Non-Cohen-Macaulay symbolic blow-ups for space monomial curves and counterexamples to Cowsik’s question
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- by Shiro Goto, Koji Nishida and Keiichi Watanabe
- Proc. Amer. Math. Soc. 120 (1994), 383-392
- DOI: https://doi.org/10.1090/S0002-9939-1994-1163334-9
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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.References
- R. C. Cowsik, Symbolic powers and number of defining equations, Algebra and its applications (New Delhi, 1981) Lecture Notes in Pure and Appl. Math., vol. 91, Dekker, New York, 1984, pp. 13–14. MR 750839
- Shiro Goto, Koji Nishida, and Yasuhiro Shimoda, The Gorensteinness of symbolic Rees algebras for space curves, J. Math. Soc. Japan 43 (1991), no. 3, 465–481. MR 1111598, DOI 10.2969/jmsj/04330465
- Shiro Goto, Koji Nishida, and Yasuhiro Shimoda, The Gorensteinness of the symbolic blow-ups for certain space monomial curves, Trans. Amer. Math. Soc. 340 (1993), no. 1, 323–335. MR 1124166, DOI 10.1090/S0002-9947-1993-1124166-4
- Shiro Goto, Koji Nishida, and Yasuhiro Shimoda, Topics on symbolic Rees algebras for space monomial curves, Nagoya Math. J. 124 (1991), 99–132. MR 1142978, DOI 10.1017/S0027763000003792
- Jürgen Herzog, Generators and relations of abelian semigroups and semigroup rings, Manuscripta Math. 3 (1970), 175–193. MR 269762, DOI 10.1007/BF01273309
- Jürgen Herzog and Bernd Ulrich, Self-linked curve singularities, Nagoya Math. J. 120 (1990), 129–153. MR 1086575, DOI 10.1017/S0027763000003305
- Craig Huneke, Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987), no. 2, 293–318. MR 894879, DOI 10.1307/mmj/1029003560
- Mayumi Morimoto and Shiro Goto, Non-Cohen-Macaulay symbolic blow-ups for space monomial curves, Proc. Amer. Math. Soc. 116 (1992), no. 2, 305–311. MR 1095226, DOI 10.1090/S0002-9939-1992-1095226-6
- Paul C. Roberts, A prime ideal in a polynomial ring whose symbolic blow-up is not Noetherian, Proc. Amer. Math. Soc. 94 (1985), no. 4, 589–592. MR 792266, DOI 10.1090/S0002-9939-1985-0792266-5
- Paul Roberts, An infinitely generated symbolic blow-up in a power series ring and a new counterexample to Hilbert’s fourteenth problem, J. Algebra 132 (1990), no. 2, 461–473. MR 1061491, DOI 10.1016/0021-8693(90)90141-A
- Jean-Pierre Serre, Valeurs propres des endomorphismes de Frobenius (d’après P. Deligne), Séminaire Bourbaki, 26ème année (1973/1974), Lecture Notes in Math., Vol. 431, Springer, Berlin, 1975, pp. Exp. No. 446, pp. 190–204 (French). MR 0463177
- A. Simis and Ngô Việt Trung, The divisor class group of ordinary and symbolic blow-ups, Math. Z. 198 (1988), no. 4, 479–491. MR 950579, DOI 10.1007/BF01162869
Bibliographic Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 120 (1994), 383-392
- MSC: Primary 13A30; Secondary 13E15, 13H10
- DOI: https://doi.org/10.1090/S0002-9939-1994-1163334-9
- MathSciNet review: 1163334