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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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The Gorensteinness of the symbolic blow-ups for certain space monomial curves
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by Shiro Goto, Koji Nishida and Yasuhiro Shimoda PDF
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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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 340 (1993), 323-335
  • MSC: Primary 13A30; Secondary 13H10, 14M05
  • DOI: https://doi.org/10.1090/S0002-9947-1993-1124166-4
  • MathSciNet review: 1124166