $\min \{ {n_1},{n_2},{n_3}\} = 4$ and ${\mathbf {p}} = {\mathbf {p}}(m,m + 1,m + 4)$ with $m \ne 9,13$ . The rings ${R_s}({\mathbf {p}})$ for ${\mathbf {p}} = {\mathbf {p}}(9,10,13)$ and ${\mathbf {p}} = {\mathbf {p}}(13,14,17)$ are Noetherian but non-Cohen-Macaulay, if $\operatorname {ch} k = 3$ .
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